Optimal Demixing of Nonparametric Densities

Motivated by applications in statistics and machine learning, we consider a problem of unmixing convex combinations of nonparametric densities. Suppose we observe n groups of samples, where the ith group consists of Ni independent samples from a d-variate density fi(x)=sumk=1K πi(k)gk(x). Here, each gk(x) is a nonparametric density, and each πi is a K-dimensional mixed membership vector. We aim to estimate g1(x), ldots,gK(x). This problem generalizes topic modeling from discrete to continuous variables and finds its applications in LLMs with word embeddings. In this paper, we propose an estimator for the above problem, which modifies the classical kernel density estimator by assigning group-specific weights that are computed by topic modeling on histogram vectors and de-biased by U-statistics. For any β>0, assuming that each gk(x) is in the Nikol'ski class with a smooth parameter β, we show that the sum of integrated squared errors of the constructed estimators has a convergence rate that depends on n, K, d, and the per-group sample size N. We also provide a matching lower bound, which suggests that our estimator is rate-optimal.