# Sparsity Order Estimation from a Single Compressed Observation Vector

## TL;DR

This paper introduces a novel method for estimating the sparsity order from a single compressed observation vector by designing specific measurement matrices that enable direct inference without sparse recovery.

## Contribution

It demonstrates that measurement matrices based on Khatri-Rao and Vandermonde structures can reliably infer sparsity order from a single snapshot, with new designs for low-coherence Vandermonde matrices.

## Key findings

- Effective rank of measurement matrix indicates sparsity order in multiple snapshots.
- Khatri-Rao and Vandermonde matrices enable single snapshot sparsity estimation.
- New Vandermonde matrix design achieves low coherence, improving measurement quality.

## Abstract

We investigate the problem of estimating the unknown degree of sparsity from compressive measurements without the need to carry out a sparse recovery step. While the sparsity order can be directly inferred from the effective rank of the observation matrix in the multiple snapshot case, this appears to be impossible in the more challenging single snapshot case. We show that specially designed measurement matrices allow to rearrange the measurement vector into a matrix such that its effective rank coincides with the effective sparsity order. In fact, we prove that matrices which are composed of a Khatri-Rao product of smaller matrices generate measurements that allow to infer the sparsity order. Moreover, if some samples are used more than once, one of the matrices needs to be Vandermonde. These structural constraints reduce the degrees of freedom in choosing the measurement matrix which may incur in a degradation in the achievable coherence. We thus also address suitable choices of the measurement matrices. In particular, we analyze Khatri-Rao and Vandermonde matrices in terms of their coherence and provide a new design for Vandermonde matrices that achieves a low coherence.