Cortex-like learning machine for temporal and hierarchical pattern recognition

[0001] This application claims the benefit of provisional patent application Ser. No. 61/128,499, filed 2008 May 22 by the present inventor.

[0002] In the terminology of pattern recognition, neural networks and machines learning, a feature vector is a transformation of a measurement vector, whose components are measurements or sensor outputs. This invention is mainly concerned with processing feature vectors and sequences of feature vectors for detecting and recognizing spatial and temporal causes (e.g., objects in images/video, words in speech, and characters in handwriting). This is what pattern recognition, neural networks and machines learning are essentially about. It is also a typical problem in the fields of computer vision, signal processing, system control, telecommunication, and data mining. Example applications that can be formulated as such a problem are handwritten character classification, face recognition, fingerprint identification, DNA sequence identification, speech recognition, machine fault detection, baggage/container examination, video monitoring, text/speech understanding, automatic target recognition, medical diagnosis, prosthesis control, robotic arm control, and vehicle navigation.

[0003] A good introduction to the prior art in pattern classification, neural networks and machine learning can be found in Simon Haykin, Neural Networks and Learning Machines, Third Edition, Pearson Education, New Jersey, 2009; Christopher M. Bishop, Pattern Recognition and Machine Learning, Springer Science, New York, 2006; Neural Networks for Pattern Recognition, Oxford University Press, New York, 1995; B. D. Ripley, Pattern Recognition and Neural Networks, Cambridge University Press, New York, 1996; S. Theodoridis and K. Koutroumbas, Pattern Recognition, Second Edition, Academic Press, New York, 2003; Anil K. Jain, Robert P. W. Duin and Jianchang Mao, “Statistical Pattern Recognition: A Review,” in IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 22, No. 1, January 2000; R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, second edition, John Wiley & Sons, New York, 2001; and Bernhard Scholkopf and Alexander J. Smota, Learning with Kernels, The MIT Press, Cambridge, Mass., 2002.

[0004] Commonly used pattern classifiers include template matching, nearest mean classifiers, subspace methods, 1-nearest neighbor rule, k-nearest neighbor rule, Bayes plug-in, logistic classifiers, Parzen classifiers, Fisher linear discriminants, binary decision trees, multilayer perceptrons, radial basis networks, and support vector machines. They each are suitable for some classification problems. However, in general, they all suffer from some of such shortcomings as difficult training/design, much computation/memory requirement, ad hoc character of the penalty function, or poor generalization/performance. For example, the relatively more powerful multilayer perceptrons and support vector machines are difficult to train, especially if the dimensionality of the feature vectors is large. After training, if new training data is to be learned, the trained multilayer perceptron or support vector machine is usually discarded and new one is trained over again. Its decision boundaries are determined by exemplary patterns from all classes. Furthermore, if there are a great many classes or if there are no or not enough exemplary patterns for some “confuser classes” such as for target and face recognition, training an MLP or SVM either is impractical or incurs a high misclassification rate. Camouflaged targets or occluded faces not included in the training data are known to also cause high misclassification rates.

[0005] A pattern classification approach, that is relatively seldom mentioned in the pattern recognition literature, is the correlation matrix memories, or CMMs, which have been studied essentially in the neural networks community (T. Kohonen, Self-Organization and Associative Memory, second edition, Springer-Verlag, 1988; R. Hecht-Nielsen, Neurocomputing, Addison-Wesley, 1990; Branko Soucek and The Iris Group, Fuzzy, Holographic, and Parallel Intelligence—The Sixth-Generation Breakthrough, edited, John Wiley and Sons, 1992; James A. Anderson, An Introduction to Neural Networks, The MIT Press, 1995; S. Y. Kung, Digital Neural Networks, Pearson Education POD, 1997; D. J. Willshaw, P. P. Buneman and H. C. Longet-Higgins, “Non-holographic associative memory,” Nature, 222, pp. 960-962, 1969; D. J. Willshaw and H. C. Longet-Higgins, “Associative memory models,” Machine Intelligence, vol. 5, edited by B. Meltzer & O. Michie, Edinburgh University Press, 1970; K. Nagano, “Association—a model of associative memory,” IEEE Transactions on Systems, Man and Cybernetics, vol. SMC-2, pp. 68-70, 1972; G. Palm, “On associative memory,” Biological Cybernetics, vol. 36, pp. 19-31, 1980; E. Gardner, “The space of interactions in neural network models,” Journal of Physics, vol. A21, pp. 257-270, 1988; S. Amari, “Characteristics of sparsely encoded associative memory,” Neural Networks, vol. 2(6), pp. 451-457, 1989; J. Buckingham and D. Willshaw, “On setting unit thresholds in an incompletely connected associative net,” Network, vol. 4, pp. 441-459, 1993; M. Turner and J. Austin, “Matching Performance of Binary Correlation Matrix Memories,” Neural Networks, 1997). The training of CMMs, which are associative memories, is easy and fast even if they have a very high dimensional input. If new training data is to be learned or if the dimensionality of a trained CMM is to be modified, the CMM is not discarded, but can be easily updated or expanded.

[0006] Two types of CMM are noteworthy. They are the holographic neural nets (John Sutherland, “Artificial neural device utilizing phase orientation in the complex number domain to encode and decode stimulus response patterns,” U.S. Pat. No. 5,214,745, May 25, 1993; John Sutherland, “Neural networks,” U.S. Pat. No. 5,515,477, May 7, 1996) and the binary CMMs in the aforementioned papers by Willshaw and Longuet-Higgins (1970), Palm (1980), Gardner (1988), S. Amari (1989), M. Turner and J. Austin (1997), and the references therein.

[0007] The main idea of holographic neural nets (HNets) is representing real numbers by phase angle orientations on a complex number plane through the use of a sigmoidal transformation such as a hyperbolic tangent function. After each component of the input stimuli and output responses is converted into a complex number whose phase angle orientation (i.e. argument) represents the component, the correlation matrix is constructed in the standard manner. A holographic neural cell comprises essentially such a correlation matrix. If the dimensionality of the stimulus is large enough, augmented if necessary, and if the phase angle orientations of the stimuli and responses are more or less statistically independent and uniformly distributed on the unit circles in the complex number plane, the “signal part” in the response to an input stimulus is hopefully much greater than the “interference part” in the response to the same input stimulus during its retrieval because of self-destruction of those stored stimuli that are out of phase with said input stimulus like the self-destruction of a random walk on the complex number plane. This idea allows more stimuli to be stored in a complex correlation matrix than does the earlier versions of the correlation matrix.

[0008] However, the holographic neural cell approach suffers from the following shortcomings. First, to avoid ambiguity at the point, (−1, 0)=−1+0i, in the complex plane, a neighborhood of (−1, 0) must be excluded in the range of the sigmoidal transformation. This prevents the mentioned uniform distribution required for good self-destruction of the interference part. Second, it is not clear how to augment the stimuli without introducing much correlations among the stimuli, which again may reduce self-destruction of the interference part. Third, the argument of a complex number on the unit circle ranges from −π to π. To pack more stimuli on it, better self-destruction of the interference part is needed, which in turn requires a higher dimensionality of the stimuli. Such a higher dimensionality means a higher dimensionality of the correlation matrix, requiring more memory space to hold the matrix.

[0009] Binary CMMs have feature vectors encoded either into unipolar binary vectors with components equal to 1 or 0 or into bipolar binary vectors with components equal to 1 or −1. Bipolar binary vectors were used in most of the earlier work on binary CMMs. Superiority of sparse unipolar binary encoding (with most of the components of encoded feature vectors being 0 and only a few being 1) to nonsparse unipolar binary encoding and bipolar binary encoding was remarked and proved in the mentioned papers by Willshaw and Longuet-Higgins (1970), Palm (1980), Gardner (1988), and S. Amari (1989). Sparsely encoded CMMs are easy to implement (J. Austin and J. Kennedy, “A hardware implementation of a binary neural network,” MicroNeuro, IEEE Computer Press, 1994), and have found many applications. Nevertheless, sparsely encoded CMMs have quite a few shortcomings: (a) A large sparse correlation matrix has very low “information density” and takes much memory space. (b) A multistage sparsely encoded CMMs is often necessary. (c) There is no systematic way to determine the dimensionality of the sparse unipolar binary vectors to represent the feature vectors. (d) There is no systematic way to determine the number of stages or the number of neurons in each stage in a multistage sparsely encoded CMM. (e) There is no systematic way to determine whether a sparsely encoded CMM has a minimum misclassification probability for the given CMM architecture. (f) The mapping from the feature vectors to their sparse binary vectors representations must be stored in some memory space, further reducing the overall memory density of the CMM.

[0010] Judging from the foregoing shortcomings of the commonly used pattern classifiers, the holographic neural nets, and the sparsely encoded CMMs, there remains a need for alternatives to existing pattern classifiers in the prior art for recognizing patterns.

[0011] In this invention disclosure, a cortex-like learning machine, called a probabilistic associative memory (PAM), is disclosed that processes feature vectors or sequence of feature vectors, each feature vector being a ternary feature vector.

[0012] A PAM is a network of processing units (PUs). It can be viewed as a new neural network paradigm or a new type of learning machine. Each PU generates a representation of a subjective probability distribution of the label of a feature subvector or a sequence of feature subvectors that are received by the PU. Some PUs convert such representations into ternary vectors, which are included in feature subvectors input to other PUs. Weights in a PU learn an input feature subvector with or without supervision by a Hebb rule of learning. Some advantages of PAMs are the following:

• • [0013] 1. As opposed to most of commonly used pattern recognizers, a PAM generalizes not by only a single holistic similarity criterion for the entire input exogenous feature vector, which noise, erasure, distortion and occlusion can easily defeat, but by a large number of similarity criteria for feature subvectors input to a large number of PUs (processing units) in different layers. These criteria contribute individually and collectively to generalization for single and multiple causes. Example 1: smiling; putting on a hat; growing or shaving beard; or wearing a wig can upset a single similarity criterion used for recognizing a face in a mug-shot photograph. However, a face can be recognized by each of a large number of feature subvectors of the face. If one of them is recognized to belong to a certain face, the face is recognized. Example 2: a typical kitchen contains a refrigerator, a counter top, sinks, faucets, stoves, fruit and vegetable on a table, etc. The kitchen is still a kitchen if a couple of items, say the stoves and the table with fruit and vegetable, are removed.
  • [0014] 2. Masking matrices in a PU eliminate effects of corrupted ternary components of the feature subvector input to the PU, and thereby enable maximal generalization capability of the PU, and in turn that of the PAM.
  • [0015] 3. PAMs are neural networks, but are no more blackboxes with “fully connected” layers much criticized by opponents of such neural networks as multilayer perceptrons (MLPs) and recurrent MLPs, whose weights are iteratively determined through minimizing an error criterion and have no interpretation in the context of their applications. In a PU of a PAM, weights are correlations between orthogonal expansions of subvectors of the PU's input feature subvectors and the labels of these feature subvectors. Each PU has a receptive region in the exogenous feature vector input to the PAM and classifies any cause within the receptive region. Such interpretations can be used to help select the architecture (i.e., layers, PUs, connections, feedback structures, etc.) of a PAM for the application.
  • [0016] 4. The weights in each PU of a PAM learn by a Hebb rule and thus the PAM has a “photographic memory.” No iterative optimization such as that involved in local-search training methods using backpropagation or backpropagation through time is needed for learning. This allows easy learning of a large number of large exogenous feature vectors in reasonable time as well as easy online adaptive learning.
  • [0017] 5. A PU can learn with or without supervision. This allows a PAM to (1) perform unsupervised deep learning in lower layers and supervised learning in higher layers; (2) perform supervised learning when a label is provided from outside the PAM and unsupervised learning when not; and (3) perform autonomous learning.
  • [0018] 6. A PAM may have some capability of recognizing rotated, translated and scaled patterns. Moreover, easy learning and retrieving by a PAM allow it to learn translated, rotated and scaled versions of an input image with ease.
  • [0019] 7. PUs generate representations of probability distributions of the labels of their input feature subvectors. Such representations of probability distributions of a common label can be combined into a single representation of probability distributions of the common label.
  • [0020] 8. PAMs with hierarchical and feedback structures can detect and recognize multiple and hierarchical causes in a spatial or temporal exogenous feature vector.
  • [0021] 9. The weight matrices (e.g., expansion correlation matrices) in different PUs can be added to combine the learned knowledge at virtually no additional cost.
  • [0022] 10. The architecture of a PAM can be adjusted without discarding learned knowledge in the PAM. This allows enlargement of the feature subvectors, increase of the number of layers, and even increase of feedback connections.
  • [0023] 11. Only a small number of algorithmic steps of parallel computing are needed for retrieval, which are suitable for massive parallelization at the bit level and by VLSI implememtation.

