Multivariate Exponential Analysis from the Minimal Number of Samples
This provides a theoretically optimal sample complexity for multivariate exponential analysis, a fundamental problem in signal processing and sparse interpolation.
The paper reduces the number of samples required for multivariate exponential analysis to the absolute minimum of (d+1)n, where d is dimension and n is the number of exponential terms, by combining a one-dimensional method with linear systems.
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.