  Multivariate generalized Gamma convolutions are distributions defined by a
convolutional semi-parametric structure. Their flexible dependence structures,
the marginal possibilities and their useful convolutional expression make them
appealing to the practitioner. However, fitting such distributions when the
dimension gets high is a challenge. We propose stochastic estimation procedures
based on the approximation of a Laguerre integrated square error via (shifted)
cumulants approximation, evaluated on random projections of the dataset.
Through the analysis of our loss via tools from Grassmannian cubatures, sparse
optimization on measures and Wasserstein gradient flows, we show the
convergence of the stochastic gradient descent to a proper estimator of the
high dimensional distribution. We propose several examples on both low and
high-dimensional settings.