[0024] The present invention is embodied in systems, learning machines or methods. In this Section: SUMMARY, embodiments in a system are stated. Replacing the term, “system”, in these statements of system embodiments with the term, “learning machine”, yields statements of embodiments in a learning machine. It is understood that these statements of embodiments in a learning machine are incorporated in this Section, although they are not explicitly provided.

[0025] An objective of the present invention is to provide a system that learns to recognize spatial or temporal patterns or causes.

[0026] Another objective of the present invention is to provide a system that can learn and recognize partially erased, smeared, noise-corrupted, covered, distorted, or altered patterns.

[0027] Still another objective of the present invention is to provide a system that can perform supervised or unsupervised learning or both.

[0028] Still another objective of the present invention is to provide a system whose architecture can be adjusted without discarding already learned knowledge.

[0029] Still another objective of the present invention is to provide a system that produces representations of probability distributions of labels of feature vectors and subvectors.

[0030] Still another objective of the present invention is to provide a system with a hierarchical structure for recognizing hierarchical causes (or patterns) at different levels such as line segments in a letter, or ears, nose, mouth and eyes on a face, or apples and oranges in baskets on a table.

[0031] Still another objective of the present invention is to provide a system with feedback structures for processing sequences of related exogenous feature vectors such as those obtained from examining one single image for a plurality of consecutive time points, images of an object from different angles, consecutive frames in a video or movie, and handwritten letters in a word, words in a sentence, and sentences in a paragraph.

[0032] Still another objective of the present invention is to provide a system that can recognize rotated, translated and/or scaled versions of images that have been learned by the system.

[0033] Still another objective of the present invention is to provide a system that learns autonomously.

[0034] The foregoing objectives, as well as other objectives of the present invention that will become apparent from the discussion below, are achieved by the present invention with the following preferred embodiments.

[0035] A first major embodiment of the present invention disclosed herein is a system (or learning machine) for processing feature vectors. Such feature vectors that are input to the system are also called exogenous feature vectors to distinguish them from feature vectors that contain components generated internally by said system. The first major embodiment comprises at least one processing unit that comprises

• • [0036] expansion means for generating, in response to a first feature vector input to said processing unit, a first orthogonal expansion of a subvector of said first feature vector, said first orthogonal expansion comprising components of said subvector of said first feature vector and a plurality of products of said components of said subvector of said first feature vector;
  • [0037] at least one first expansion correlation matrix that is a weighted sum of outer products, each being an outer product of a linear combination (or weighted sum) of a vector with components all equal to 1 and a label of a second feature vector input to said processing unit and a second orthogonal expansion of a subvector of said second feature vector, said second orthogonal expansion comprising components of said subvector of said second feature vector and a plurality of products of said components of said subvector of said second feature vector; and
  • [0038] estimation means for using at least said at least one first expansion correlation matrix and at least one third orthogonal expansion of a subvector of a third feature vector input to said processing unit, said third orthogonal expansion being generated by said expansion means in response to said third feature vector, to compute a representation of a probability distribution of a label of said third feature vector.

[0039] Note that an orthogonal expansion {hacek over (v)} of a vector v with m components is displayed in (2). A way to generate {hacek over (v)} is to start with {hacek over (v)}(1)=[1 v₁]′ and then keep using the recursive formula (I) until {hacek over (v)}={hacek over (v)}(1, . . . , m) is obtained.

[0040] Note that said second expansion correlation matrix is a weighted sum of outer products, each being an outer product (c₁I+c₂r_t(n)){hacek over (x)}_t′(n(u)) of a linear combination (or weighted sum) c₁I+c₂r_t(n) of a vector I=[1 . . . 1]′ with all components equal to 1 and a label r_t(n) of a feature vector x_t(n) input to said processing unit and an orthogonal expansion {hacek over (x)}_t(n(u)) of the subvector x_t(n(u)) of the feature vector x_t(n), where t=1, . . . , T for some positive integer T, and c₁ and c₂ are real-valued weights. Expansion correlation matrices with specific values of c₁ and c₂ are shown in FIG. 4 and (3), (4), (5) and (6). Note that p_τ(n) is also a representation of the probability distribution.

[0041] Note that a label of a feature vector input to a processing unit is defined as follows: A PU in a PAM has a “receptive field” in the exogenous feature vector and a “receptive field” in the measurement vector. These two receptive fields can be found by tracing the feedforward connections in the PAM backforward from a feature vector input to the PU (or the feature subvector index of the PU) to an exogenous feature vector (or the input terminals) of the PAM, and then tracing the transformation, that maps the measurement vector into the exogenous feature vector, backward from the exogenous feature vector to the measurement vector. The components of the measurement vector that can be reached by this backward tracing from a PU to the exogenous feature vector and then to the measurement vector are called the “receptive field” of the PU in the measurement vector. The components of the exogenous feature vector that can be reached by this backward tracing from a PU to the exogenous feature vector are called the “receptive field” of the PU in the exogenous feature vector. The label of a feature vector input to a PU is the label of the corresponding components of measurement vector in the receptive field of the PU in the measurement vector. The label of the corresponding components of the exogenous feature vector in the receptive field of the PU in the exogenous feature vector is also this label.

[0042] Note that the weights in said weighted sum of outer products are ΛW_t(n(u), T), t=1, . . . , T. If the matrix W, (n(u), T) is a diagonal matrix with equal entries, then the weights in the weighted sum of outer products are actually scalar weights. Two examples are W_t(n(u), T)=λ^T-tI and W_t(n(u), T)=I/√{square root over (T)}. Therefore, the weights in the weighted sum of outer products in the first major embodiment are either matrix weights or scalar weights.

[0043] Note that an estimation means for generating a representation y_τ(n) of a probability distribution p_τ(n) of the label r_τ(n) of x_τ(n) is shown in FIG. 10 and described in Subsection 5.3, “Representations of Probability Distributions”.

[0044] Another embodiment of the present invention is the first major embodiment, wherein said processing unit further comprises at least one masking matrix that is a sum of an identity matrix and at least one summand masking matrix multiplied by a weight, said summand masking matrix setting certain components of a fourth orthogonal expansion of a subvector of a fourth feature vector input to said processing unit equal to zero, as said masking matrix is multiplied to said fourth orthogonal expansion, said fourth orthogonal expansion comprising components of said subvector of said fourth feature vector and a plurality of products of said components of said subvector of said fourth feature vector, wherein said estimation means also uses said at least one masking matrix in computing a representation of a probability distribution of a label of said third feature vector.

[0045] Note that a masking matrix M(n(u)) is displayed in FIG. 7 and (26). The masking matrix M(n(u)) is a sum of an identity matrx I and

[0046] $\sum _{j=1}^{J\left(n\left(u\right)\right)}\sum _{i_j=j}^{\dim n\left(u\right)}\dots \sum _{i_2=2}^{i_3-1}\sum _{i_1=1}^{i_2-1}{2}^{-8j}{2}^j\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-},i_2^{-},\dots ,i_j^{-}\right)\right).$
Because diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) appears in a summand 2^−8j2^jdiag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) in (26) that defines the masking matrix M(n(u)), the matrix diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) is called a summand masking matrix in M(n(u)). A summand 2^−8j2^j diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) of M(n(u)) in (26) is a summand masking matrix diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) multiplied by a weight 2^−8j2^j. When the masking matrix M(n(u)) is multiplied to an orthogonal expansion {hacek over (x)}_τ(n(u)) of a subvector x_τ(n(u)) of x_τ, each diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) in FIG. 7 or (26) is multiplied to {hacek over (x)}_τ(n(u)) to get diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) {hacek over (x)}_τ(n(u)), in which the components of {hacek over (x)}_τ(n(u)) that involve the i₁-th, i₂-th, . . . , and i_j-th components of x_τ(n(u)) are set equal to 0. This is explained in Subsection 5.4.

[0047] Another embodiment of the present invention are the first major embodiment of the present invention, wherein c₁=0 and c₂=1 in the linear combination or weighted sum c₁I+c₂r_t(n). In this embodiment the second expansion correlation matrix is D(n(u)) displayed in FIG. 4 and (3).

[0048] Another embodiment of the present invention is the first major embodiment of the present invention, wherein c₁=1 and c₂=0 in the linear combination c₁I+c₂r_t(n). In this embodiment the second expansion correlation matrix is C(n(u)) displayed in FIG. 4 and (4).

[0049] Another embodiment of the present invention is the first major embodiment of the present invention, wherein c₁=1 and c₂=1 in the linear combination c₁I+c₂r_t(n). In this embodiment the second expansion correlation matrix is C(n(u)) displayed in (5).

[0050] Another embodiment of the present invention is the first major embodiment of the present invention, wherein c₁=1 and c₂=−1 in the linear combination c₁I+c₂r_t(n). In this embodiment the second expansion correlation matrix is B(n(u)) displayed in (6).

[0051] Another embodiment of the present invention is the first major embodiment wherein weights in said weighted sum of outer products are equal.

[0052] Another embodiment of the present invention is the first major embodiment, wherein at least one expansion correlation matrix is an expansion correlation matrix on a rotation/translation/scaling (RTS) suite of a subvector of a feature subvector index.

[0053] Note that such expansion correlation matrices are shown in FIGS. 19 and (40)-(43), where the rotation/translation/scaling suite is denoted by Ω(n). These expansion correlation matrices help the above embodiment recognize rotated, translated and scaled causes or objects.

[0054] Still another embodiment of the present invention is the first major embodiment, said processing unit further comprising supervised learning means for adjusting, in response to a fifth feature vector input to said processing unit, said at least one first expansion correlation matrix by using at least an outer product of a linear combination of a vector with components all equal to 1 and a label of said fifth feature vector input to said processing unit and a fifth orthogonal expansion of a subvector of said fifth feature vector, said fifth orthogonal expansion comprising components of said subvector of said fifth feature vector and a plurality of products of said components of said subvector of said fifth feature vector, wherein said label of said fifth feature vector is provided from outside said learning machine.

[0055] Note that supervised learning is discussed in Subsection 5.6, “Processing Units and Supervised/Unsupervised Learning”. Examples of adjusting expansion correlation matrices, D(n(u)) and C(n(u)), in supervised learning by using outer products, r_t(n){hacek over (x)}_t′(n(u)) and I{hacek over (x)}_t′(n(u)), respectively, are shown in FIG. 5 and FIG. 6, where the label r_t(n) of the feature vector x_t(n) input to said processing unit is provided from outside the system or learning machine.

[0056] Still another embodiment of the present invention is the first major embodiment, said processing unit further comprising conversion means for converting said representation of said probability distribution produced by said estimation means into a vector being output from said processing unit as a label of said third feature vector.

[0057] Note that conversion of a representation of a probability distribution is discussed in Subsection 5.5. Two conversion means are described in the Section and are shown in FIG. 11 and FIG. 12.

[0058] Still another embodiment of the present invention, called embodiment 1, is the first major embodiment, said processing unit further comprising a pseudo-random vector generating means for generating a pseudo-random vector in accordance with said probability distribution produced by said estimation means, said pseudo-random vector being output from said processing unit as a label of said third feature vector.

[0059] Note that the pseudo-random vector generating means is shown in FIG. 12 and is the second conversion method described in Subsection 5.5.

[0060] Still another embodiment of the present invention is embodiment 1, said processing unit further comprising unsupervised learning means for adjusting, in response to a sixth feature vector input to said processing unit, said at least one first expansion correlation matrix by using at least one outer product of a linear combination of a vector with components all equal to 1 and a label of said sixth feature vector and a sixth orthogonal expansion of a subvector of said sixth feature vector, said sixth orthogonal expansion comprising components of said subvector of said sixth feature vector and a plurality of products of said components of said subvector of said sixth feature vector, wherein said label of said sixth feature vector is a pseudo-random vector generated by said pseudo-random vector generating means as a label of said sixth feature vector.

