Hosted by: Huy Huynh and Yao Li
of combinatorial structures has contributed to a renaissance in research
in finite Markov chains in the last two decades.
Applications are wide-ranging from sophisticated card shuffles,
deciphering simple substitution ciphers (of
prison inmates in the California state prison), estimating the volume of
a high-dimensional convex body,
and to understanding the speed of Gibbs sampling heuristics in
statistical physics. More recent applications include rigorous estimates
on J.M. Pollard's (1979) classical Rho and Kangaroo algorithms for the
discrete logarithm problem in finite cyclic groups.
The lecture will be a brief (mostly self-contained) introduction to the
Markov Chain Monte Carlo (MCMC) methodology and applications, and will
include some open problems.
Factorization (NMF) has attracted much attention during the past
decade as a dimension reduction method in machine learning and data
analysis. NMF provides a lower rank approximation of a nonnegative
high dimensional matrix by factors whose elements are also
nonnegative. Numerous success stories were reported in application
areas including text clustering, computer vision, and cancer class
this talk, we present novel algorithms for NMF and NTF (nonnegative
tensor factorization) based on the alternating non-negativity
constrained least squares (ANLS) framework. Our new algorithm for NMF
is built upon the block principal pivoting method for the
non-negativity constrained least squares problem that overcomes some
limitations of the classical active set method. The proposed NMF
algorithm can naturally be extended to obtain highly efficient NTF
algorithm for PARAFAC (PARAllel FACtor) model. Our algorithms
inherit the convergence theory of the ANLS framework and can easily
be extended to other NMF formulations such as sparse NMF and NTF with
L1 norm constraints. Comparisons of algorithms using various data
sets show that the proposed new algorithms outperform existing ones
in computational speed as well as the solution quality.
is a joint work with Jingu Kim and Krishnakumar Balabusramanian.
This is part 1 of a two part talk. The second part will continue next week.
which relates the A-polynomial and the colored Jones polynomial of a
knot in the 3-sphere. Then I will verify it for the trefoil and the
figure 8 knots (due to Garoufalidis) and torus knots (due to Hikami) by
over k. We discuss Hasse principle for existence of rational points
on homogeneous spaces under connected linear algebraic groups.
We illustrate how a positive answer to Hasse principle leads for instance to the result:
every quadratic form in nine variables over K has a nontrivial zero.