Approximation of Nonnegative Systems by Finite Impulse Response Convolutions

TL;DR

This paper addresses the nonparametric approximation of scalar nonnegative systems using finite impulse response convolutions, employing structured nonnegative matrix factorization and I-divergence minimization, with theoretical guarantees and an iterative algorithm.

Contribution

It introduces a structured nonnegative matrix factorization approach for system approximation, with conditions for uniqueness and an algorithm for I-divergence minimization.

Findings

Existence and uniqueness conditions for the approximation problem.

An alternating minimization algorithm for I-divergence minimization.

Asymptotic analysis and large sample properties of the proposed method.

Abstract

We pose the deterministic, nonparametric, approximation problem for scalar nonnegative input/output systems via finite impulse response convolutions, based on repeated observations of input/output signal pairs. The problem is converted into a nonnegative matrix factorization with special structure for which we use Csisz\'ar's I-divergence as the criterion of optimality. Conditions are given, on the input/output data, that guarantee the existence and uniqueness of the minimum. We propose a standard algorithm of the alternating minimization type for I-divergence minimization, and study its asymptotic behavior. We also provide a statistical version of the minimization problem and give its large sample properties.