Small variance asymptotics and bayesian nonparametrics for dictionary learning
Résumé
Bayesian nonparametric (BNP) is an appealing framework to infer the complexity of a model along with the parameters. To this aim, sampling or variational methods are often used for inference. However, these methods come with numerical disadvantages for large-scale data. An alternative approach is to relax the probabilistic model into a non-probabilistic formulation which yields a scalable algorithm. One limitation of BNP approaches can be the cost of Monte-Carlo sampling for inference. Small-variance asymptotic (SVA) approaches paves the way to much cheaper though approximate methods for inference by taking benefit from a fruitful interaction between Bayesian models and optimization algorithms. In brief, SVA lets the variance of the noise (or residual error) distribution tend to zero in the optimization problem corresponding to a MAP estimator with finite noise variance for instance. We propose such an SVA analysis of a BNP dictionary learning (DL) approach that automatically adapts the size of the dictionary or the subspace dimension in an efficient way. Numerical experiments illustrate the efficiency of the proposed method.
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