TY - GEN

T1 - For-all sparse recovery in near-optimal time

AU - Gilbert, Anna C.

AU - Li, Yi

AU - Porat, Ely

AU - Strauss, Martin J.

PY - 2014

Y1 - 2014

N2 - An approximate sparse recovery system in ℓ1 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 x̂ = R(Φx), which must satisfy ∥ x̂-x∥1 ≤ (1 + ε)∥x - x k∥1. We consider the "for all" model, in which a single matrix Φ is used for all signals x. The best existing sublinear algorithm by Porat and Strauss (SODA'12) uses O(ε-3 klog(N/k)) measurements and runs in time O(k1-α Nα) for any constant α>0. In this paper, 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(k1+β poly(logN,1/ε)), with a modest restriction that k ≤ N 1-α and ε ≤ (logk/logN)γ, for any constants α, β,γ > 0. With no restrictions on ε, we have an approximation recovery system with m = O(k/εlog(N/k)((logN/logk) γ +1/ε)) measurements. The algorithmic innovation is a novel encoding procedure that is reminiscent of network coding and that reflects the structure of the hashing stages.

AB - An approximate sparse recovery system in ℓ1 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 x̂ = R(Φx), which must satisfy ∥ x̂-x∥1 ≤ (1 + ε)∥x - x k∥1. We consider the "for all" model, in which a single matrix Φ is used for all signals x. The best existing sublinear algorithm by Porat and Strauss (SODA'12) uses O(ε-3 klog(N/k)) measurements and runs in time O(k1-α Nα) for any constant α>0. In this paper, 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(k1+β poly(logN,1/ε)), with a modest restriction that k ≤ N 1-α and ε ≤ (logk/logN)γ, for any constants α, β,γ > 0. With no restrictions on ε, we have an approximation recovery system with m = O(k/εlog(N/k)((logN/logk) γ +1/ε)) measurements. The algorithmic innovation is a novel encoding procedure that is reminiscent of network coding and that reflects the structure of the hashing stages.

UR - http://www.scopus.com/inward/record.url?scp=84904160633&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-43948-7_45

DO - 10.1007/978-3-662-43948-7_45

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AN - SCOPUS:84904160633

SN - 9783662439470

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 538

EP - 550

BT - Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings

PB - Springer Verlag

T2 - 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014

Y2 - 8 July 2014 through 11 July 2014

ER -