A concave regularization technique for sparse mixture models

Latent variable mixture models are a powerful tool for exploring the structure in large datasets. A common challenge for interpreting such models is a desire to impose sparsity, the natural assumption that each data point only contains few latent features. Since mixture distributions are constrained in their L1 norm, typical sparsity techniques based on L1 regularization become toothless, and concave regularization becomes necessary. Unfortunately concave regularization typically results in EM algorithms that must perform problematic non-concave M-step maximizations. In this work, we introduce a technique for circumventing this difficulty, using the so-called Mountain Pass Theorem to provide easily verifiable conditions under which the M-step is well-behaved despite the lacking concavity. We also develop a correspondence between logarithmic regularization and what we term the pseudo-Dirichlet distribution, a generalization of the ordinary Dirichlet distribution well-suited for inducing sparsity. We demonstrate our approach on a text corpus, inferring a sparse topic mixture model for 2,406 weblogs.