Signal recovery from a few linear measurements of its high-order spectra

TL;DR

This paper demonstrates that a high-order spectrum of a signal can be uniquely recovered using only N+1 linear measurements, significantly reducing the data needed compared to the full spectrum.

Contribution

It introduces a method to recover signals from minimal linear measurements of their high-order spectra, reducing computational and storage demands.

Findings

Signal can be recovered from N+1 linear measurements of its high-order spectrum.

The approach is validated through numerical experiments.

The proof uses algebraic geometry techniques.

Abstract

The $q$-th order spectrum is a polynomial of degree $q$ in the entries of a signal $x\in\mathbb{C}^N$, which is invariant under circular shifts of the signal. For $q\geq 3$, this polynomial determines the signal uniquely, up to a circular shift, and is called a high-order spectrum. The high-order spectra, and in particular the bispectrum ($q=3$) and the trispectrum ($q=4$), play a prominent role in various statistical signal processing and imaging applications, such as phase retrieval and single-particle reconstruction. However, the dimension of the $q$-th order spectrum is $N^{q-1}$, far exceeding the dimension of $x$, leading to increased computational load and storage requirements. In this work, we show that it is unnecessary to store and process the full high-order spectra: a signal can be characterized uniquely, up to symmetries, from only $N+1$ linear measurements of its high-order spectra. The proof relies on tools from algebraic geometry and is corroborated by numerical experiments.