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26
Sparse coding with an overcomplete basis set: a strategy employed by V1
 Vision Research
, 1997
"... The spatial receptive fields of simple cells in mammalian striate cortex have been reasonably well described physiologically and can be characterized as being localized, oriented, and ban@ass, comparable with the basis functions of wavelet transforms. Previously, we have shown that these receptive f ..."
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Cited by 591 (7 self)
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The spatial receptive fields of simple cells in mammalian striate cortex have been reasonably well described physiologically and can be characterized as being localized, oriented, and ban@ass, comparable with the basis functions of wavelet transforms. Previously, we have shown that these receptive field properties may be accounted for in terms of a strategy for producing a sparse distribution of output activity in response to natural images. Here, in addition to describing this work in a more expansive fashion, we examine the neurobiological implications of sparse coding. Of particular interest is the case when the code is overcompletei.e., when the number of code elements is greater than the effective dimensionality of the input space. Because the basis functions are nonorthogonal and not linearly independent of each other, sparsifying the code will recruit only those basis functions necessary for representing a given input, and so the inputoutput function will deviate from being purely linear. These deviations from linearity provide a potential explanation for the weak forms of nonlinearity observed in the response properties of cortical simple cells, and they further make predictions about the expected interactions among units in
Parallel Networks that Learn to Pronounce English Text
 COMPLEX SYSTEMS
, 1987
"... This paper describes NETtalk, a class of massivelyparallel network systems that learn to convert English text to speech. The memory representations for pronunciations are learned by practice and are shared among many processing units. The performance of NETtalk has some similarities with observed h ..."
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Cited by 456 (5 self)
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This paper describes NETtalk, a class of massivelyparallel network systems that learn to convert English text to speech. The memory representations for pronunciations are learned by practice and are shared among many processing units. The performance of NETtalk has some similarities with observed human performance. (i) The learning follows a power law. (;i) The more words the network learns, the better it is at generalizing and correctly pronouncing new words, (iii) The performance of the network degrades very slowly as connections in the network are damaged: no single link or processing unit is essential. (iv) Relearning after damage is much faster than learning during the original training. (v) Distributed or spaced practice is more effective for longterm retention than massed practice. Network models can be constructed that have the same performance and learning characteristics on a particular task, but differ completely at the levels of synaptic strengths and singleunit responses. However, hierarchical clustering techniques applied to NETtalk reveal that these different networks have similar internal representations of lettertosound correspondences within groups of processing units. This suggests that invariant internal representations may be found in assemblies of neurons intermediate in size between highly localized and completely distributed representations.
A Theory of Networks for Approximation and Learning
 Laboratory, Massachusetts Institute of Technology
, 1989
"... Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, t ..."
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Cited by 194 (24 self)
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Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nonlinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. Wedevelop a theoretical framework for approximation based on regularization techniques that leads to a class of threelayer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the wellknown Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods suchasParzen windows and potential functions and to several neural network algorithms, suchas Kanerva's associative memory,backpropagation and Kohonen's topology preserving map. They also haveaninteresting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.
Sparse Codes and Spikes
 PROBABILISTIC MODELS OF THE BRAIN: PERCEPTION AND NEURAL FUNCTION
, 2001
"... ..."
Principles of image representation in visual cortex,” in The Visual Neurosciences
, 2003
"... The visual cortex is responsible for most of our conscious perception of the visual world, yet we remain largely ignorant of the principles underlying its function despite progress on many fronts of neuroscience. ..."
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Cited by 19 (2 self)
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The visual cortex is responsible for most of our conscious perception of the visual world, yet we remain largely ignorant of the principles underlying its function despite progress on many fronts of neuroscience.
Complexity Issues in Discrete Hopfield Networks
, 1994
"... We survey some aspects of the computational complexity theory of discretetime and discretestate Hopfield networks. The emphasis is on topics that are not adequately covered by the existing survey literature, most significantly: 1. the known upper and lower bounds for the convergence times of Hopfi ..."
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Cited by 18 (4 self)
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We survey some aspects of the computational complexity theory of discretetime and discretestate Hopfield networks. The emphasis is on topics that are not adequately covered by the existing survey literature, most significantly: 1. the known upper and lower bounds for the convergence times of Hopfield nets (here we consider mainly worstcase results); 2. the power of Hopfield nets as general computing devices (as opposed to their applications to associative memory and optimization); 3. the complexity of the synthesis ("learning") and analysis problems related to Hopfield nets as associative memories. Draft chapter for the forthcoming book The Computational and Learning Complexity of Neural Networks: Advanced Topics (ed. Ian Parberry).
The Problem of Sparse Image Coding
 Journal of Mathematical Imaging and Vision
, 2001
"... Linear expansions of images nd many applications in image processing and computer vision. ..."
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Cited by 16 (0 self)
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Linear expansions of images nd many applications in image processing and computer vision.
Analogy Retrieval and Processing With Distributed Vector Representations
, 1998
"... : Holographic Reduced Representations (HRRs) are a method for encoding nested relational structures in fixed width vector representations. HRRs encode relational structures as vector representations in such a way that the superficial similarity of the vectors reflects both superficial and structural ..."
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Cited by 10 (2 self)
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: Holographic Reduced Representations (HRRs) are a method for encoding nested relational structures in fixed width vector representations. HRRs encode relational structures as vector representations in such a way that the superficial similarity of the vectors reflects both superficial and structural similarity of the relational structures. HRRs also support a number of operations that could be very useful in psychological models of human analogy processing: fast estimation of superficial and structural similarity via a vector dotproduct; finding corresponding objects in two structures; and chunking of vector representations. Although similarity assessment and discovery of corresponding objects both theoretically take exponential time to perform fully and accurately, with HRRs one can obtain approximate solutions in constant time. The accuracy of these operations with HRRs mirrors patterns of human performance on analog retrieval and processing tasks. Keywords: neural networks, distributed representations, binding, analogy, analog retrieval, structure, chunking, systematicity 1