# Derandomizing compressed sensing with combinatorial design

## TL;DR

This paper shows how to reduce randomness in compressed sensing measurement designs by using structured combinatorial objects, achieving reliable sparse signal recovery with fewer random measurements.

## Contribution

It introduces derandomization techniques using orthogonal arrays and mutually unbiased bases to improve measurement design in compressed sensing.

## Key findings

- Uniform s-sparse reconstruction guarantees with $C s \,\log(n)$ measurements.
- Measurements chosen from structured combinatorial designs.
- Imitation of random vectors using highly structured families.

## Abstract

Compressed sensing is the art of reconstructing structured $n$-dimensional vectors from substantially fewer measurements than naively anticipated. A plethora of analytic reconstruction guarantees support this credo. The strongest among them are based on deep results from large-dimensional probability theory that require a considerable amount of randomness in the measurement design. Here, we demonstrate that derandomization techniques allow for considerably reducing the amount of randomness that is required for such proof strategies. More, precisely we establish uniform s-sparse reconstruction guarantees for $C s \log (n)$ measurements that are chosen independently from strength-four orthogonal arrays and maximal sets of mutually unbiased bases, respectively. These are highly structured families of $\tilde{C} n^2$ vectors that imitate signed Bernoulli and standard Gaussian vectors in a (partially) derandomized fashion.