On deterministic sketching and streaming for sparse recovery and norm estimation

Abstract

We study classic streaming and sparse recovery problems using deterministic linear sketches, including ℓ 1/ℓ 1 and ℓ ∞/ℓ 1 sparse recovery problems, norm estimation, and approximate inner product. We focus on devising a fixed matrix A ∈ ℝ mxn and a deterministic recovery/estimation procedure which work for all possible input vectors simultaneously. We contribute several improved bounds for these problems. - A proof that ℓ ∞/ ℓ 1 sparse recovery and inner product estimation are equivalent, and that incoherent matrices can be used to solve both problems. Our upper bound for the number of measurements is m = O(ε -2 min {log n, (log n/log(1/ε)) 2}). We can also obtain fast sketching and recovery algorithms by making use of the Fast Johnson-Lindenstrauss transform. Both our running times and number of measurements improve upon previous work. We can also obtain better error guarantees than previous work in terms of a smaller tail of the input vector. - A new lower bound for the number of linear measurements required to solve ℓ 1/ℓ 1 sparse recovery. We show Ω(k/ε 2+k log(n/k)/ε) measurements are required to recover an x′ with ∥x - x′∥ 1 ≤ (1 + ε)∥x tail(k)∥ 1, where x tail(k) is x projected onto all but its largest k coordinates in magnitude. - A tight bound of m = Θ(ε -2log(ε 2n)) on the number of measurements required to solve deterministic norm estimation, i.e., to recover ∥x∥ 2 ± ε∥x∥ 1. For all the problems we study, tight bounds are already known for the randomized complexity from previous work, except in the case of ℓ 1/ℓ 1 sparse recovery, where a nearly tight bound is known. Our work thus aims to study the deterministic complexities of these problems. © 2012 Springer-Verlag.