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

TL;DR

This paper introduces adaptive estimators for low-rank density estimation in both discrete and continuous settings, achieving near-optimal convergence rates and handling unknown parameters efficiently.

Contribution

It proposes novel estimators that are adaptive to rank, smoothness, and support, with algorithms that are computationally efficient.

Findings

Achieves minimax optimal convergence rates up to logarithmic factors.

Estimator is adaptive to unknown rank, smoothness, and support.

Provides efficient algorithms for practical implementation.

Abstract

We study 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 $\beta$-H{\"o}lder 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 $L_1$ rate $\min((K/n)^{\beta/(2\beta+1)}, n^{-\beta/(2\beta+2)})$ up to logarithmic factors, and it is adaptive to the unknown support as well as to the smoothness $\beta$ and to the unknown number of separable components $K$. We present efficient algorithms for computing our estimators.