############################################################
Robust Subspace Mixture Models using :math:`t`-distributions
############################################################

Motivation(s)
=============

Mixture models have been used to approximate a manifold using a relatively small
number of localized subspaces (e.g. PCA, factor analysis).  One outstanding
problem is robustness to outliers.  To address this, recent research have
explored mixture of t-distributions and PPCA.

Proposed Solution(s)
====================

The authors assert that for manifold learning, finding an exact local PCA
solution is not necessary, as long as the main axes of the densities are aligned
with the manifold.  The Gaussian prior can be replaced by the t-distribution,
which in turn is a convolution of a Gaussian and the gamma prior.

Evaluation(s)
=============

MTS beat MPPCA in the synthetic noisy data and digit recognition experiments.
MTS was able to learn the manifold structure despite a higher level of noise.
The proposed EM algorithm assumes that the convolution of two t-distributions is
a t-distribution, which is a good approximation only for large :math:`\nu` or
for small :math:`\sigma^2`.

Future Direction(s)
===================

- Instead of an approximation, derive the true EM algorithm and compare its
  performance.

Question(s)
===========

- While I like the usage of indicator variables to remove a summation, I'm not
  sure if (9) is valid probabilistically.  Then again, if one doesn't have to
  define such a distribution, then maybe it's alright?

Analysis
========

Hidden indicator variables can be used to reformulate summations for easier
learning.

The idea is great, but the reasoning for not going through with a full
derivation is weak.  The interesting thing is that :doc:`this overview </blog/2016/01/08/robust-mixture-modelling-using-the-t-distribution>`
is clearer than this paper itself.

.. rubric:: References

.. bibliography:: refs.bib
   :all: