UTCS Colloquium/FACULTY CANDIDATE: Lek-Heng Lim - FACULTY CANDIDATE Stanford University Ten Ways to Decompose a Tensor ACES 2.302 Thursday April 5 2007 at 11:00 a.m.

Contact Name: 
Jenna Whitney
Date: 
Apr 5, 2007 11:00am - 12:00pm

There is a signup schedule for this event.
<

br>Speaker Name: Lek-Heng Lim - FACULTY CANDIDATE

Speaker Affiliati

on: Stanford University

Date: April 5 2007

Start Time: 11

:00a.m.

Location: ACES 2.302

Host: Inderjit Dhillon

Talk Title: Ten Ways to Decompose a Tensor

Talk Abstract:

In
scientific and statistical computing one often reduces the
problem at
hand -- be it a problem involving differential
equations or nonlinear

optimization or parameter estimation
-- to a simpler problem (or a sequ

ence of these) that requires
nothing more than linear algebra. In fact
the solution of linear
systems alone accounts for more than 70% of all
supercomputing
time in the world.

However with the use of incre

asingly sophisticated sensor
devices experimental methodologies and m

athematical
models we now see a new generation of problems in
scie

ntific and statistical computing that cannot be reduced
to standard pro

blems in numerical linear algebra. It is thus
pertinent to enlarge the

arsenal of computational tools available
at our disposal. Among the var

ious plausible extensions
to numerical linear algebra one will find th

e capability of
dealing with multilinearity to be among the most natura

l
desirable and powerful -- if we could do tensor computations
(n

umerical multilinear algebra) as effectively as matrix
computations (nu

merical linear algebra) then we would be
able to address many of the n

ew problems arising in modern
scientific and statistical computing.

It is not coincidental that the decompositional approach
to matrix

computations has been named one of the Top 10
Algorithms of the 20th Ce

ntury. If numerical linear algebra
is the foundation of scientific comp

uting then matrix decompositions
may be considered to be the foundatio

n of numerical linear
algebra. So the development of numericalmultiline

ar algebra
ought to begin with a few basic tensor decompositions.
In this talk we will present ten decompositions of tensors
-- the sp

eaker has developed four of these and contributed
to studies of the oth

er six. Our list will include the tensorial
generalizations of LU/LDU

decomposition QR/complete
orthogonal factorization eigenvalue decompo

sition (EVD)
singular value decomposition (SVD) nonnegative matrix factorization (NMF) Kronecker product decomposition
and more. We wi

ll discuss the similarities and differences
of these decompositions wit

h their matrix counterparts
as well as the various challenges in numer

ical multilinear
algebra of which these tensor decompositions form a <

br>cornerstone.

To every tensor decomposition there is an associate

d approximation
problem. We will see how these may be applied to multil

inear
statistical models that generalize vector space models
indep

endent component analysis graphical models/Bayesian
networks and mode

l reduction. We will illustrate these with selected
applications in bioi

nformatics computer vision signal processing
spectroscopy and sensor
location.