For-All Sparse Recovery in Near-Optimal Time

Abstract

An approximate sparse recovery system in ℓ 1 norm consists of parameters k , ϵ, N ; an m -by- N measurement Φ; and a recovery algorithm R . Given a vector, x , the system approximates x by xˆ = R (Φ x ), which must satisfy ‖ xˆ- x ‖ 1 ≤ (1+ϵ)‖ x - x k ‖ 1 . We consider the “for all” model, in which a single matrix Φ, possibly “constructed” non-explicitly using the probabilistic method, is used for all signals x . The best existing sublinear algorithm by Porat and Strauss [2012] uses O (ϵ −3 k log ( N / k )) measurements and runs in time O ( k 1 − α N α ) for any constant α > 0. In this article, we improve the number of measurements to O (ϵ − 2 k log ( N / k )), matching the best existing upper bound (attained by super-linear algorithms), and the runtime to O ( k 1+β poly(log N ,1/ϵ)), with a modest restriction that k ⩽ N 1 − α and ϵ ⩽ (log k /log N ) γ for any constants α, β, γ > 0. When k ⩽ log c N for some c > 0, the runtime is reduced to O ( k poly( N ,1/ϵ)). With no restrictions on ϵ, we have an approximation recovery system with m = O ( k /ϵlog ( N / k )((log N /log k ) γ + 1/ϵ)) measurements. The overall architecture of this algorithm is similar to that of Porat and Strauss [2012] in that we repeatedly use a weak recovery system (with varying parameters) to obtain a top-level recovery algorithm. The weak recovery system consists of a two-layer hashing procedure (or with two unbalanced expanders for a deterministic algorithm). The algorithmic innovation is a novel encoding procedure that is reminiscent of network coding and that reflects the structure of the hashing stages. The idea is to encode the signal position index i by associating it with a unique message m i , which will be encoded to a longer message m ′ i (in contrast to Porat and Strauss [2012] in which the encoding is simply the identity). Portions of the message m ′ i correspond to repetitions of the hashing, and we use a regular expander graph to encode the linkages among these portions. The decoding or recovery algorithm consists of recovering the portions of the longer messages m ′ i and then decoding to the original messages m i , all the while ensuring that corruptions can be detected and/or corrected. The recovery algorithm is similar to list recovery introduced in Indyk et al. [2010] and used in Gilbert et al. [2013]. In our algorithm, the messages { m i } are independent of the hashing, which enables us to obtain a better result.