[0061] Note that unsupervised learning is discussed in Subsection 5.6, “Processing Units and Supervised/Unsupervised Learning”. Examples of adjusting expansion correlation matrices, D(n(u)) and C(n(u)), by using outer products, r_t(n){hacek over (x)}_t′(n(u)) and I{hacek over (x)}_t′(n(u)), respectively, are shown in FIG. 5 and FIG. 6, where the label r_t(n) of said sixth feature vector x_t(n) is said pseudo-random vector generated by said pseudo-random vector generating means as a label of said sixth feature vector.

[0062] Still another embodiment of the present invention is embodiment 1, wherein a plurality of components of a pseudo-random vector that is output from a processing unit are components of a feature vector that is input to another processing unit.

[0063] Note that in this embodiment, there are at least 2 processing units. A plurality of components of a pseudo-random vector output from one of the processing units are components of a feature vector input to another processing unit. Said at least 2 processing units form a network of processing units, which may be a multilayer network of processing units. A multilayer network of processing units is shown in FIG. 15.

[0064] Still another embodiment of the present invention is embodiment 1, wherein a plurality of said at least one processing unit form a network with a plurality of ordered layers of said processing units; each exogenous feature vector is input to layer 1 of said network, which is the lowest-ordered layer of said network; and components of a feature vector input to a processing unit in layer l of said network, where l>1, are components of at least one label that is output from at least one processing unit in layer l-1 of said network.

[0065] Note that a multilayer network of processing units is shown in FIG. 15.

[0066] Still another embodiment of the present invention is embodiment 1 for processing exogenous feature vectors in sequences of exogenous feature vectors, wherein a plurality of components of a pseudo-random vector that is output from a processing unit in processing a certain exogenous feature vector in a sequence of exogenous feature vectors are included as components, after a time delay, in a feature vector that is input to a processing unit in processing an exogenous feature vector subsequent to said certain exogenous feature vector in said sequence.

[0067] Note that in this embodiment, a plurality of components of a pseudo-random vector output from a processing unit are, after a time delay, components of a feature vector input to another processing unit. Because of the time delay, said at least 2 processing units can form a network of processing units with “feedback connections”, which is a dynamical system by itself.

[0068] Still another embodiment of the present invention is embodiment 1 for processing exogenous feature vectors in sequences of exogenous feature vectors, wherein at least one component of a label that is output from a processing unit in layer j in processing a certain exogenous feature vector in a sequence is included as a component, after a time delay, in a feature vector that is input to a processing unit in layer k, where k≦j, in processing an exogenous feature vector subsequent to said certain exogenous feature vector in said sequence. A multilayer network of processing units with feedbacks is shown in FIG. 16.

[0069] A second major embodiment is a method for processing feature vectors, said method comprising:

• • [0070] an expanding step of expanding a subvector of a first feature vector into a first orthogonal expansion that comprises components of said subvector of said first feature vector and a plurality of products of said components of said subvector of said first feature vector, and
  • [0071] an estimating step of using
    • [0072] 1. at least one orthogonal expansion of a subvector of said first feature vector produced by said expanding step; and
    • [0073] 2. at least one expansion correlation matrix that is a weighted sum of outer products, each being an outer product of a weighted sum of a vector with components all equal to 1 and a label of a second feature vector and a second orthogonal expansion of a subvector of said second feature vector, said second orthogonal expansion comprising components of said subvector of said second feature vector and a plurality of products of said components of said subvector of said second feature vector;
      to compute a representation of a probability distribution of a label of said first feature vector.

[0074] Note that all the terms used in the above second major embodiment are those used in the first major embodiment, which are briefly described for the first major embodiment. Note also that the terms used in all the embodiments below are those used in the embodiments above following the first major embodiment.

[0075] Another embodiment of the present invention is the second major embodiment, wherein said estimating step also uses at least one masking matrix that is a sum of an identity matrix and at least one summand masking matrix multiplied by a weight, said summand masking matrix setting certain components of a third orthogonal expansion of a subvector of a third feature vector equal to zero, as said masking matrix is multiplied to said third orthogonal expansion, to compute a representation of a probability distribution of a label of said first feature vector, said third orthgonal expansion comprising components of said subvector of said third feature vector and a plurality of products of said components of said subvector of said third feature vector.

[0076] Another embodiment of the present invention is the second major embodiment, wherein said weighted sum of a vector with components all equal to 1 and a label of a second feature vector is said label of said second feature vector.

[0077] Another embodiment of the present invention is the second major embodiment, wherein said weighted sum of a vector with components all equal to 1 and a label of a second feature vector is said vector with components all equal to 1.

[0078] Another embodiment of the present invention is the second major embodiment, wherein said weighted sum of a vector with components all equal to 1 and a label of a second feature vector is a sum of said vector with components all equal to 1 and said label of said second feature vector

[0079] Another embodiment of the present invention is the second major embodiment, wherein weights in said weighted sum of outer products are equal.

[0080] Another embodiment of the present invention, called embodiment 2, is the second major embodiment, further comprising a generating step of generating a pseudo-random vector in accordance with said probability distribution as a label of said first feature vector.

[0081] Another embodiment of the present invention is embodiment 2, further comprising a feedforward step of including a plurality of components of a pseudorandom vector generated by said generating step as a label of said first feature vector as components in a fourth feature vector and processing said fourth feature vector by said expanding step and said estimating step.

[0082] Another embodiment of the present invention is embodiment 2, further comprising a feedback step of including, after a time delay, a plurality of components of a pseudorandom vector generated by said generating step as a label of said first feature vector as components in a fifth feature vector and processing said fifth feature vector by said expanding step and said estimating step.

[0083] Another embodiment of the present invention is embodiment 2, further comprising an unsupervised learning step of adjusting said expansion correlation matrix by using at least one outer product of a weighted sum of a vector with components all equal to 1 and a label of a sixth feature vector and an orthogonal expansion of a subvector of said sixth feature vector produced by said expanding step, wherein said label of said sixth feature vector is a pseudo-random vector generated by said generating step as a label of said sixth feature vector.

[0084] Another embodiment of the present invention is embodiment 2, further comprising a supervised learning step of adjusting said expansion correlation matrix by using at least an outer product of a weighted sum of a vector with components all equal to 1 and a label of a seventh feature vector and an orthogonal expansion of a subvector of said seventh vector produced by said expanding step, wherein said label of said seventh feature vector is provided.

[0146] In the terminology of pattern recognition, a feature vector is a transformation of a measurement vector, whose components are measurements or sensor outputs, and a label of said feature vector is a label of said measurement vector. A subvector of a feature vector is called a feature subvector. A feature subvector is itself a feature vector. As a special case, the transformation is the identity transformation, and the feature vector is the measurement vector. Example measurement vector are digital pictures, frames of a video, segments of speech, handwritten characters/words. This invention is mainly concerned with processing feature vectors and sequences of related feature vectors for detecting and recognizing spatial and temporal causes or patterns.

[0147] In this invention disclosure, a cortex-like learning machine, called a probabilistic associative memory (PAM), is disclosed that processes feature vectors or sequence of feature vectors, each feature vector being a ternary feature vector. Such feature vectors input to a PAM are called exogenous feature vectors. A PAM can be viewed as a new neural network paradigm, a new type of learning machine, or a new type of pattern recognizer. A PAM is a network of processing units (PUs). In a multilayer PAM with or without feedback connections, the vector input to a layer is a feature vector, because it is a transformation of exogenous feature vectors input to the PAM, and in turn a transformation of the measurement vector. In a PAM with (delayed) feedback connections (or feedback means), called a recurrent PAM, a feature vector input to layer/comprises a vector output from layer l-1 and vectors output and feedbacked from PUs in other layers. For example, if there is a feedback connection to layer 1, then an exogenous feature vector is not an entire feature vector input to layer 1, but only a subvector of said entire feature vector.

[0148] A PU may comprise expansion means, estimation means, conversion means, adjustment means, feedback means, supervised learning means, unsupervised learning means, and/or storage means. A feature vector input to a PU is usually a subvector of a feature vector input to the layer to which said PU belongs. The subvector input to a PU is called a feature subvector to distinguish it from the feature vector input to the layer to which said PU belongs. If no confusion is likely, the vector input to a PU is still called a feature vector. A label of a feature vector (or subvector) input to a PU in a PAM is a label of the subvector of the exogenous feature vector that is transformed in the PAM into said feature vector (or subvector) input to the PU. A PU may have one or both of two functions—retrieving the label of a feature subvector from the memory (i.e., expansion correlation memories or general expansion correlation memories disclosed in this invention) and learning a feature subvector and its label that is either provided from outside the PU (in supervised learning) or generated by the PU itself (in unsupervised learning). In performing retrieval, a feature subvector input to a PU is first expanded into a general orthogonal expansion by the expansion means (to be described later on). The general orthogonal expansion is then processed by the estimation means, using the memory from the storage, into a representation of a probability distribution of the label of said feature subvector. The conversion means converts said representation into a ternary vector, which is an output of the PU. If said representation is needed for use outside the PU, it is also output from the PU.

[0149] There are three types of PU according to how they learn—supervised PUs, unsupervised PUs, and supervised/unsupervised PU. A supervised PU performs supervised learning, if a label of a feature subvector input to the PU is provided from outside the PU (or the PAM) and learning is wanted. An unsupervised PU performs unsupervised learning if a label of a feature subvector input to the PU is not provided from outside the PU but learning the feature subvector is wanted. A supervised/unsupervised PU can perform both supervised learning and unsupervised learning. Both supervised and unsupervised learning follow a Hebb rule of learning. During the process of learning, supervised or unsupervised, the PU is said to be performing learning.

[0150] A PU with a general masking matrix (to be described later on) has good generalization capability. A PU that has learned feature subvectors on a rotation/translation/scaling suite (to be described later on) has good capability for recognizing rotated, translated and scaled patterns.

[0151] In this invention disclosure, prime ′ denotes matrix transposition. Vectors whose components are 0's, 1's and −1's are called ternary vectors. Thus, the components of ternary vectors are elements of the ternary set, {−1, 0, 1}. Bipolar binary vectors are vectors whose components are elements of the binary set, {−1, 1}. Unipolar binary vectors are vectors whose components are elements of the binary set, {0, 1}. Since {−1, 1} and {0, 1} are subsets of the ternary set, bipolar and unipolar binary vectors are ternary vectors. For example, the bipolar binary vector [1 −1 1 −1]′ and the unipolar binary vector [1 0 1 0]′ are ternary vectors.

[0152] In the present invention disclosure, exogenous feature vectors input to a PAM are ternary vectors. 0's are usually used to represent unknown, unavailable, or corrupted part or parts of exogenous feature vectors. A vector whose components are the numberings (or subscripts) of the components of a feature vector that constitute a feature subvector and are ordered by the magnitudes of the numberings is called the feature subvector index of the feature subvector. For example, [v₂ v₄]′ is a feature subvector of the feature vector v=[v₁ v₂ v₃ v₄]′, and the feature subvector index of the feature subvector is [2 4]′. As usual, one of the subvectors of a vector is said vector itself. Labels in the present invention disclosure are also ternary vectors. 0's are usually used to represent unavailable, unknown or unused part or parts of labels. For instance, if we start out with labels with different dimensionalities (or different numbers of components) in an application, the dimensionalities of those with smaller dimensionalities can be increased by inserting 0's at the tops or bottoms as additional components so that all the labels have the same dimensionality in the application.

[0153] An orthogonal expansion and a general orthogonal expansion of a ternary vector are described in the next subsection. A general orthogonal expansion has, as its block column(s), at least one orthogonal expansion. An orthogonal expansion is a special case of a general orthogonal expansion. A sum or weighted sum of products of values of a vector-valued function evaluated at labels of feature subvectors and transposes of general orthogonal expansions of these feature subvectors (with the same feature subvector index) is called an expansion correlation matrix (on the feature subvector index). Note that if the vector-valued function of the label is one-dimensional, the feature subvector expansion correlation matrix is a vector, sometimes called a expansion correlation vector.

[0154] In the present invention disclosure, probabilities are usually subjective probabilities, and therefore variances; estimations; distributions and statements based on probabilities are usually those based on subjective probabilities, whether the word “subjective” is used or not, unless indicated otherwise.

[0155] There are many ways to convert discrete numbers into bipolar binary vectors. A standard way to convert a base-10 number into a unipolar binary number is to convert the base-10 representation into a base-2 representation. For instance, (51)₁₀=(110011)₂. A unipolar binary number can be converted into a bipolar binary vector by changing every 0 to −1. For instance, (110011)₂ is converted into [1 1 −1 −1 1 1]′. An example of converting a representation of a probability distribution, which is a R-vector with real-valued components, is illustrated in FIG. 11, wherein each component is approximated by a 3-component unipolar binary number before being converted into a bipolar binary number by changing every 0 to −1.

