Generalized multi-view model: Adaptive density estimation under low-rank constraints

Westudy the problem of bivariate discrete or continuous probability density estimation under low-rank constraints. For discrete distributions, we assume that the two-dimensional array to estimate is a low-rank 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 β-H¨older and can be decomposed as a sum of K components, each of which is a product of one-dimensional functions. In both settings, 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 continuous case, our estimator converges with the L1 rate min((K/n)β/(2β+1),n−β/(2β+2)) up to logarithmic factors, and it is adaptive to the un known support as well as to the smoothness β and to the unknown number of separable components K. We present efficient algorithms to compute our estimators.