# Multi-view models and adaptive density estimation under low-rank constraints

This talk considers the problem of estimating discrete and continuous probability densities under low-rank constraints. For discrete distributions, we assume that the two-dimensional array to estimate is a rank K probability matrix. In the continuous case, we assume that the density with respect to the Lebesgue measure satisfies a generalized multi-view model, meaning that it is Hölder smooth and can be decomposed as a sum of K components, each of which is a product of one-dimensional functions. We propose estimators that achieve, up to logarithmic factors, the minimax optimal convergence rates under such low-rank constraints. In the discrete case, the proposed estimator is adaptive to the rank K. In the continuos case, the estimator is adaptive to the unknown support as well as to the smoothness, and to the unknown number of separable components K. We propose efficient algorithms for computing the estimators.

This is a joint work with Julien Chhor and Olga Klopp.