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arXiv · 1903.12135

An Improved Lower Bound for Sparse Reconstruction from Subsampled Walsh Matrices

Jarosław Błasiok·Patrick Lopatto·Kyle Luh·Jake Marcinek·Shravas Rao
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Abstract

We give a short argument that yields a new lower bound on the number of subsampled rows from a bounded, orthonormal matrix necessary to form a matrix with the restricted isometry property. We show that a matrix formed by uniformly subsampling rows of an $N \times N$ Walsh matrix contains a $K$-sparse vector in the kernel, unless the number of subsampled rows is $\Omega(K \log K \log (N/K))$ -- our lower bound applies whenever $\min(K, N/K) > \log^C N$. Containing a sparse vector in the kernel precludes not only the restricted isometry property, but more generally the application of those matrices for uniform sparse recovery.

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Jarosław Błasiok, Patrick Lopatto, Kyle Luh, Jake Marcinek, Shravas Rao. 2019-03-28. An Improved Lower Bound for Sparse Reconstruction from Subsampled Walsh Matrices. https://arxiv.org/abs/1903.12135

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