Signal Recovery From Product of Two Vandermonde Matrices
Dzevdan Kapetanovic
TL;DR
The paper investigates recovering $s$-sparse signals that are structured in Vandermonde/ Fourier bases under both phase-aware and phase-less measurements. It develops deterministic Vandermonde-based measurement designs and polynomial-time recovery algorithms, leveraging rational- and Laurent-polynomial representations to establish identifiability and efficient reconstruction. Key results include exact recovery from $m=2s$ Vandermonde measurements when $n\ge 2s$ (R1/R2), and phase-less guarantees with sample counts as low as $4s-1$ to $8s-3$ (R4–R5, with R3 extending the framework); in certain regimes an extra random measurement resolves remaining ambiguities, achieving $O(\mathrm{poly}(s))$ time. Collectively, these contributions yield deterministic, scalable recovery guarantees for structured sparse signals in both phase-aware and phase-less settings, with practical implications for frequency-domain sensing and robust phase retrieval.
Abstract
In this work, we present some new results for compressed sensing and phase retrieval. For compressed sensing, it is shown that if the unknown $n$-dimensional vector can be expressed as a linear combination of $s$ unknown Vandermonde vectors (with Fourier vectors as a special case) and the measurement matrix is a Vandermonde matrix, exact recovery of the vector with $2s$ measurements and $O(\mathrm{poly}(s))$ complexity is possible when $n \geq 2s$. Based on this result, a new class of measurement matrices is presented from which it is possible to recover $s$-sparse $n$-dimensional vectors for $n \geq 2s$ with as few as $2s$ measurements and with a recovery algorithm of $O(\mathrm{poly}(s))$ complexity. In the second part of the work, these results are extended to the challenging problem of phase retrieval. The most significant discovery in this direction is that if the unknown $n$-dimensional vector is composed of $s$ frequencies with at least one being non-harmonic, $n \geq 4s - 1$ and we take at least $8s-3$ Fourier measurements, there are, remarkably, only two possible vectors producing the observed measurement values and they are easily obtainable from each other. The two vectors can be found by an algorithm with only $O(\mathrm{poly}(s))$ complexity. An immediate application of the new result is construction of a measurement matrix from which it is possible to recover almost all $s$-sparse $n$-dimensional signals (up to a global phase) from $O(s)$ magnitude-only measurements and $O(\mathrm{poly}(s))$ recovery complexity when $n \geq 4s - 1$.