[0156] The Hamming distance between the standard unipolar binary representations of two integers is not “consistent” with their real value distance in the sense that a larger Hamming distance may correspond to a smaller real-value distance. For instance, consider (10000)₂=(16)₁₀, (01111)₂=(15)₁₀, and (00000)₂=(0)₁₀. The Hamming distance between 10000 and 01111 is 5, and the real-value distance between 15 and 16 is only 1. However, the Hamming distance between 00000 and 01111 is 4, but the real-value distance between 15 and 0 is 15.

[0157] In some applications of the disclosed invention, the “consistency” with Hamming distance is important to ensure that the disclosed pattern classifier has better generalization ability. For “consistency” with Hamming distance, “grey level unipolar binary representations” can be used. For instance, the integers, 6 and 4, are represented by the grey level representations, 00111111 and 00001111, instead of the unipolar binary numbers, 110 and 100, respectively. The 8-dimensional bipolar binary vectors representing these grey level unipolar binary representations are, respectively,

• • [0158] [−1 −1 1 1 1 1 1 1]′
  • [0159] [−1 −1 −1 −1 1 1 1 1]′
    n-dimensional bipolar binary vectors are also called n-component bipolar binary vectors. For example, the above two 8-dimensional vectors are also called 8-component bipolar binary vectors. An obvious disadvantage of such a bipolar binary vector representation is the large number of components required.

[0160] For reducing this disadvantage, the well-known Gray encoding can be used (John G. Proakis, Digital Communication, Third Edition, McGraw-Hill, 1995). The Gray code words of two adjacent integers differ by one component. For example, the Gray code words of the integers, 0 to 15, are, respectively, 0000, 0001, 0011, 0010, 0110, 0111, 0101, 0100, 1100, 1101, 1111, 1110, 1010, 1011, 1001, 1000. The corresponding bipolar binary vector representations are easily obtained as before. For example, the Gray code word of the integer 12 is 1010, and the bipolar binary vector representation of it is [1, −1, 1, −1]. Gray code is not completely “consistent” with the Hamming distance. For instance, the Hamming distance between the Gray code words of the integers, 0 and 2, is 2, but the Hamming distance between those of the integers, 0 and 3, is only 1. However, compared with the grey level representation, the representation from Gray encoding requires much smaller number of components.

[0161] There are other methods of transforming measurement vectors or feature vectors that are not ternary vectors into bipolar binary feature vectors, which can then be transformed into ternary feature vectors. Feature vectors used by PAMs are ternary feature vectors.

[0162] For simplicity and clarity, all feature vectors are ternary feature vectors in the rest of this invention disclosure unless indicated otherwise; column and row vectors are special matrices and also called matrices; and a matrix is considered to consist of row or column vector(s).

5.1 Orthogonal Expansion of Ternary Vectors

[0163] We now show how ternary vectors are expanded into orthogonal ternary vectors by a method recently discovered by this inventor. In this invention disclosure, the transpose of a matrix or a vector is denoted by an apostrophe ′ (i.e., a prime).

[0164] Given an m-dimensional ternary vector v=[v₁ v₂ . . . v_m]′, the first-stage expansion of v is defined as {hacek over (v)}(1)=[1 v₁]′, and the second-stage expansion is defined as

[0165] $\begin{array}{c}\stackrel{⋓}{v}\left(1,2\right)={\left[\begin{array}{cccc}{\stackrel{⋓}{v}}^{\prime }& \left(1\right)& v_2{\stackrel{⋓}{v}}^{\prime }& \left(1\right)\end{array}\right]}^{\prime }\\ ={\left[\begin{array}{cccc}1& v_1& v_2& v_2{v}_1\end{array}\right]}^{\prime }\end{array}$
In general, the (j+1)-th-stage expansion is recursively defined as
{hacek over (v)}(1, . . . ,j+1)=[{hacek over (v)}(1, . . . ,j)v_j+1{hacek over (v)}′(1, . . . ,j)]′  (1)
The m-th stage expansion, which includes all the different powers of the components of v, is a 2^m-dimensional ternary vector:
{hacek over (v)}[1v₁v₂v₂v₁v₃v₃v₁v₃V₂v₃v₂v₁ . . . v_m . . . v₁]′  (2)
which is called the orthogonal expansion of v. Reordering the components of {hacek over (v)} in accordance with the powers of the components, we obtain an alternative orthogonal expansion:
{hacek over (v)}[1v₁ . . . v_mv₁v₂ . . . v₁v_mv₂v₃ . . . v₁ . . . v_m]′
which can also be used in this invention disclosure. In fact, many other orthogonal expansions of v are possible by different orderings of the components, but are all denoted by {hacek over (v)}. The use of the same symbol {hacek over (v)}is not expected to cause confusion. The components of any orthogonal expansion of v form the set,
{v₁ⁱ¹v₂ⁱ² . . . v_m^im|i_jε{0,1},j=1,2, . . . ,m}
which has 2^m elements. The components in {hacek over (v)}are actually the terms in the expansion of (1+v₁)(1+v₂) . . . (1+v_m). Subvectors of orthogonal expansions are sometimes used instead for reducing storage or memory space and/or computation requirements.
Theorem 1.

[0166] Let a=[a₁ . . . a_m]′ and b=[b₁ . . . b_m]′ be two m-dimensional ternary vectors. Then the inner product {hacek over (a)}′{hacek over (b)} of their orthogonal expansions, {hacek over (a)} and {hacek over (b)}, can be expressed as follows:

[0167] ${\stackrel{⋓}{a}}^{\prime }\stackrel{⋓}{b}=\prod _{j=1}^m\left(1-a_j{b}_j\right)$
The following properties are immediate consequences of this formula:

[0168] 1. If a_kb_k=−1 for some kε{1, . . . , m}, then {hacek over (a)}′{hacek over (b)}=0.

[0169] 2. If a_kb_k=0 for some k in {1, . . . , m}, then

[0170] ${\stackrel{⋓}{a}}^{\prime }\stackrel{⋓}{b}=\prod _{j=1,j\ne k}^m\left(1-a_j{b}_j\right).$

[0171] 3. If {hacek over (a)}/{hacek over (b)}≠0, then {hacek over (a)}′{hacek over (b)}=2^a′^b.

[0172] 4. If a and b are bipolar binary vectors, then {hacek over (a)}′{hacek over (b)}=0 if a≠b; and {hacek over (a)}′{hacek over (b)}=2^m if a=b. Proof. Applying the recursive formula (I), we obtain

[0173] ${\stackrel{⋓}{a}}^{\prime }\left(1,\dots ,j+1\right)\stackrel{⋓}{b}\left(1,\dots ,j+1\right)={\left[{\stackrel{⋓}{a}}^{\prime }\left(1,\dots ,j\right)a_{j+1}{\stackrel{⋓}{a}}^{\prime }\left(1,\dots ,j\right)\right]\left[{\stackrel{⋓}{b}}^{\prime }\left(1,\dots ,j\right)b_{j+1}{\stackrel{⋓}{b}}^{\prime }\left(1,\dots ,j\right)\right]}^{\prime }={\stackrel{⋓}{a}}^{\prime }\left(1,\dots ,j\right)\stackrel{⋓}{b}\left(1,\dots ,j\right)+a_{j+1}{b}_{j+1}{\stackrel{⋓}{a}}^{\prime }\left(1,\dots ,j\right)\stackrel{⋓}{b}\left(1,\dots ,j\right)={\stackrel{⋓}{a}}^{\prime }\left(1,\dots ,j\right)\stackrel{⋓}{b}\left(1,\dots ,j\right)\left(1+a_{j+1}{b}_{j+1}\right)$
It follows that {hacek over (a)}′{hacek over (b)}=(1+a₁b₁)(1+a₂b₂) . . . (1+a_mb_m). The four properties above are easy consequences of this formula.

[0174] We remark that if some components of a are set equal to zero to obtain a vector c and the nonzero components of c are all equal to their corresponding components in b, then we still have {hacek over (c)}′{hacek over (b)}≠0. This property is used by learning machines disclosed herein to learn and recognize corrupted, distorted and occluded patterns and to facilitate generalization on such patterns.

[0175] The following notations and terminologies are used in this invention disclosure: For v=[v₁ v₂ . . . v_m]′ considered above, let n=[n₁ . . . n_k] be a vector whose components are different integers from the set {1, . . . , m} such that 1≦n₁< . . . <n_k≦m. The vector v(n)=[v_n1 . . . v_nk]′ is a subvector, called a k-component or k-dimensional subvector, of the vector v. The vector n is called a subvector index. v(n) is said to be on the subvector index n or have the subvector index n. {hacek over (v)}(n) denotes the orthogonal expansion of v(n).

5.2 Expansion Correlation Matrices

[0176] A PAM is a network of PUs (processing units) organized in one or more layers. A typical layer is shown in FIG. 14. The feature vector input to layer l at time or numbering τ is denoted by x_τ^l-1, and the output from the layer at τ is denoted by x{y_τ^l}, which is a point estimate of the label of the feature vector x_τ^l-1. The symbols, y_τ^l and x{y_τ^l}, are defined and explained in detail below. A multilayer PAM without feedbacks is shown in FIG. 15 and is called a hierarchical PAM (HPAM). Note that the feature vector x_τ⁰ input to layer l of an HPAM is the exogenous feature vector x_τ^ex, and that for l>1, the feature vector x_τ^l-1 input to layer l is the output x{y_τ^l-1} from layer l-1. An example multilayer PAM with feedbacks is shown in FIG. 16 and is called a temporal HPAM (THPAM). Note that the feature vector x_τ⁰ input to layer 1 of an THPAM comprises the exogenous feature vector x_τ^ex and the feedbacks from the same or higher-ordered layers (e.g., x{y_τ-1¹} and x{y_τ-1²} in FIG. 16) and that the feature vector x_τ^l-1 input to layer l comprises the output x{y_τ^l-1} from layer l-1 and feedbacks from the same or higher-ordered layers (e.g., x_τ² comprises x{y_τ²}, x{y_τ-1⁵} and x{y_τ-1³} in FIG. 16). HPAMs and THPAMs are described in more detail in the subsection on “Multilayer and Recurrent Networks.” Before said subsection, this and the next 6 subsections describe essentially the PUs (processing units) in those networks. For notational simplicity, the superscript l-1 in x_t^l-1 and dependencies on l-1 or l in other symbols are usually suppressed in these 7 subsections when no confusion is expected.

[0177] Let x_t, t=1, 2, . . . , denote a sequence of M-dimensional feature vectors x_t=[x_t1 . . . x_tM]′, whose components are ternary numbers. The feature vectors x_t, t=1, 2, . . . , are not necessarily different. The ternary entry x_tm is called the m-th component of the feature vector x_t. Let n=[n₁ . . . n_k]′ be a subvector [1 . . . M]′ such that n₁< . . . <n_k. The subvector x_t(n)=[x_tn1 . . . x_tnk]′ is a feature subvector of the feature vector x_t. n is called a feature subvector index (FSI), and x_t(n) is said to be a feature subvector on the FSI n or have the FSI n. However, we stress that x_t(n) is itself a feature vector. When a PU is discussed regardless of the layer of the PAM that the PU belongs, an input vector to the PU is referred to as a feature vector. Each PU is associated with a fixed FSI n and denoted by PU(n). Using these notations, the sequence of subvectors of x_t, t=1, 2, . . . , that is input to PU(n) is x_t(n), t=1, 2, . . . . An example of a group of ternary “pixels” that is identified with a feature subvector index n is shown in FIG. 3. An FSI n of a PU usually has subvectors, n(u), u=1, . . . , U, on which subvectors x_t(n(u)) of x_t(n) are separately processed by PU(n) at first. The subvectors, n(u), u=1, . . . , U, are not necessarily disjoint, and their components are usually randomly selected from those of n. An example of such a subvector n(u) is shown in FIG. 3 and indicated by 46.

