TY - GEN

T1 - Sublinear time, measurement-optimal, sparse recovery for all

AU - Porat, Ely

AU - Strauss, Martin J.

PY - 2012

Y1 - 2012

N2 - An approximate sparse recovery system in ℓ1 norm makes a small number of measurements of a noisy vector with at most k large entries and recovers those heavy hitters approximately. Formally, it consists of parameters N, k, ε, an m-by-N measurement matrix, Φ, and a decoding algorithm, D. Given a vector, x, where xk denotes the optimal k-term approximation to x, the system approximates x by x̂ = D(Φx), which must satisfy ∥x̂ - x∥1 ≤ (1 + ε) ∥x - x k∥1. Among the goals in designing such systems are minimizing the number m of measurements and the runtime of the decoding algorithm, V. We consider the "forall" model, in which a single matrix Φ, possibly "constructed" non-explicitly using the probabilistic method, is used for all signals x. Many previous papers have provided algorithms for this problem. But all such algorithms that use the optimal number m = O(k log(N/k)) of measurements require superlinear time Ω(N log(N/k)). In this paper, we give the first algorithm for this problem that uses the optimum number of measurements (up to constant factors) and runs in sublinear time o(N) when k is sufficiently less than N. Specifically, for any positive integer ℓ, our approach uses time O(ℓ5 ε-3 k(N/k) 1/ℓ) and uses m = O(ℓ8 ε-3 k log(N/k)) measurements, with access to a data structure requiring space and preprocessing time O(ℓN k0.2/ε).

AB - An approximate sparse recovery system in ℓ1 norm makes a small number of measurements of a noisy vector with at most k large entries and recovers those heavy hitters approximately. Formally, it consists of parameters N, k, ε, an m-by-N measurement matrix, Φ, and a decoding algorithm, D. Given a vector, x, where xk denotes the optimal k-term approximation to x, the system approximates x by x̂ = D(Φx), which must satisfy ∥x̂ - x∥1 ≤ (1 + ε) ∥x - x k∥1. Among the goals in designing such systems are minimizing the number m of measurements and the runtime of the decoding algorithm, V. We consider the "forall" model, in which a single matrix Φ, possibly "constructed" non-explicitly using the probabilistic method, is used for all signals x. Many previous papers have provided algorithms for this problem. But all such algorithms that use the optimal number m = O(k log(N/k)) of measurements require superlinear time Ω(N log(N/k)). In this paper, we give the first algorithm for this problem that uses the optimum number of measurements (up to constant factors) and runs in sublinear time o(N) when k is sufficiently less than N. Specifically, for any positive integer ℓ, our approach uses time O(ℓ5 ε-3 k(N/k) 1/ℓ) and uses m = O(ℓ8 ε-3 k log(N/k)) measurements, with access to a data structure requiring space and preprocessing time O(ℓN k0.2/ε).

UR - http://www.scopus.com/inward/record.url?scp=84860176453&partnerID=8YFLogxK

U2 - 10.1137/1.9781611973099.96

DO - 10.1137/1.9781611973099.96

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AN - SCOPUS:84860176453

SN - 9781611972108

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1215

EP - 1227

BT - Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012

PB - Association for Computing Machinery

T2 - 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012

Y2 - 17 January 2012 through 19 January 2012

ER -