All-pole centroids in the Wasserstein metric with applications to clustering of spectral densities

TL;DR

This paper introduces a novel method for computing spectral barycenters within the all-pole model class using the Wasserstein-2 metric, resulting in compact and interpretable spectral centroids suitable for tasks like clustering and classification.

Contribution

The paper presents a gradient descent approach for all-pole spectral barycenters in the Wasserstein metric, offering a parametric, low-dimensional alternative to non-parametric barycenters.

Findings

The method produces low-dimensional spectral centroids.

Application to phoneme classification demonstrates effectiveness.

The approach allows quantification of sub-optimality.

Abstract

In this work, we propose a method for computing centroids, or barycenters, in the spectral Wasserstein-2 metric for sets of power spectral densities, where the barycenters are restricted to belong to the set of all-pole spectra with a certain model order. This may be interpreted as finding an autoregressive representative for sets of second-order stationary Gaussian processes. While Wasserstein, or optimal transport, barycenters have been successfully used earlier in problems of spectral estimation and clustering, the resulting barycenters are non-parametric and the complexity of representing and storing them depends on, e.g., the choice of discretization grid. In contrast, the herein proposed method yields compact, low-dimensional, and interpretable spectral centroids that can be used in downstream tasks. Computing the all-pole centroids corresponds to solving a non-convex optimization problem in the model parameters, and we present a gradient descent scheme for addressing this. Although convergence to a globally optimal point cannot be guaranteed, the sub-optimality of the obtained centroids can be quantified. The proposed method is illustrated on a problem of phoneme classification.