[0178] A PU in a PAM has a “receptive field” in the exogenous feature vector and a “receptive field” in the measurement vector. These receptive fields can be found by tracing the feedforward connections in the PAM backforward from a feature vector input to the PU (or the feature subvector index of the PU) to an exogenous feature vector (or the input terminals) of the PAM, and then tracing the transformation, that maps the measurement vector into the exogenous feature vector, backwards from the exogenous feature vector to the measurement vector. The components of the measurement vector that can be reached by this backward tracing from a PU to the exogenous feature vector and then to the measurement vector are called the “receptive field” of the PU in the measurement vector. The components of the exogenous feature vector that can be reached by this backward tracing from a PU to the exogenous feature vector are called the “receptive field” of the PU in the exogenous feature vector. The label of a feature vector input to a PU is the label of the corresponding components of the measurement vector in the receptive field of the PU in the measurement vector. The label of the corresponding components of the exogenous feature vector in the receptive field of the PU in the exogenous feature vector is also this label.

[0179] Let a label of the feature vector x_t(n) be denoted by r_t(n), which is an R-dimensional ternary vector. If R is 1, r_t(n) is real-valued. All subvectors, x_t(n(u)), u=1, . . . , U, of x_t(n) share the same label r_t(n). In supervised learning by PU(n), r_t(n) is provided from outside the PAM, and in unsupervised learning by PU(n), r_t(n) is generated by the PU itself.

[0180] The pairs (x_t(n(u)), r_t(n)), t=1, 2, . . . , are learned by the PU to form two of expansion correlation matrices, ECMs, D(n(u)), C(n(u)), A (n(u)), B(n(u)) on n(u). After the first T pairs are learned, these matrices are

[0181] $\begin{array}{cc}D\left(n\left(u\right)\right)=\Lambda \sum _{t=1}^T{W}_t\left(n\left(u\right),T\right)r_t\left(n\right){\stackrel{⋓}{x}}_t^{\prime }\left(n\left(u\right)\right)& \left(3\right)\\ C\left(n\left(u\right)\right)=\Lambda \sum _{t=1}^T{W}_t\left(n\left(u\right),T\right)I{\stackrel{⋓}{x}}_t^{\prime }\left(n\left(u\right)\right)& \left(4\right)\\ A\left(n\left(u\right)\right)=\frac{1}{2}\Lambda \sum _{t=1}^T{W}_t\left(n\left(u\right),T\right)\left(I+r_t\left(n\right)\right){\stackrel{⋓}{x}}_t^{\prime }\left(n\left(u\right)\right)& \left(5\right)\\ B\left(n\left(u\right)\right)=\frac{1}{2}\Lambda \sum _{t=1}^T{W}_t\left(n\left(u\right),T\right)\left(I-r_t\left(n\right)\right){\stackrel{⋓}{x}}_t^{\prime }\left(n\left(u\right)\right)& \left(6\right)\end{array}$
where {hacek over (x)}_t(n(u)) are orthogonal expansions of x_t(n(u)), I=[1 . . . 1]′ with R components, Λ is a scaling constant that is selected to keep all numbers involved in an application of a PAM manageable, W_t(n(u), T) is a weight matrix, which is usually an diagonal matrix, diag(w_t1(n(u), T) w_t2(u), T) . . . w_tR(n(u), T)), that is selected to place emphases on components of the label, place emphases on (x_t(n(u)), r_t(n)) of different numberings t, and keep the entries in the ECMs bounded. For example, W_t(n(u), T)=λ^T-t2^{−dim n(u)}h(n(u))I, where λ (0<λ<1) is a forgetting factor, 2^{−dim n(u)} eliminates the constant 2^{dim n(u)} arising from {hacek over (x)}_t(n(u)){hacek over (x)}_t(n(u))=2^{dim n(u)}, and h(n(u)) assigns emphases to subvectors x_t(n(u)) on n(u). There are many other possible weight matrices, depending on applications of the present invention.

[0182] Any two of the above four expansion correlation matrices (ECMs) can be obtained from the other two. We usually use D(n(u)) and C(n(u)) in this invention disclosure. They can be combined into one ECM [C′(n(u)) D′(n(u))]′. If W_t(n(u), T)=w_t(n(u), T) I, where w_t(n(u), T) is a real-valued function, all rows of C(n(u)) are equal, and only one needs to be stored.

[0183] ECMs, D(n(u)) and C(n(u)), can be adjusted to learn a pair (x_τ(n(u)), r_τ(n)). For example, if W_t(n(u), T)=λ^T-tI, the ECMs are adjusted as follows: If r_τ(n) 0, D(n(u)) and C(n(u)) are replaced respectively with λD(n(u))+Λr_τ(n){hacek over (x)}_t(n(u)) and λC(n(u))+Λ{hacek over (x)}_t′(n(u)), which are each a weighted sum of a ECM and r_τ(n){hacek over (x)}_t′(n(u)) or {hacek over (x)}_t′(n(u)). If r_τ(n)=0, then D(n(u)) and C(n(u)) are unchanged. Note that W_t(n(u), T) is a diagonal matrix, and C(n(u)) is a row vector here. This adjustment of the ECMs is shown in FIG. 5.

[0184] If W_t(n(u), T)=I/√{square root over (T)}, the ECMs are adjusted as follows: If r_τ(n)≠0, D(n(u)) and C(n(u)) are replaced respectively with (√{square root over (T)}D(n(u))+Λr_τ(n){hacek over (x)}_t′(n(u)))/√{square root over (T+1)} and (√{square root over (T)}C(n(u))+Λ{hacek over (x)}_t′(n(u)))/√{square root over (T+1)}, which are each a weighted sum of a ECM and r_τ(n){hacek over (x)}_t′(n(u)) or {hacek over (x)}_t′(n(u)), and √{square root over (T)} is replaced with √{square root over (T+1)}. If r_τ(n)=0, then D(n(u)) and C(n(u)) are unchanged. Note that W_t(n(u), T) is a diagonal matrix, and C(n(u)) is a row vector here. This adjustment of the ECMs is shown in FIG. 6.

[0185] Orthogonal expansions (OEs) {hacek over (x)}_t(n(u)) and ECMs, D(n(u)), C(n(u)), A (n(u)), B(n(u)), u=1, . . . , U, are assembled into a general orthogonal expansion (GOE) {hacek over (x)}_t(n) and general expansion correlation matrices (GECMs), D(n), C(n), A(n), B(n), for PU(n) (the PU on the feature subvector index n) as follows:
{hacek over (x)}_t(n)=[{hacek over (x)}_t′(n(1)){hacek over (x)}_t′(n(2)) . . . {hacek over (x)}_t′(n(U))]′  (7)
D(n)=[D(n(1))D(n(2)) . . . D(n(U))]  (8)
C(n)=[C(n(1))C(n(2)) . . . C(n(U))]  (9)
A(n)=[A(n(1))A(n(2)) . . . A(n(U))]  (10)
B(n)=[B(n(1))B(n(2)) . . . B(n(U))]  (11)
The GOE {hacek over (x)}_t(n) and the GECMs, D(n) and C(n), are shown in FIG. 8.
5.3 Representations of Probability Distributions

[0186] Let us define the symbols a_τ(n(u)), b_τ(n(u)), c_τ(n(u)), d_τ(n(u)) by
a_τ(n(u)):=A(n(u)){hacek over (x)}_τ(n(u))  (12)
b_τ(n(u)):=B(n(u)){hacek over (x)}_τ(n(u))  (13)
c_τ(n(u)):=C(n(u)){hacek over (x)}_τ(n(u))  (14)
d_τ(n(u)):=D(n(u)){hacek over (x)}_τ(n(u))  (15)
and the symbols a_τ(n), b_τ(n), c_τ(n), d_τ(n) by

[0187] $\begin{array}{cc}{a}_{\tau }\left(n\right):=A\left(n\right){\stackrel{⋓}{x}}_{\tau }\left(n\right)=\sum _{u=1}^{U}A\left(n\left(u\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)=\sum _{u=1}^U{a}_{\tau }\left(n\left(u\right)\right)& \left(16\right)\\ b_{\tau }\left(n\right):=B\left(n\right){\stackrel{⋓}{x}}_{\tau }\left(n\right)=\sum _{u=1}^{U}B\left(n\left(u\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)=\sum _{u=1}^U{b}_{\tau }\left(n\left(u\right)\right)& \left(17\right)\\ c_{\tau }\left(n\right):=C\left(n\right){\stackrel{⋓}{x}}_{\tau }\left(n\right)=\sum _{u=1}^{U}C\left(n\left(u\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)=\sum _{u=1}^U{c}_{\tau }\left(n\left(u\right)\right)& \left(18\right)\\ d_{\tau }\left(n\right):=D\left(n\right){\stackrel{⋓}{x}}_{\tau }\left(n\right)=\sum _{u=1}^{U}D\left(n\left(u\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)=\sum _{u=1}^U{d}_{\tau }\left(n\left(u\right)\right)& \left(19\right)\end{array}$
where {hacek over (x)}_t(n) is a general orthogonal expansion (GOE) and D(n), C(n), A(n) and B(n) are general expansion correlation matrices (GECMs) for PU(n). It is easy to see that d_τ(n(u))=2a_τ(n(u))−c_τ(n(u)), and d_τ(n)=2a_τ(n)−c_τ(n).

[0188] To illustrate the use of a_τ(n), b_τ(n), c_τ(n), d_τ(n), two examples are given in the following: Example 1. Assume that x_t(n) and x_τ(n) are bipolar vectors, and W_t(n(u), T)=I/√{square root over (T)}. By Theorem 1, {hacek over (x)}_t(n(u)) {hacek over (x)}_τ(n(u))=0 or {hacek over (x)}_t(n(u)){hacek over (x)}_τ(n(u))=2^{dim n(u)}, depending on whether x_k(n(u))≠x_τ(n(u)) or x_k(n(u))=x_τ(n(u)). It follows that
a_τj(n(u))=Λ2^dimn(u)|G_τj(n(u),+)|/√{square root over (T)}  (20)
b_τj(n(u))=Λ2^dimn(u)|G_τj(n(u),−)|/√{square root over (T)}  (21)
G_τj(n(u),+)={tε[0,T]|x_t(n(u))=x_τ(n(u)),r_tj(n)=1}  (22)
G_τj(n(u),−)={tε[0,T]|x_t(n(u))=x_τ(n(u)),r_tj(n)=−1}  (23)
where |G_τj(n(u),+)| and |G_τj(n(u),−)| are the numbers of elements in the sets G_τj(n(u),+) and G_τj(n(u),−), respectively. If a_τj(n(u))=b_τj(n(u)) (i.e., c_τj(n(u))=0, or c_τj(n(u))≠0, but a_τj(n(u))=b_τj(n(u))), the ECMs for n(u) have no information or no preference about the value of the component r_τj(n) of the label r_τ(n) of the feature subvector x_τ(n).

[0189] It follows that

[0190] $a_{\tau j}\left(n\right)=\frac{\Lambda }{\sqrt{T}}\sum _{u=1}^U{2}^{\dim n\left(u\right)}G_{\tau j}\left(n\left(u\right),+\right)$ $b_{\tau j}\left(n\right)=\frac{\Lambda }{\sqrt{T}}\sum _{u=1}^U{2}^{\dim n\left(u\right)}G_{\tau j}\left(n\left(u\right),-\right)$ $c_{\tau j}\left(n\right)=\frac{\Lambda }{\sqrt{T}}\sum _{u=1}^U{2}^{\dim n\left(u\right)}\left(G_{\tau j}\left(n\left(u\right),+\right)+G_{\tau j}\left(n\left(u\right),-\right)\right)$ $d_{\tau j}\left(n\right)=2a_{\tau j}\left(n\right)-c_{\tau j}\left(n\right)$
(This shows that if dim n(u₁)=dim n(u₂)+k, then a_τj(n(u₁)) and c_τj(n(u₁)) are weighted 2^k times as much as a_τj(n(u₂)) and c_τj(n(u₂)) in a_τj(n) and c_τj(n). This weighting can be changed by using W_t(n(u), T)=2^{−dim n(u)}h(n(u))I/√{square root over (T)}.)

[0191] For simplicity and clarity, assume that dim n(u), u=1, . . . , U, are all the same. If c_τj(n)≠0, then

[0192] $\begin{array}{cc}\frac{a_{\tau j}\left(n\right)}{c_{\tau j}\left(n\right)}=\frac{\sum _{u=1}^UG_{\tau j}\left(n\left(u\right),+\right)}{\sum _{u=1}^U\left(G_{\tau j}\left(n\left(u\right),+\right)+G_{\tau j}\left(n\left(u\right),-\right)\right)}& \left(24\right)\end{array}$
can be viewed as a probability that r_τj(n)=1. This probability is a subjective probability, because it is based on “experience” represented by the GECMs C(n) and A(n), or C(n) and D(n)=2A(n)−C(n), which are learned from (or constructed with) (x_t(n(u)), r_t(n)), t=1, 2, . . . , T.

