Sketching for Large-Scale Learning of Mixture Models

Learning parameters from voluminous data can be prohibitive in terms of memory and computational requirements. Furthermore, new challenges arise from modern database architectures, such as the requirements for learning methods to be amenable to streaming, parallel and distributed computing. In this context, an increasingly popular approach is to first compress the database into a representation called a linear sketch, that satisfies all the mentioned requirements, then learn the desired information using only this sketch, which can be significantly faster than using the full data if the sketch is small. In this thesis, we introduce a generic methodology to fit a mixture of probability distributions on the data, using only a sketch of the database. The sketch is defined by combining two notions from the reproducing kernel literature, namely kernel mean embedding and Random Features expansions. It is seen to correspond to linear measurements of the underlying probability distribution of the data, and the estimation problem is thus analyzed under the lens of Compressive Sensing (CS in which a (traditionally finite-dimensional) signal is randomly measured and recovered. We extend CS results to our infinite-dimensional framework and give generic conditions for successful estimation. Our analysis is based on the construction of random sketching operators such that a Restricted Isometry Property (RIP) condition holds in the Banach space of finite signed measures with high probability. In a second part we introduce a flexible heuristic greedy algorithm to estimate mixture models from a sketch. We apply it on synthetic and real data on three problems: the estimation of centroids from a sketch, for which it is seen to be significantly faster than k-means, Gaussian Mixture Model estimation, for which it is more efficient than Expectation-Maximization, and the estimation of mixtures of multivariate stable distributions, for which, to our knowledge, it is the only algorithm capable of performing such a task.