Wasserstein metric-driven Bayesian inversion with applications to signal processing

TL;DR

This paper introduces a Wasserstein metric-driven Bayesian inversion framework that improves signal processing by handling complex noise, avoiding local extrema, and capturing amplitude and phase differences, leading to better inverse uncertainty quantification.

Contribution

The paper presents a novel exponential likelihood function based on the Wasserstein metric for Bayesian inversion, enhancing robustness and accuracy in signal processing applications.

Findings

Better convergence to true posteriors with MCMC sampling.

Effective handling of complex noise structures.

Improved signal inversion capturing amplitude and phase differences.

Abstract

We present a Bayesian framework based on a new exponential likelihood function driven by the quadratic Wasserstien metric. Compared to conventional Bayesian models based on Gaussian likelihood functions driven by the least-squares norm ($L_2$ norm), the new framework features several advantages. First, the new framework does not rely on the likelihood of the measurement noise and hence can treat complicated noise structures such as combined additive and multiplicative noise. Secondly, unlike the normal likelihood function, the Wasserstein-based exponential likelihood function does not usually generate multiple local extrema. As a result, the new framework features better convergence to correct posteriors when a Markov Chain Monte Carlo sampling algorithm is employed. Thirdly, in the particular case of signal processing problems, while a normal likelihood function measures only the amplitude differences between the observed and simulated signals, the new likelihood function can capture both the amplitude and the phase differences. We apply the new framework to a class of signal processing problems, that is, the inverse uncertainty quantification of waveforms, and demonstrate its advantages compared to Bayesian models with normal likelihood functions.