[0193] Assume that all x_t(n) and x_τ(n) are bipolar binary vectors, and W_t(n(u), T)=λ^T-tI.

[0194] $a_{\tau j}\left(n\left(u\right)\right)=\Lambda \sum _{t\in G_{\tau j}\left(n\left(u\right),+\right)}{2}^{\dim n\left(u\right)}{\lambda }^{T-t}$ $b_{\tau j}\left(n\left(u\right)\right)=\Lambda \sum _{t\in G_{\tau j}\left(n\left(u\right),-\right)}{2}^{\dim n\left(u\right)}{\lambda }^{T-t}$
Assume further that dim n(u), u=1, U are all the same. Then if c_τ(n)≠0,

[0195] $\begin{array}{cc}\frac{a_{\tau j}\left(n\right)}{c_{\tau j}\left(n\right)}=\frac{\sum _{u=1}^U\sum _{t\in G_{\tau j}\left(n\left(u\right),+\right)}{\lambda }^{T-t}}{\sum _{u=1}^U\sum _{t\in G_{\tau j}\left(n\left(u\right),+\right)\bigcup G_{\tau j}\left(n\left(u\right),-\right)}{\lambda }^{T-t}}& \left(25\right)\end{array}$
can be viewed as a subjective probability that the label r (n)=1, based on the GECMs C(n) and A(n), or C(n) and D(n)=2A(n)−C(n), which are constructed with pairs (x_t(n(u)), r_t(n)), t=1, 2, . . . , T. The forgetting factor λ de-emphasizes past pairs gradually.

[0196] In both of the above examples, W_t(n(u), T) is a diagonal matrix with equal diagonal entries. In this case, all components c_τj(n) of C_τ(n) are equal and all rows of C(n) are equal. Therefore, only one row of C(n), say C₁(n), and one component of C_τ(n), say c_τ1(n), are sufficient. If only C₁(n) is used, it is also denoted by C(n).

[0197] In general, the ratio a_τj(n)/c_τj(n) can be viewed as a subjective probability that r_τj(n)=1 based on the pairs (x_t(n), r_t(n)) that have been learned by PU(n) and the weight matrices W_t(n(u), T). All the statements concerning a probability in this invention disclosure are statements concerning a subjective probability, and the word “subjective” is usually omitted. If c_τj(n)≠0, then a_τj(n)/c_τj(n) is the probability p_τj(n) that the j-th component r (n) of the label r_τ(n) of x_τ(n) is +1 based on D(n) and C(n). If c_τj(n)=0, then we set p_τj(n)=1/2. The vector
p_τ(n)=[p_τ1(n)p_τ2(n) . . . p_τR(n)]′
is a representation of a probability distribution of the label r_τ(n) of the feature subvector x_τ(n) input to PU(n). Since D(n)=2A(n)−C(n), if c_τj(n)≠0, the ratio d_τj(n)/c_τj(n) is equal to 2p_τj(n)−1. If c_τj(n)=0, set 2p_τj(n)−1=0. Denote 2p_τj(n)−1 by y_τj(n). Then the vector r_τ(n)=2p_τ(n)−I satisfies

[0198] $\begin{array}{c}{y}_{\tau }\left(n\right)={\left[\begin{array}{ccc}2p_{\tau 1}\left(n\right)-1& \cdots & 2p_{\tau R}\left(n\right)-1\end{array}\right]}^{\prime }\\ ={\left[\begin{array}{cccc}{d}_{\tau 1}\left(n\right)/c_{\tau 1}\left(n\right)& d_{\tau 2}\left(n\right)/c_{\tau 2}\left(n\right)& \cdots & d_{\tau R}\left(n\right)/c_{\tau R}\left(n\right)\end{array}\right]}^{\prime }\end{array}$
and is also a representation of a probability distribution of the label r_τ(n) of the feature subvector x_τ(n). Here, I=[1 1 . . . 1]′.
5.4 Masking Matrices

[0199] Let a subvector x_τ(n(u)) be a slightly different (e.g., corrupted, modified, deviated) version of x_ξ(n(u)), which is one of the subvectors, x_t(n(u)), t=1, 2, . . . , T, stored in ECMs, D(n(u)) and C(n(u)) (or any two of the four ECMs, D(n(u)), C(n(u)), A (n(u)) and B(n(u)), on n(u)). Assume that x_τ(n(u)) is very different from other subvectors stored in the ECMs. Since {hacek over (x)}_ξ′(n(u)){hacek over (x)}_τ(n(u))=0, the information stored in D(n(u)) and C(n(u)) about the label r (n) cannot be obtained from d(n(u))=D(n(u)){hacek over (x)}_τ(n(u)) and c (n(u))=C(n(u)){hacek over (x)}_τ(n(u)). This is viewed as failure of d (n(u)) and c (n(u)) or the ECMs to generalize or adequately generalize on x_τ(n(u)). Because of property 2 in Theorem 1, if the corrupted, distorted and occluded components in x_τ(n(u)) are set equal to zero, then the information stored in the ECMs about the label r_ξ(n) can be obtained in part from the remaining components of x_τ(n(u)). This observation motivated masking matrices described in this section.

[0200] Let us denote the vector v=[v₁ v₂ . . . v_n]′ with its i₁-th, i₂-th, . . . , and i_i-th components set equal to 0 by v(i₁⁻, i₂⁻, . . . , i_j⁻), where 1≦i₁<i₂< . . . <i_j≦<n. For example, if v=[1 −1 −1 1]′, then v(2⁻, 4⁻)=[1 0 −1 0]′. Denoting the n-dimensional vector [1 1 . . . 1]′ by I and denoting the orthogonal expansion of v(i₁⁻, i₂⁻, . . . , i_j⁻) by {hacek over (v)}(i₁⁻, i₂⁻, . . . , i_j⁻), we note that v(i₁⁻, i₂⁻, . . . , i_j⁻)=diag(I(i₁⁻, i₂⁻, . . . , i_j⁻)) v and {hacek over (v)}(i₁⁻, i₂⁻, . . . , i_j⁻)=diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) {hacek over (v)}, were {hacek over (v)}(i₁⁻, i₂⁻, . . . , i_j⁻) and {hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻) denote the orthogonal expansions of v(i₁⁻, i₂⁻, . . . , i_j⁻) and I(i₁⁻, i₂⁻, . . . , i_j⁻) respectively (not the orthogonal expansions of v and I with their i₁-th, i₂-th, . . . , and i_j-th components set equal to 0).

[0201] Using these notations, a feature subvector x (n(u)) with its i₁-th, i₂-th, . . . , and i_j-th components set equal to 0 is x_t(n(u)) (i₁⁻, i₂⁻, . . . , i_j⁻), and the orthogonal expansion of x_t(n(u)) (i₁⁻, i₂⁻, . . . , i_j⁻) is diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)){hacek over (x)}_t(n(u)). Hence, the matrix diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻), as a matrix transformation, sets the i₁-th, i₂-th, . . . , and i_j-th components of x_t(n(u)) equal to zero in transforming {hacek over (x)}_t(n(u)) (i.e., diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)){hacek over (x)}_t(n(u))). Therefore, diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) is called a masking matrix.

[0202] An important property of the masking matrix diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) is the following: If
diag({hacek over (I)}(i₁⁻,i₂⁻, . . . ,i_j⁻)){hacek over (x)}_t(n(u))=diag({hacek over (I)}(i₁⁻,i₂⁻, . . . ,i_j⁻)){hacek over (x)}_τ(n(u))
then
{hacek over (x)}_t(n(u))diag({hacek over (I)}(i₁⁻,i₂⁻, . . . ,i_j⁻)){hacek over (x)}_τ(n(u))=2^dimn(u)-j.
If

• • [0203] diag({hacek over (I)}(i₁⁻,i₂⁻, . . . , i_j⁻)) {hacek over (x)}_t(n(u))≠diag ({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) {hacek over (x)}_τ(n(u)) then (n(u))diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) {hacek over (x)}_τ(n(u))=0.

[0204] Using this property, we combine all such masking matrices that set less than or equal to a selected positive integer J (n(u)) of components of x_t(n(u)) equal to zero into the following masking matrix

[0205] $\begin{array}{cc}M\left(n\left(u\right)\right)=I+\sum _{j=1}^{J\left(n\left(u\right)\right)}\sum _{i_j=1}^{\dim n\left(u\right)}\cdots \sum _{i_2=2}^{i_3-1}\sum _{i_1=1}^{i_2-1}{2}^{-8j}{2}^j\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_i^{-},i_2^{-},\dots ,i_j^{-}\right)\right)& \left(26\right)\end{array}$
where 2^j is used to compensate for the factor 2^−j in 2^{dim n(u)-j} in the important property stated above, and 2^−8j is an example weight selected to differentiate between different levels j of maskings. Some other examples are 2^−6j, 2^−7j, 2^−9j, 10⁻², etc. The weight should be selected to suit the application. M(n(u)) is shown in FIG. 7 and FIG. 9. Because diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) appears in a summand 2^−8j2^jdiag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) in (26) for the masking matrix M(n(u)), the matrix diag({hacek over (I)}(i₁⁻, i₂⁻, . . . , i_j⁻)) is called a summand masking matrix in M(n(u)).

[0206] Let us denote M(n(u)) by M here for abbreviation. Note that for j=1, . . . , R, we have the following:

[0207] If C_j(n(u)){hacek over (x)}_τ(n(u))≠0, then
D_j(n(u)){hacek over (x)}_τ(n(u))≈D_j(n(u))M{hacek over (x)}_τ(n(u))
C_j(n(u)){hacek over (x)}_τ(n(u))≈C_j(n(u))M{hacek over (x)}_τ(n(u))
A_j(n(u)){hacek over (x)}_τ(n(u))≈A_j(n(u))M{hacek over (x)}_τ(n(u))
B_j(n(u)){hacek over (x)}_τ(n(u))≈B_j(n(u))M{hacek over (x)}_τ(n(u))

[0208] If C_j(n(u)){hacek over (x)}_τ(n(u))=0, but

[0209] $C_j\left(n\left(u\right)\right)\sum _{i_1=1}^{\dim n\left(u\right)}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\ne 0,$
then

[0210] $D_j\left(n\left(u\right)\right)\sum _{i_1=1}^{\dim n\left(u\right)}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\approx D_j\left(n\left(u\right)\right)M{\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)$ $C_j\left(n\left(u\right)\right)\sum _{i_1=1}^{\dim n\left(u\right)}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\approx C_j\left(n\left(u\right)\right)M{\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)$ $A_j\left(n\left(u\right)\right)\sum _{i_1=1}^{\dim n\left(u\right)}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\approx A_j\left(n\left(u\right)\right)M{\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)$ $B_j\left(n\left(u\right)\right)\sum _{i_1=1}^{\dim n\left(u\right)}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\approx B_j\left(n\left(u\right)\right)M{\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right).$

[0211] If C_j(n(u)) x_τ(n(u))=0,

[0212] $C_j\left(n\left(u\right)\right)\sum _{i_1=1}^{\dim n\left(u\right)}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)=0,$
but

[0213] $C_j\left(n\left(u\right)\right)\sum _{i_2=2}^{\dim n\left(u\right)}\sum _{i_1=1}^{i_2-1}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-},i_2^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\ne 0,$
then

[0214] $D_j\left(n\left(u\right)\right)\sum _{i_2=2}^{\dim n\left(u\right)}\sum _{i_1=1}^{i_2-1}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-},i_2^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\approx D_j\left(n\left(u\right)\right)M{\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)$ $C_j\left(n\left(u\right)\right)\sum _{i_2=2}^{\dim n\left(u\right)}\sum _{i_1=1}^{i_2-1}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-},i_2^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\approx C_j\left(n\left(u\right)\right)M{\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)$ $A_j\left(n\left(u\right)\right)\sum _{i_2=2}^{\dim n\left(u\right)}\sum _{i_1=1}^{i_2-1}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-},i_2^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\approx A_j\left(n\left(u\right)\right)M{\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)$ $B_j\left(n\left(u\right)\right)\sum _{i_2=2}^{\dim n\left(u\right)}\sum _{i_1=1}^{i_2-1}\mathrm{diag}\left(\stackrel{⋓}{I}\left(i_1^{-},i_2^{-}\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)\approx B_j\left(n\left(u\right)\right)M{\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right).$

[0215] Continuing in this manner, it is seen that D_j(n(u)) M{hacek over (x)}_τ(n(u)), C_j(n(u)) M{hacek over (x)}_τ(n(u)), A_j(n(u)) M{hacek over (x)}_τ(n(u)), B_j(n(u)) M{hacek over (x)}_τ(n(u)) always use the greatest number of uncorrupted, undeviated or unmodified components of x_τ(n(u)) in estimating d_τj(n(u)), c_τj(n(u)), a_τj(n(u)), b_τj(n(u)), respectively.

