  Latent Variable Models (LVMs) are a large family of machine learning models
providing a principled and effective way to extract underlying patterns,
structure and knowledge from observed data. Due to the dramatic growth of
volume and complexity of data, several new challenges have emerged and cannot
be effectively addressed by existing LVMs: (1) How to capture long-tail
patterns that carry crucial information when the popularity of patterns is
distributed in a power-law fashion? (2) How to reduce model complexity and
computational cost without compromising the modeling power of LVMs? (3) How to
improve the interpretability and reduce the redundancy of discovered patterns?
To addresses the three challenges discussed above, we develop a novel
regularization technique for LVMs, which controls the geometry of the latent
space during learning to enable the learned latent components of LVMs to be
diverse in the sense that they are favored to be mutually different from each
other, to accomplish long-tail coverage, low redundancy, and better
interpretability. We propose a mutual angular regularizer (MAR) to encourage
the components in LVMs to have larger mutual angles. The MAR is non-convex and
non-smooth, entailing great challenges for optimization. To cope with this
issue, we derive a smooth lower bound of the MAR and optimize the lower bound
instead. We show that the monotonicity of the lower bound is closely aligned
with the MAR to qualify the lower bound as a desirable surrogate of the MAR.
Using neural network (NN) as an instance, we analyze how the MAR affects the
generalization performance of NN. On two popular latent variable models ---
restricted Boltzmann machine and distance metric learning, we demonstrate that
MAR can effectively capture long-tail patterns, reduce model complexity without
sacrificing expressivity and improve interpretability.
