Latent feature sharing: an adaptive approach to linear decomposition models

Latent feature models are canonical tools for exploratory analysis in classical and modern multivariate statistics. Many high-dimensional data can be approximated using a union of low-dimensional subspaces or factors. The allocation of data points to these latent factors itself typically uncovers key relationships in the input and helps us represent hidden causes explaining the data. A widely adopted view is to model feature allocation with discrete latent variables, where each data point is associated with a binary vector indicating latent features possessed by this data point. In this work we revise some of the issues with existing parametric and Bayesian nonparametric processes for feature allocation modelling and propose a novel framework that can capture wider set of feature allocation distributions. This new framework allows for explicit control over the number of features used to express each point and enables a more flexible set of allocation distributions including feature allocations with different sparsity levels. We use this approach to derive a novel adaptive Factor analysis (aFA), as well as, an adaptive probabilistic principle component analysis (aPPCA) capable of flexible structure discovery and dimensionality reduction in a wide case of scenarios. We derive both standard a Gibbs sampler, as well as, an expectation-maximization inference algorithms for aPPCA and aFA that converge orders of magnitude faster to a point estimate. We demonstrate that aFA can handle richer feature distributions, when compared to widely used sparse FA models and nonparametric FA models. We show that aPPCA and aFA can infer interpretable high level features both when applied on raw MNIST, when applied for interpreting autoencoder features. We also demonstrate an application of the aPPCA to more robust blind source separation for functional magnetic resonance imaging (fMRI).