  Ubiquitous linear Gaussian exploratory tools such as principle component
analysis (PCA) and factor analysis (FA) remain widely used as tools for:
exploratory analysis, pre-processing, data visualization and related tasks.
However, due to their rigid assumptions including crowding of high dimensional
data, they have been replaced in many settings by more flexible and still
interpretable latent feature models. The Feature allocation is usually modelled
using discrete latent variables assumed to follow either parametric
Beta-Bernoulli distribution or Bayesian nonparametric prior. In this work we
propose a simple and tractable parametric feature allocation model which can
address key limitations of current latent feature decomposition techniques. The
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. This approach is
used 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 Gibbs sampler, as well as, an expectation-maximization
inference algorithms that converge orders of magnitude faster to a reasonable
point estimate solution. The utility of the proposed aPPCA model is
demonstrated for standard PCA tasks such as feature learning, data
visualization and data whitening. We show that aPPCA and aFA can infer
interpretable high level features both when applied on raw MNIST and 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).
