Sample Complexity Using Infinite Multiview Models

TL;DR

This paper introduces the NL-spectrum as a new measure of probability density complexity, enabling finite sample bounds and convergence rates for a broad class of densities beyond multiview models.

Contribution

It proposes the NL-spectrum to characterize any pdf's complexity, extending convergence analysis beyond restrictive multiview assumptions.

Findings

Finite sample bounds depend on the NL-spectrum.

Dimension-independent convergence rates are derived.

The NL-spectrum allows for fast convergence under certain conditions.

Abstract

Recent works have demonstrated that the convergence rate of a nonparametric density estimator can be greatly improved by using a low-rank estimator when the target density is a convex combination of separable probability densities with Lipschitz continuous marginals, i.e. a multiview model. However, this assumption is very restrictive and it is not clear to what degree these findings can be extended to general pdfs. This work answers this question by introducing a new way of characterizing a pdf's complexity, the non-negative Lipschitz spectrum (NL-spectrum), which, unlike smoothness properties, can be used to characterize virtually any pdf. Finite sample bounds are presented that are dependent on the target density's NL-spectrum. From this dimension-independent rates of convergence are derived that characterize when an NL-spectrum allows for a fast rate of convergence.