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We propose a greedy variational method for decomposing a non-negativemultivariate signal as a weighted sum of Gaussians, which, borrowing theterminology 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. asinput. (2) The method can handle general (i.e. ellipsoidal) Gaussians. (3) Noprior assumption on the number of mixture components is needed. To the best ofour knowledge, no previous method for Gaussian mixture model decompositionsimultaneously enjoys all these features. We also prove an upper bound, whichcannot be improved by a global constant, for the distance from any mode of aGaussian mixture model to the set of corresponding means. For mixtures ofspherical Gaussians with common variance $\sigma^2$, the bound takes the simpleform $\sqrt{n}\sigma$. We evaluate our method on one- and two-dimensionalsignals. Finally, we discuss the relation between clustering and signaldecomposition, and compare our method to the baseline expectation maximizationalgorithm.

Gaussian mixture model decomposition of multivariate signals