An Improved Lower Bound for Sparse Reconstruction from Subsampled Hadamard Matrices

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 $N \times N$ Hadamard 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.Comment: Improved exposition and added an autho

Isometric sketching of any set via the Restricted Isometry Property

In this paper we show that for the purposes of dimensionality reduction certain class of structured random matrices behave similarly to random Gaussian matrices. This class includes several matrices for which matrix-vector multiply can be computed in log-linear time, providing efficient dimensionality reduction of general sets. In particular, we show that using such matrices any set from high dimensions can be embedded into lower dimensions with near optimal distortion. We obtain our results by connecting dimensionality reduction of any set to dimensionality reduction of sparse vectors via a chaining argument.Comment: 17 page