Talk by Dr. Masahiro Nakano, NTT Communication Science Laboratories

Talk by Dr. Masahiro Nakano, NTT Communication Science Laboratories

Title: Support of Infinitely Exchangeable Rectangular Partitioning

Abstract: Bayesian nonparametric machine learning is generally to analyze infinite
data via infinite-dimensional probabilistic models. As a telling
example, we focus on the stochastic block models (SBMs) for relational
data analysis, and tackle an open question whether or not we can
construct any probabilistic models on arbitrary rectangular partitioning
of a matrix with infinite size. The problem can be regarded as whether
or not there exists any stochastic rectangular partitioning which
satisfy the exchangeability and projectivity conditions, led by the
Aldous-Hoover representation theorem [1979, 1981] and Kolmogorov's
extension theorem [1955], respectively. We introduce two reasonable
definitions of rectangular partitioning of a matrix. For the first
definition, we completely answer the question: We show
that, if any probabilistic models for rectangular partitioning satisfy
exchangeability and projectivity, then positive probabilities
are concentrated on a small subset of arbitrary rectangular
partitioning. For the second definition, the problem is still open. We
also clarify the relationship between our result and conventional
Bayesian nonparametric SBMs, including the infinite relational model
[AAAI, 2006], the Mondrian process [NIPS, 2009], and the rectangular
tiling process [ICML, 2014].