[0216] Corresponding to {hacek over (x)}_t(n), D(n), C(n), A(n), B(n) defined in (7), (8), (9), (10), (11), a general masking matrix is defined as follows:
M(n)=diag[M(n(1))M(n(2)) . . . M(n(U))]  (27)
where the right side is a matrix with M(n(u)), u=1, 2, . . . , U, as diagonal blocks and zero elsewhere. M(n) is shown in FIG. 9.

[0217] If the masking matrix M(n(u)) is used, the symbols a_τ(n(u)), b_τ(n(u)), c_τ(n(u)), d_τ(n(u)) are defined as follows:
a_τ(n(u)):=A(n(u))M(n(u)){hacek over (x)}_τ(n(u))  (28)
b_τ(n(u)):=B(n(u))M(n(u)){hacek over (x)}_τ(n(u))  (29)
c_τ(n(u)):=C(n(u))M(n(u)){hacek over (x)}_τ(n(u))  (30)
d_τ(n(u)):=D(n(u))M(n(u)){hacek over (x)}_τ(n(u))  (31)
If the masking matrix M(n) is used, the symbols a_τ(n), b_τ(n), C_τ(n), d_τ(n) are defined as follows:

[0218] $\begin{array}{cc}{a}_{\tau }\left(n\right):=A\left(n\right)M\left(n\right){\stackrel{⋓}{x}}_{\tau }\left(n\right)=\sum _{u=1}^{U}A\left(n\left(u\right)\right)M\left(n\left(u\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)& \left(32\right)\\ b_{\tau }\left(n\right):=B\left(n\right)M\left(n\right){\stackrel{⋓}{x}}_{\tau }\left(n\right)=\sum _{u=1}^{U}B\left(n\left(u\right)\right)M\left(n\left(u\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)& \left(33\right)\\ c_{\tau }\left(n\right):=C\left(n\right)M\left(n\right){\stackrel{⋓}{x}}_{\tau }\left(n\right)=\sum _{u=1}^{U}C\left(n\left(u\right)\right)M\left(n\left(u\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)& \left(34\right)\\ d_{\tau }\left(n\right):=D\left(n\right)M\left(n\right){\stackrel{⋓}{x}}_{\tau }\left(n\right)=\sum _{u=1}^{U}D\left(n\left(u\right)\right)M\left(n\left(u\right)\right){\stackrel{⋓}{x}}_{\tau }\left(n\left(u\right)\right)& \left(35\right)\end{array}$
where {hacek over (x)}_t(n) is a general orthogonal expansion (GOE) and D(n), C(n), A(n) and B(n) are general expansion correlation matrices (GECMs) for PU(n). It follows that

[0219] $\begin{array}{cc}{a}_{\tau }\left(n\right)=\sum _{u=1}^U{a}_{\tau }\left(n\left(u\right)\right)& \left(36\right)\\ b_{\tau }\left(n\right)=\sum _{u=1}^U{b}_{\tau }\left(n\left(u\right)\right)& \left(37\right)\\ c_{\tau }\left(n\right)=\sum _{u=1}^U{c}_{\tau }\left(n\left(u\right)\right)& \left(38\right)\\ d_{\tau }\left(n\right)=\sum _{u=1}^U{d}_{\tau }\left(n\left(u\right)\right)& \left(39\right)\end{array}$
It is easy to see that d_τ(n(u))=2a_τ(n(u))−c_τ(n(u)), and d_τ(n)=2a_τ(n)−c_τ(n). If c_τj(n)=0, then we set d_τj(n)/c_τj(n)=0. If c_τj(n)≠0, then d_τj(n)/c_τj(n)=2p_τj(n)−1, where p_τj(n) is the probability that the j-th component r_τj(n) of the label r_τ(n) of x_τ(n) is +1 based on D(n) and C(n). It follows that
2p_τ(n)−1=[d_τ1(n)/c_τ1(n)d_τ2(n)d_τ2(n)/c_τ2(n) . . . d_τR(n)/c_τR(n)]′  (40)
is a representation of a probability distribution of the label r_τ(n) of x_τ(n).

[0220] An estimation means for generating this representation of a probability distribution is shown in FIG. 10: For j=1, 2, . . . , R, if c_τj(n)=0, then set y_τj(n)=0, else set y_τj(n)=d_τj(n)/c_τj(n). The output of the estimation means is y_τ(n)=[y_τ1(n) y_τ2(n) . . . y_τR(n)]′, which is a representation 2p_τ(n)−1 of a probability distribution of the label r_τ(n) of x_τ(n).

5.5 Conversion of Probabilities into Ternary Numbers

[0221] In a multilayer PAM, a feature vector into layer l is a ternary vector denoted by x_t^l-1, where t denotes a numbering of the feature vector or a time instant. In this subsection, two methods of converting a representation of a probability distribution, y_τ(n)=[y_τ1(n)y _τ2(n) . . . y_τR(n)]′, generated by a PU (processing unit) into a ternary vector are described. Recall y_τk(n)=2p_τk(n)−1, where P_τk(n) is a probability that r_τk(n) is +1. p_τ(n)=[p_τ1(n) p_τ2(n) . . . p_τR(n)]′ is an alternative representation of probability distribution of p_τ(n). y_τ(n) and p_τ(n) are related by y_τ(n)=2p_τ(n)−I.

[0222] FIG. 11 shows an example conversion means 13a for converting y_τ(n) into a ternary vector x{y_τ(n)}. Every component y_τk(n) of y_τ(n) is converted into a one-dimensional ternary vector (i.e., a ternary number)×{y_τk(n)} by the following steps: For k=1, . . . , R, set y_τk(n)=2p_τk(n)−1, and generate a pseudo-random number in accordance with the probability distribution of a random variable v: P (v=1)=p_τk(n) and P (v=−1)=1−p_τk(n), and set x{y_τk(n)} equal to the resultant pseudo-random number. Assemble x{y_τk(n)}, k=1, . . . , R, into a vector x{y_τ(n)}=[×{y_τ1(n)} x{y_τ2(n)} . . . x{y_τR(n)}]′. Note that this vector is a bipolar binary vector, which is a ternary vector.

[0223] FIG. 12 shows an alternative conversion means 13b for converting a representation y_τ(n)=2p_τ(n)−I of a probability distribution into a ternary vector x{y_τ(n)}. Assume that each component y_τk(n) of y_τ(n) is to be converted into a three-dimensional ternary vector. Recall that −1≦y_τk(n)≦¹. If y_τk(n) is very close to 0, the probability p_τk(n) is very close to ½ and contains little information about the label r_τk(n). To eliminate it from further processing, the conversion means converts it into x{y_τk(n)}=[0 0 0]. If y_τk(n) is not very close to 0, we convert it into a 3-component ternary vector x{y_τk(n)} as shown in FIG. 12. The output x{y_τ(n)} of the converter is a 3R-dimensional concatenation of x{y_τk(n)}, k=1, . . . , R. The method of converting a component y_τk(n) of y_τ(n) into a 3-dimensional ternary vector can easily be generalized to a method of converting y_τk(n) into a ternary vector of any dimensionality.

5.6 Processing Units and Supervised/Unsupervised Learning

[0224] An example PU(n) (processing unit on a feature subvector index n) is shown in FIG. 13. The PU comprises expansion means 2, estimation means 54, conversion means 13, adjustment means 9 and storage means 56. A PU has essentially two functions, retrieving an estimate of the label of a feature subvector from the memory (i.e., ECMs or GECMs) and learning a feature subvector and its label that is either provided from outside the PU (in supervised learning) or generated by itself (in unsupervised learning). In performing retrieval, a feature subvector x_τ(n) on the FSI n is first expanded into a general orthogonal expansion {hacek over (x)}_τ(n) by the expansion means 2. {hacek over (x)}_τ(n) is then processed by the estimation means, using the general expansion correlation matrices, C(n) and D(n), from the storage means 56, into a representation of a probability distribution y_τ(n) of the label of x_τ(n). The conversion means converts y_τ(n) into a ternary vector x{y_τ(n)}, which is an output of the PU. If a representation of a probability distribution of x_τ(n) is needed for use outside the PU, y_τ(n) is also output by the PU. The dashed line in the arrow 55 indicates “output as needed.” y_τ(n) and x{y_τ(n)} are the products of retrieval.

[0225] C(n) and D(n) model the synaptic weights in a cortex. The estimation means and conversion means constitute a processing node, which models a group of R neurons in a cortex.

[0226] Recall that a PU in a PAM has a “receptive field” in the exogenous feature vector and a “receptive field” in the measurement vector. These receptive fields can be found by tracing the feedforward connections in the PAM backforward from a feature subvector input to the PU (or the feature subvector index of the PU) to an exogenous feature vector (or the input terminals) of the PAM, and then tracing the transformation, that maps the measurement vector into the exogenous feature vector, backward from the exogenous feature vector to the measurement vector. The components of the measurement vector that can be reached by this backward tracing from a PU to the exogenous feature vector and then to the measurement vector are called the “receptive field” of the PU in the measurement vector. The components of the exogenous feature vector that can be reached by this backward tracing from a PU to the exogenous feature vector are called the “receptive field” of the PU in the exogenous feature vector. The label of a feature vector input to a PU is the label of the corresponding components of the measurement vector in the receptive field of the PU in the measurement vector. The label of the corresponding components of the exogenous feature vector in the receptive field of the PU in the exogenous feature vector is also this label

[0227] If a label r_τ(n) of x_τ(n) from outside the PU is available for learning, supervised learning can be performed by the PU. In the supervised learning mode, the label r_τ(n) is received through a lever represented by a thick solid line with a solid dot in the position 48 by a general expansion correlation matrix (GECM) adjustment means 9, which receives also {hacek over (x)}_τ(n) and uses a method of adjusting ECMs such as those depicted in FIG. 5 and FIG. 6 and assembles the resultant ECMs
C(n)=[C(n(1))C(n(2)) . . . C(n(U))]
D(n)=[D(n(1))D(n(2)) . . . D(n(U))]
These C(n) and D(n) are then stored, after a one-numbering delay (or a unit-time delay) 33, in the storage 56, from which they are sent to the estimation means 54. The one-numbering delay is usually a time delay that is long enough for the estimation means to finish using current C(n) and D(n) in generating and outputting y_τ(n), but short enough for getting the next C(n) and D(n) generated by the adjustment means available for the estimation means to use for processing the next orthogonal expansion or general orthogonal expansion from the expansion means.

[0228] Supervised learning means of the PU comprises adjustment means 9 for adjusting at least one GECM (general expansion correlation matrix) by receiving a GOE (general orthogonal expansion) {hacek over (x)}_τ(n) generated by expansion means 2 and a label r_τ(n) of x_τ(n) provided from outside the PAM and replacing said at least one GECM with a weighted sum of said at least one GECM and a product of said label r_τ(n) and the transpose of said GOE {hacek over (x)}_τ(n).

[0229] If a label r_τ(n) of x_τ(n) from outside the PU is unavailable, unsupervised learning can be performed by the PU. In this case, the lever (shown in position 48 in FIG. 13) should be in the position 50. The feature subvector x_τ(n) is first processed by the expansion means 2, estimation means 54, conversion means 13 as in performing retrieval described above. The resultant ternary vector x{y_τ(n)} is received, through the lever in position 50, and used by the adjustment means 9 as the label r_τ(n) of x_τ(n). The adjustment means 9 uses r_τ(n)=x{y_τ(n)} and {hacek over (x)}_τ(n) to adjust C(n) and D(n) and store the resultant C(n) and D(n) in the storage 56 after a one-numbering delay (or a unit-time delay) 33.

