  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.
