Ramsey theory reveals the conditions when sparse coding on subsampled data is unique

Details Ramsey theory reveals the conditions when sparse coding on subsampled data is unique Ramsey theory reveals the conditions when sparse coding on subsampled data is unique

2011-06-18

arxiv.org/abs/1106.3616v2

Biology

Quantitative Biology

Neurons and Cognition

8 pages, 2 figures

Scientific paper

Sparse coding or dictionary learning has been widely used to reveal the sparse underlying structure of many kinds of sensory data. A related advance in signal processing is compressed sensing, a theory explaining how sparse data can be subsampled below the Nyquist-Shannon limit and then efficiently recovered from these subsamples. Here we study whether the conditions for recovery in compressed sensing are sufficient for dictionary learning to discover the original sparse causes of subsampled data. Using combinatorial Ramsey theory, we completely characterize when the learned dictionary matrix and sparse representations of subsampled data are unique (up to the natural equivalences of permutation and scaling). Surprisingly, uniqueness is shown to hold without any assumptions on the learned dictionaries or inferred sparse codes. Our result has implications for the learning of overcomplete dictionaries from subsampled data and has potential applications in data analysis and neuroscience. For instance, it identifies sparse coding as a possible learning mechanism for establishing lossless communication through severe bottlenecks, which might explain how different brain regions communicate through axonal fiber projections.

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