Sparse coding

The sparse code is a kind of neural code in which each item is encoded by the strong activation of a relatively small set of neurons. For each item to be encoded, this is a different subset of all available neurons.

As a consequence, sparseness may be focused on temporal sparseness ("a relatively small number of time periods are active") or on the sparseness in an activated population of neurons. In this latter case, this may be defined in one time period as the number of activated neurons relative to the total number of neurons in the population. This seems to be a hallmark of neural computations since compared to traditional computers, information is massively distributed across neurons. A major result in neural coding from Olshausen et al. is that sparse coding of natural images produces wavelet-like oriented filters that resemble the receptive fields of simple cells in the visual cortex.

Overview

Given a potentially large set of input patterns, sparse coding algorithms attempt to automatically find a small number of representative patterns which, when combined in the right proportions, reproduce the original input patterns. The sparse coding for the input then consists of those representative patterns. For example, the very large set of English sentences can be encoded by a small number of symbols (i.e. letters, numbers, punctuation, and spaces) combined in a particular order for a particular sentence, and so a sparse coding for English would be those symbols.

More formally, given a k-dimensional set of real-numbered input vectors ${\vec {\xi }}\in \mathbb {R} ^{k}$, the goal of sparse coding is to determine n k-dimensional basis vectors ${\vec {b_{1}}},\ldots ,{\vec {b_{n}}}\in \mathbb {R} ^{k}$ along with a sparse n-dimensional vector of weights or coefficients ${\vec {s}}\in \mathbb {R} ^{n}$ for each input vector, so that a linear combination of the basis vectors with proportions given by the coefficients results in a close approximation to the input vector: ${\vec {\xi }}\approx \sum _{j=1}^{n}s_{j}{\vec {b}}_{j}$.

A trivial solution is for each basis vector to be equal to one of the input vectors. The coefficients for a given input vector would then be zero everywhere except one for the basis vector which equals the input vector. However, we generally want the number of basis vectors n to be much less than the number of input vectors.

See also

• Rate coding
• Independent-spike coding
• Correlation coding
• Population coding
• Grandmother cell

Bibliography

• Földiák P, Endres D, Sparse coding, Scholarpedia, 3(1):2984, 2008.
• Dayan P & Abbott LF. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. Cambridge, Massachusetts: The MIT Press; 2001. ISBN 0-262-04199-5
• Rieke F, Warland D, de Ruyter van Steveninck R, Bialek W. Spikes: Exploring the Neural Code. Cambridge, Massachusetts: The MIT Press; 1999. ISBN 0-262-68108-0
• B. A. Olshausen and D. J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381(6583):607–9, jun 1996.