# Geometry of error amplification in solving Prony system with near-colliding nodes

## TL;DR

This paper investigates how errors are amplified when reconstructing spike-train signals with near-colliding nodes using Prony's method, revealing algebraic geometric structures that influence reconstruction accuracy.

## Contribution

It introduces the concept of Prony varieties to describe error amplification geometry and provides bounds on reconstruction errors near node collisions.

## Key findings

- Error amplification is governed by algebraic varieties called Prony varieties.
- Bounds on worst-case reconstruction errors are established.
- Geometry of Prony varieties can be used to improve reconstruction accuracy.

## Abstract

We consider a reconstruction problem for ``spike-train'' signals $F$ of an a priori known form $F(x)=\sum_{j=1}^{d}a_{j}\delta\left(x-x_{j}\right),$ from their moments $m_k(F)=\int x^kF(x)dx.$ We assume that the moments $m_k(F)$, $k=0,1,\ldots,2d-1$, are known with an absolute error not exceeding $\epsilon > 0$. This problem is essentially equivalent to solving the Prony system $\sum_{j=1}^d a_jx_j^k=m_k(F), \ k=0,1,\ldots,2d-1.$ We study the ``geometry of error amplification'' in reconstruction of $F$ from $m_k(F),$ in situations where the nodes $x_1,\ldots,x_d$ near-collide, i.e. form a cluster of size $h \ll 1$. We show that in this case, error amplification is governed by certain algebraic varieties in the parameter space of signals $F$, which we call the ``Prony varieties''. Based on this we produce lower and upper bounds, of the same order, on the worst case reconstruction error. In addition we derive separate lower and upper bounds on the reconstruction of the amplitudes and the nodes. Finally we discuss how to use the geometry of the Prony varieties to improve the reconstruction accuracy given additional a priori information.