Multivariate Exponential Analysis from the Minimal Number of Samples

TL;DR

This paper introduces a novel approach to multivariate exponential analysis that reduces the required samples to the theoretical minimum by combining 1D exponential methods with linear systems, improving efficiency.

Contribution

It presents a new method that achieves the minimal sample count for multivariate exponential analysis by integrating 1D techniques with linear algebra.

Findings

Sample count reduced to (d+1)n, the theoretical minimum.

Method effectively combines 1D exponential analysis with linear systems.

Demonstrates improved efficiency over existing approaches.

Abstract

The problem of multivariate exponential analysis or sparse interpolation has received a lot of attention, especially with respect to the number of samples required to solve it unambiguously. In this paper we show how to bring the number of samples down to the absolute minimum of $(d+1)n$ where $d$ is the dimension of the problem and $n$ is the number of exponential terms. To this end we present a fundamentally different approach for the multivariate problem statement. We combine a one-dimensional exponential analysis method such as ESPRIT, MUSIC, the matrix pencil or any Prony-like method, with some linear systems of equations because the multivariate exponents are inner products and thus linear expressions in the parameters.