Compressed sensing of data with a known distribution

Abstract

Abstract. Compressed sensing is a technique for recovering an unknown sparse signal from a small number of linear measurements. When the measurement matrix is random, the number of measurements required for perfect recovery exhibits a phase transition: there is a threshold on the number of measurements after which the probability of exact recovery quickly goes from very small to very large. In this work we are able to reduce this threshold by incorporating statistical information about the data we wish to recover. Our algorithm works by minimizing a suitably weighted 1 -norm, where the weights are chosen so that the expected statistical dimension of the corresponding descent cone is minimized. We also provide new discrete-geometry-based Monte Carlo algorithms for computing intrinsic volumes of such descent cones, allowing us to bound the failure probability of our methods.

Distribution-Aware Block-Sparse Recovery via Convex Optimization

“…Therefore, Proposition 1 generalizes the results of [20] to the block-sparse case. However, our approach to reach this generalized result is different from and somewhat simpler than [20].…”

“…(Prior work) The special case k b " 1 (x is sparse instead of block-sparse) reduces (7) to the weighted 1 minimization which is studied in [20]. Therefore, Proposition 1 generalizes the results of [20] to the block-sparse case. However, our approach to reach this generalized result is different from and somewhat simpler than [20].…”

“…In presence of a block distribution, we obtain the optimal weights in P η 1,2,w . This result can be considered as an extension of [20] to the block-sparse (and joint-sparse 2 ) setting. However, the derivation of the optimal weights in this case is non-trivial and rather challenging.…”

“…Their approach is based on calculating the internal and external angles of a weighted cross polytope. With a different approach, [20] has investigated a discrete measure for Bayesian information (a special case of Model 1 with k " 1). They relate the weights of weighted 1 minimization to the discrete probability distribution by minimizing the expected intrinsic volumes of a weighted cone.…”

“…However, the derivation of the optimal weights in this case is non-trivial and rather challenging. Further, our derivation approach is different from [20]. 2) Optimally exploiting multiple estimates.…”