Nonuniform Sparse Recovery with Gaussian Matrices

Compressive sensing predicts that sufficiently sparse vectors can be recovered from highly incomplete information. Efficient recovery methods such as 1-minimization find the sparsest solution to certain systems of equations. Random matrices have become a popular choice for the measurement matrix. Indeed, near-optimal uniform recovery results have been shown for such matrices. In this note we focus on nonuniform recovery using Gaussian random matrices and 1-minimization. We provide a condition on the number of samples in terms of the sparsity and the signal length which guarantees that a fixed sparse signal can be recovered with a random draw of the matrix using 1minimization. The constant 2 in the condition is optimal, and the proof is rather short compared to a similar result due to Donoho and Tanner.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Ulas Ayaz, Holger Rauhut
• Keywords: CORR 2010, Education, Efficient Recovery Methods, Gaussian Random Matrices, Random Matrices