On Deterministic Sketching and Streaming for Sparse Recovery and Norm Estimation

We study classic streaming and sparse recovery problems using deterministic linear sketches, including `1/`1 and `∞/`1 sparse recovery problems, norm estimation, and approximate inner product. We focus on devising a fixed matrix A ∈ Rm×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 `∞/`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{logn, (logn/ log(1/ε))}). 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 `1/`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)‖xtail(k)‖1, where xtail(k) is x projected onto all but its largest k coordinates in magnitude. – A tight bound of m = Θ(ε−2 log(εn)) 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 `1/`1 sparse recovery, where a nearly tight bound is known. Our work thus aims to study the deterministic complexities of these problems.