  In this paper, we describe a method for estimating the joint probability
density from data samples by assuming that the underlying distribution can be
decomposed as a mixture of product densities with few mixture components. Prior
works have used such a decomposition to estimate the joint density from
lower-dimensional marginals, which can be estimated more reliably with the same
number of samples. We combine two key ideas: dictionaries to represent 1-D
densities, and random projections to estimate the joint distribution from 1-D
marginals, explored separately in prior work. Our algorithm benefits from
improved sample complexity over the previous dictionary-based approach by using
1-D marginals for reconstruction. We evaluate the performance of our method on
estimating synthetic probability densities and compare it with the previous
dictionary-based approach and Gaussian Mixture Models (GMMs). Our algorithm
outperforms these other approaches in all the experimental settings.
