Signed Cumulative Distribution Transform for Parameter Estimation of 1-D Signals

TL;DR

This paper introduces the signed cumulative distribution transform (SCDT), a novel signal representation based on optimal transport, enabling efficient and globally optimal parameter estimation for 1-D signals through simple linear least squares.

Contribution

It extends the cumulative distribution transform (CDT) to signed signals, allowing Wasserstein distance minimization via linear least squares for arbitrary signal classes.

Findings

SCDT enables global minimization in parameter estimation.

The method outperforms traditional $L_p$ minimization techniques.

Linear least squares suffice for Wasserstein-type distance minimization.

Abstract

We describe a method for signal parameter estimation using the signed cumulative distribution transform (SCDT), a recently introduced signal representation tool based on optimal transport theory. The method builds upon signal estimation using the cumulative distribution transform (CDT) originally introduced for positive distributions. Specifically, we show that Wasserstein-type distance minimization can be performed simply using linear least squares techniques in SCDT space for arbitrary signal classes, thus providing a global minimizer for the estimation problem even when the underlying signal is a nonlinear function of the unknown parameters. Comparisons to current signal estimation methods using $L_p$ minimization shows the advantage of the method.