On algebraic properties of low rank approximations of Prony systems

TL;DR

This paper investigates the algebraic structure of low-rank approximations in Prony systems, providing bounds and geometric insights into the stability and uniqueness of reconstructing spike train signals from noisy moments.

Contribution

It derives lower bounds for moment differences in low-rank approximations and characterizes the geometry of non-generic cases where fewer nodes can match moments.

Findings

Lower bounds for moments difference between signals with different node counts.

Complete description of the geometry for the case of one node difference and specific moments.

References for the case where moments match up to a certain order, extending previous work.

Abstract

We consider the reconstruction of spike train signals of the form $$F(x) = \sum_{i=1}^d a_i \delta(x-x_i),$$ from their moments measurements $m_k(F)=\int x^k F(x) dx = \sum_{i=1}^d a_ix^k$. When some of the nodes $x_i$ near collide the inversion becomes unstable. Given noisy moments measurements, a typical consequence is that reconstruction algorithms estimate the signal $F$ with a signal having fewer nodes, $\tilde{F}$. We derive lower bounds for the moments difference between a signal $F$ with $d$ nodes and a signal $\tilde{F}$ with strictly less nodes, $l$. Next we consider the geometry of the non generic case of $d$ nodes signals $F$, for which there exists an $l<d$ nodes signal $\tilde{F}$, with moments \begin{align*} m_0(\tilde{F})=m_{0}(F),\ldots,m_{p}(\tilde{F})=m_{p}(F),&& p>2l-1 . \end{align*} We give a complete description for the case of a general $d$, $l=1$ and $p=2$. We give a reference for the case $p=2l-1$ which can be inferred from earlier work.