Gaussian mixture model decomposition of multivariate signals

TL;DR

This paper introduces a greedy variational approach for decomposing multivariate signals into Gaussian mixtures, capable of handling various data types and Gaussian shapes without prior component number assumptions.

Contribution

It presents a novel method that simultaneously supports multivariate signals, general Gaussian shapes, and does not require pre-specifying the number of components.

Findings

Method effectively decomposes signals into Gaussian mixtures.

Provides an upper bound on mode-to-mean distance in Gaussian mixtures.

Outperforms baseline expectation maximization in evaluations.

Abstract

We propose a greedy variational method for decomposing a non-negative multivariate signal as a weighted sum of Gaussians, which, borrowing the terminology from statistics, we refer to as a Gaussian mixture model. Notably, our method has the following features: (1) It accepts multivariate signals, i.e. sampled multivariate functions, histograms, time series, images, etc. as input. (2) The method can handle general (i.e. ellipsoidal) Gaussians. (3) No prior assumption on the number of mixture components is needed. To the best of our knowledge, no previous method for Gaussian mixture model decomposition simultaneously enjoys all these features. We also prove an upper bound, which cannot be improved by a global constant, for the distance from any mode of a Gaussian mixture model to the set of corresponding means. For mixtures of spherical Gaussians with common variance $\sigma^2$, the bound takes the simple form $\sqrt{n}\sigma$. We evaluate our method on one- and two-dimensional signals. Finally, we discuss the relation between clustering and signal decomposition, and compare our method to the baseline expectation maximization algorithm.