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 × N Hadamard matrix contains a K-sparse vector in the kernel, unless the number of subsampled rows is Ω(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.