On scalable inference and learning in spike-and-slab sparse coding

Sparse coding is a widely applied latent variable analysis technique. The standard formulation of sparse coding assumes Laplace as a prior distribution for modeling the activations of latent components. In this work we study sparse coding with spike-and-slab distribution as a prior for latent activity. A spike-and-slab distribution has its probability mass distributed across a ’spike’ at zero and a ’slab’ spreading over a continuous range. For its capacity to induce exact zeros with a higher likelihood, a spike-and-slab prior distribution constitutes a more accurate model of sparse coding. The distribution as a prior also allows for the sparseness of latent activity to be directly inferred from observed data, which essentially makes spike-and-slab sparse coding more flexible and self-adaptive to a wide range of data distributions. By modeling the slab with a Gaussian distribution, we furthermore show that in contrast to the standard approach to sparse coding, we can indeed derive closed-form analytical expressions for exact inference and learning in linear spike-and-slab sparse coding. However, as the posterior landscape of a spike-and-slab prior turns out to be highly multi-modal with a prohibitive exploration cost, in addition to the exact method, we also develop subspace and Gibbs sampling based approximate inference techniques for scalable applications of the linear model. We contrast our approximation methods with variational approximation for scalable posterior inference in linear spike-and-slab sparse coding. We further combine the Gaussian spike-and-slab prior with a nonlinear generative model, which assumes a point-wise maximum combination rule for the generation of observed data. We analyze the model as a precise encoder of low-level features such as edges and their occlusions in visual data. We again combine subspace selection with Gibbs sampling to overcome the analytical intractability of performing exact inference in the model. We numerically analyze our methods on both synthetic and real data for their verification and comparison with other approaches. We assess the linear spike-and-slab approach on source separation and image denoising benchmarks. In most experiments we obtain competitive or state-of-the-art results, while we find that spike-and-slab sparse coding overall outperforms other comparable approaches. By extracting thousands of latent components from a large amount of training data we further demonstrate that our subspace Gibbs sampler is among the most scalable posterior inference methods for a linear sparse coding approach. For the nonlinear model we experiment with artificial and real images to demonstrate that the components learned by the model lie closer to the ground-truth and are easily interpretable as the underlying generative causes of the input. We find that in comparison to standard sparse coding, the nonlinear spike-and-slab approach can compressively encode images using naturally sparse and discernible compositions of latent components. We also demonstrate that the components inferred by the model from natural image patches are statistically more consistent with respect to their structure and distribution to the response patterns of simple cells in the primary visual cortex of the brain. This work thereby contributes novel methods for sophisticated inference and learning in spike-and-slab sparse coding, while it also empirically showcases their functional efficacy through a variety of applications.