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 ℓ1/ℓ1 and ℓ ∞ /ℓ1 sparse recovery problems, norm estimation, and approximate inner product. We focus on devising a fixed matrix A ∈ ℝ m×n 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 ∞ /ℓ1 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 {logn, (logn / 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 ℓ1/ℓ1 sparse recovery. We show Ω(k/ε 2 + klog(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 2 n)) 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 ℓ1/ℓ1 sparse recovery, where a nearly tight bound is known. Our work thus aims to study the deterministic complexities of these problems.