Sample Complexity Using Infinite Multiview Models

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.