NIPS Proceedingsβ

Resolution Limits of Sparse Coding in High Dimensions

Part of: Advances in Neural Information Processing Systems 21 (NIPS 2008)

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Recent research suggests that neural systems employ sparse coding. However, there is limited theoretical understanding of fundamental resolution limits in such sparse coding. This paper considers a general sparse estimation problem of detecting the sparsity pattern of a $k$-sparse vector in $\R^n$ from $m$ random noisy measurements. Our main results provide necessary and sufficient conditions on the problem dimensions, $m$, $n$ and $k$, and the signal-to-noise ratio (SNR) for asymptotically-reliable detection. We show a necessary condition for perfect recovery at any given SNR for all algorithms, regardless of complexity, is $m = \Omega(k\log(n-k))$ measurements. This is considerably stronger than all previous necessary conditions. We also show that the scaling of $\Omega(k\log(n-k))$ measurements is sufficient for a trivial ``maximum correlation'' estimator to succeed. Hence this scaling is optimal and does not require lasso, matching pursuit, or more sophisticated methods, and the optimal scaling can thus be biologically plausible.