  Mixture models are widely used to fit complex and multimodal datasets. In
this paper we study mixtures with high dimensional sparse latent parameter
vectors and consider the problem of support recovery of those vectors. While
parameter learning in mixture models is well-studied, the sparsity constraint
remains relatively unexplored. Sparsity of parameter vectors is a natural
constraint in variety of settings, and support recovery is a major step towards
parameter estimation. We provide efficient algorithms for support recovery that
have a logarithmic sample complexity dependence on the dimensionality of the
latent space. Our algorithms are quite general, namely they are applicable to
1) mixtures of many different canonical distributions including Uniform,
Poisson, Laplace, Gaussians, etc. 2) Mixtures of linear regressions and linear
classifiers with Gaussian covariates under different assumptions on the unknown
parameters. In most of these settings, our results are the first guarantees on
the problem while in the rest, our results provide improvements on existing
works.