[0230] It is sometimes expensive or impossible to provide labels to feature subvectors y_τ(n) especially for PUs in lower layers of a PAM. If a label r_τ(n) of x_τ(n) is not provided from outside the PU(n), unsupervised learning can be performed by the PU. In this case, the lever in position 48 should be switched to the position 50. The ternary vector x{y_τ(n)} generated by the conversion means in performing retrieval is received and used by the adjustment means 9 as the label r_τ(n) of x_τ(n). As in supervised learning described above, the adjustment means 9 uses r_τ(n) and X_τ(n) to adjust C(n) and D(n) and store the resultant C(n) and D(n) in the storage 56.

[0231] This unsupervised learning method is consistent with the Hebb rule of learning: The synaptic weight between two neurons is increased if the neurons fire at the same time, and the synaptic weight decreases otherwise. Nevertheless, the orthogonal expansion 2 of x_τ(n(u)), the masking matrix M(n(u)) 11a, the conversion 13 and the estimation 54 used in this invention are new.

[0232] If a feature subvector x_τ(n) or a slightly different version of it has not been learned by PU(n), and C(n) {hacek over (x)}_τ(n)=0, then y_τ(n)=0 and p_τ(n)=(½)I, where I=[1 1 . . . 1]′. The conversion means shown in FIG. 11 converts y_τ(n) into a purely random label r_τ(n)=x{y_τ(n)}, with the probability that x{y_τk(n)}=+1 being equal to ½ for k=1, 2, . . . , R. Once this x_τ(n) has been learned and stored in C(n) and D(n), if x_τ(n) is input to PU(n) and to be learned without supervision for the second time, then x{y_τ(n)}=r_τ(n) and one more copy of the pair (x_τ(n), r_τ(n)) is included in C(n) and D(n). Note that the conversion means 13b shown in FIG. 12 converts y_τ(n)=0 into x{y_τk(n)}=0. Hence, this conversion means 13b in FIG. 12 cannot be used for unsupervised learning.

[0233] Assume a feature subvector x_τ(n) is a noise vector and is given a label in unsupervised learning by PU(n). If the noise vector is not repeatedly fed to the PU, as is usually the case, this noise vector and other noise vectors distruct themselves in C(n) and D(n).

[0234] If no learning is to be performed by the PU, the lever represented by a thick solid line with a solid dot is placed in the position 49, through which 0 is sent as the label r_τ(n) of x_τ(n) to the adjustment means, which then keeps C(n) and D(n) unchanged or stores the same C(n) and D(n) in the storage 56 after a one-numbering delay (or a unit time delay).

[0235] There are three types of PU:

• • [0236] 1. Supervised PU. This type of PU is only capable of performing supervised learning. In a PU of this type, the position, 50, in FIG. 13 does not exist. If a label r_τ(n) of x_τ(n) from outside the PU is available and wanted for learning, the lever represented by a thick solid line with a solid dot is placed in the position 48, through which r_τ(n) is sent to the adjustment means 9. If r_τ(n) is not available or wanted for learning, the lever is placed in the position 49, through which 0 is sent to the adjustment means 9. A condition under which learning is not wanted is given below. It is understood that the “lever” is simply a symbol used here to explain which “label” to use by the adjustment means 9.
  • [0237] 2. Unsupervised PU. This type of PU is only capable of performing unsupervised learning. In a PU of this type, the position, 48, in FIG. 13 does not exist. If unsupervised learning is wanted, then the output x{y_τ(n)} is used as a label r_τ(n) of x_τ(n), and the lever represented by a thick solid line with a solid dot is placed in the position 50, through which r_τ(n) is sent as to the adjustment means 9. If no learning is wanted, the lever is placed in the position 49, through which 0 is sent to the adjustment means 9. A condition under which learning is not wanted is given below. It is understood that the “lever” is simply a symbol to explain which “label” to use by the adjustment means 9.
  • [0238] 3. Supervised/Unsupervised PU. This type of PU is capable of performing both supervised and unsupervised learning. In a PU of this type, all the three positions, 48, 49 and 50, in FIG. 13 coexist. If a label r_τ(n) of x_τ(n) from outside the PU is available and wanted for learning, the lever represented by a thick solid line with a solid dot is placed in the position 48. If a label r_τ(n) of x_τ(n) from outside the PU is unavailable, but expected to become available, and y_τ(n) generated by the PU's estimation means in its performing retrieval mode is a zero vector or sufficiently close to it by some criterion, then the lever is placed in the position 49 and no learning is performed. This avoids generating and establishing a randomly selected label in unsupervised learning, which may turn out to be difficult to “unlearn” through supervised learning. If a label r_τ(n) of x_τ(n) from outside the PU is unavailable, y_τ(n) generated by the PU's estimation means in its performing retrieval is neither a zero vector nor sufficiently close to it by some criterion, and unsupervised learning for strengthening learned the knowledge stored in the GECMs (general expansion correlation matrices) or ECMs in the PU is wanted, then the lever is placed in the position 50 and unsupervised learning is performed. A condition under which learning is not wanted is given below.

[0239] A condition under which the lever is placed in the position 49 and no learning is performed is the following: If y_τ(n) generated by a PU's estimation means in retrieving is a bipolar vector or sufficiently close to a bipolar vector by some criterion, which indicates that the input feature subvector x_τ(n) is adequately learned, then the lever is placed in the position 49 and no learning is performed. This avoids “saturating” the expansion correlation matrices with one feature subvector and its label.

[0240] It is understood that the “lever” and “lever positions” are simply a symbol used here to explain which label to use by the adjustment means 9. In application of the present invention, the “lever” and “lever positions” are implemented usually by software to select among supervised learning, unsupervised learning and no learning. Of course, hardware implementation is also possible.

[0241] Note that the learning methods for the three types of PE described above are valid for both batch learning and online learning and jointly or separately, and are suitable for semi-autonomous or autonomous learning. Note also that a PU in a PAM usually has a “receptive field” in the measurement vectors which can be found by tracing the feedforward connections in the PAM backforward from the feature subvectors input to the PU (or the feature subvector index of the PU) to the exogenous feature vectors (or the input terminals) of the PAM, and tracing the transformation, that maps the measurement vectors into exogenous feature vectors, backforward from exogenous feature vectors to measurement vectors. For supervised learning, a feature vector input to a PU is usually assigned the same label as (or is assigned a translation of) the label of the subvector of the measurement vector that appears in the receptive field of the PU. It is understood that there are other ways to assign a label to a feature subvector for supervised learning.

5.7 Learning to Recognize Rotated, Translated or Scaled Patterns

[0242] In this subsection, we describe methods for PUs (processing units) to learn to recognize rotated, translated and scaled patterns. The methods can be modified for PUs to learn to recognize trans-lated and scaled temporal patterns such as speech and music. The methods are valid for both supervised and unsupervised learning. Therefore, labels r_t(n) to be referred to may be provided from outside the PAM in supervised learning or generated by the PUs in unsupervised learning for the three types of PU, supervised PUs, unsupervised PUs and supervised/unsupervised PUs described at the end of the last Subsection.

[0243] It is assumed in this subsection that feature vectors are arrays of ternary pixels. Other types of feature vector must be converted into arrays of ternary pixels for the methods to be described to apply. For example, an image with 8-bit pixels may be converted by using a pseudo-random number generator to generate a bipolar pulse train for each pixel whose average pulse rate (i.e., the rate of +1 pulse) is proportional to 8-bit light intensity of the pixel. Another way to convert an image with 8-bit pixels is to replace an 8-bit pixel with 3 bipolar 2-bit pixels placed at the same location in considering rotation, translation and scaling. After conversion, at any instant of time, the feature vector is an array of ternary pixels.

[0244] Locations of ternary pixels in an array are assumed to be dense relative to the locations of the pixels selected as components of a feature subvector x_t(n) input to a PU. We identify the FSI (feature subvector index) n of a feature subvector with the locations of the pixels in x_t(n). In other words, the components of n are also the numberings of the locations of the pixels included as components of x_t(n).

[0245] Consider a PU with an FSI n shown in FIG. 3 and FIG. 17a. Imagine a thin rubber rectangle with small holes at the locations n of the pixels of the feature subvector with the FSI n. We translate the rectangle in some directions (e.g., 0, 15, 30, 45, . . . , 330, 345 degrees) (FIG. 17b) a number of steps (e.g., 0, 1, 2, . . . ), rotate the rectangle clockwise and counterclockwise a number (e.g., 0, 1, 2, . . . ) of angles (e.g., 0, 5, 10, 15 degrees) at each translation (FIG. 17c), and expand and compress the rubber rectangle uniformly for a number of times (e.g., 0, 1, 2, . . . ) at each translation for some percentages (e.g., 0%, 5%, 10%, . . . ) (FIG'S. 17d and 17e), to obtain other feature subvector indices of the same dimensionality as n. Note that in using the rubber rectangle to determine an FSI, if a hole in the rubber rectangle contains more than one pixel in the image, the one nearest to the center of the hole is included in the FSI.

[0246] Let Ω(n)={w(i), i=1, . . . , |Ω(n)|} be a set of FSIs w(i) identified with such rotations, translations, and scalings of n including n. Ω(n) is called a rotation/translation/scaling (RTS) suite of n, and |Ω(n)| denotes the number of elements in Ω(n). Note that an RTS suite may contain only rotations, or only translations, or only scalings, or a combination thereof. (Notice the digit 0 in the parentheses (e.g., 0, 1, 2, . . . ) in the last paragraph. It indicates a rotation, a translation, or a scaling that is the feature subvector itself.) As a special case of Ω(n), there is only one element in the set Ω(n) that is n itself. In this special case, |Ω(n)|=1.

[0247] Although w(i) is a rotation, translation, or scaling of n, this dependence on n is not indicated in the symbol w(i) for notational simplicity. As n is rotated, translated or scaled into w(i), n(u) as a subvector of n is rotated, translated or scaled into a subvector of w(i). This subvector of w(i) is denoted by n(u,w(i)). The set {n(u,w(i)), i=1, . . . , |Ω(n)|} of such subvectors of w(i), i=1, . . . , |Ω(n)|, is denoted by Ω(n(u)) and called a rotation/translation/scaling (RTS) suite of n(u). Note that |Ω(n(u))|=|Ω(n)|. The set {x_t(n(u,w(i))), i=1, . . . , |Ω(n)|}, which is also denoted by {x_t(n(u,w)), wεΩ(n)}, is called the rotation/translation/scaling (RTS) suite of x_t(n(u)) on Ω(n(u)). In generating and summing orthogonal expansions on an RTS suite Ω(n(u)), elements in the RTS suite of x_t(n(u)) on Ω(n(u)) first go through orthogonal expansion 1. The resultant orthogonal expansions {hacek over (x)}_t(n(u,w(i))) are then added up to form the sum τ_wεΩ(n){hacek over (x)}_t(n(u,w)) on the RTS suite Ω(n(u)) of n(u) (FIG. 18).

[0248] In both the supervised learning and unsupervied learning, the subvectors, x_t(n(u,w)), w E Ω(n), on Ω(n(u)) are assigned the label r_t(n) of x_t(n). ECMs (expansion correlation matrices), C(n(u)), D(n(u)), A (n(u)) and B(n(u)), on Ω(n(u)) are defined by

[0249] $\begin{array}{cc}C\left(n\left(u\right)\right)=\Lambda \sum _{t=1}^T{W}_t\left(n\left(u\right),T\right)I\sum _{\omega \in \Omega \left(n\right)}{\stackrel{⋓}{x}}_t^{\prime }\left(n\left(u,\omega \right)\right)& \left(41\right)\\ D\left(n\left(u\right)\right)=\Lambda \sum _{t=1}^T{W}_t\left(n\left(u\right),T\right)r_t\left(n\right)\sum _{\omega \in \Omega \left(n\right)}{\stackrel{⋓}{x}}_t^{\prime }\left(n\left(u,\omega \right)\right)& \left(42\right)\\ A\left(n\left(u\right)\right)=\Lambda \sum _{t=1}^T{W}_t\left(n\left(u\right),T\right)\left(1+r_t\left(n\right)\right)\sum _{\omega \in \Omega \left(n\right)}{\stackrel{⋓}{x}}_t^{\prime }\left(n\left(u,\omega \right)\right)& \left(43\right)\\ B\left(n\left(u\right)\right)=\Lambda \sum _{t=1}^T{W}_t\left(n\left(u\right),T\right)\left(1-r_t\left(n\right)\right)\sum _{\omega \in \Omega \left(n\right)}{\stackrel{⋓}{x}}_t^{\prime }\left(n\left(u,\omega \right)\right)& \left(44\right)\end{array}$
Definitions of C(n(u)) and D(n(u)) are shown in FIG. 19.