The Weizmann Institute of Science
Faculty of Mathematics and Computer Science
Vision and Robotics Seminar
Amnon Shashua
School of Engineering and Computer Science
Hebrew University, Jerusalem
will speak on
Non-Negative Tensor Factorization, EM and L2--EM,
with applications to statistics and computer vision
Abstract:
We derive two families of algorithms for finding a non-negative $n$-dimensional tensor factorization (NTF)which includes
the non-negative matrix factorization (NMF) as a particular case when $n=2$. We present two areas in statistics and computer
vision where NTF plays a crucial role. In statistics, conditionally independent statements over $l$-tuples of variables subsets
correspond to rank-1 tensor slices of the $n$-dimensional joint probability distribution and form rank-k tensor slices in the
presence of a $k$-graded latent variable. These low rank constraints form the essence of the required statistical reasoning and
an NTF is the underlying algorithm for enforcing the rank constraints. In computer vision, we introduce NTF as a tool for capturing
a local part decomposition from an image training set. The part decomposition is conventionally performed using an NMF, however,
we argue that an NMF is not the right tool for the task and instead an NTF is a more appropriate engine for extracting local part
description from an image class.
The first algorithm family is a direct approach based on a positive preserving gradient descent update rule on the desired
low-rank tensor elements. The direct approach is a tensorial version of the NMF update rule proposed by {\it Lee-Seung-nature}.
The second algorithm family is based on repeated rank-1 approximations. The Expectation-Maximization (EM) engine surfaces at
this point when relative entropy error model is used and a new algorithm, referred to as $L_2$-EM, emerges when $L_2$ error
model is used. The $L_2$-EM has certain interesting advantages over EM --- the main one being {\it sparsity}, i.e., the solutions
produced by $L_2$-EM are sparser than those produced by EM.
Joint work with Tamir Hazan.
The lecture will take place in the
Lecture Hall, Room 1, Ziskind Building
on Thursday, May 26, 2005
noon - 13:00