There were several talks and posters in Aussois involving mixtures, incl. one by Romane Giard on a proposal recently posted on arXiv. The mixture is made of multivariate Gaussian components with unknown but diagonal covariance matrices. The number s of components is unknown as well. The estimation approach is based on a form of LASSO that relies on a point process representation of the mixture parameters, an approach long defended by Peter Green (as summarised in his introductory chapter for our Handbook of Mixture Analysis. Which avoids unnecessary confusions about identifiability and label switching.
The procedure of estimation is thus (B-)LASSO, meaning a data fitting (or fidelity) term comparing a kernel density estimate (seen as a Gaussian convolution) to the smoothed predicted density. Plus a sparsity term, namely a total variation norm of the point process measure. The theoretical result requires a fixed minimum on the variances and a minimum distance between the component parameter vectors. Things get a bit more complicated (for me), given that the point process measure gets replaced by a weighted, approximate, version where each component is attributed a positive weight depending on the variance parameters uk of the Gaussians, a product of (uk²+τ²)-1/4 terms (with τ a regularising parameter). Since the paper is fully focussed on theoretical properties, it does not expand on the practical derivation of the estimator, acknowledging the numerical resolution of the minimisation problem is yet unknown, with a substitute “closer to a realistic algorithmic framework” (p20).
