Research
My research interests are as follow: machine learning, statistical pattern recognition, probability and stochastic processes,
statistical learning theory, applied probability, digital image processing, digital signal processing, information theory, and source coding.
The following is the abstract of my
Ph.D. thesis, titled "On the Convergence and Applications of Mean Shift Type Algorithms":
Abstract
Mean shift (MS) and subspace constrained mean shift (SCMS) algorithms are non-parametric,
iterative methods to ?nd a representation of a high dimensional data set on a principal curve
or surface embedded in a high dimensional space. The representation of high dimensional
data on a principal curve or surface, the class of mean shift type algorithms and their
properties, and applications of these algorithms are the main focus of this dissertation.
Although MS and SCMS algorithms have been used in many applications, a rigorous
study of their convergence is still missing. This dissertation aims to ?ll some of the gaps
between theory and practice by investigating some convergence properties of these algorithms.
In particular, we propose a suf?cient condition for a kernel density estimate with a
Gaussian kernel to have isolated stationary points to guarantee the convergence of the MS
algorithm. We also show that the SCMS algorithm inherits some of the important convergence properties of
the MS algorithm. In particular, the monotonicity and convergence of
the density estimate values along the sequence of output
values of the algorithm are shown. We also show that the distance between consecutive points of the output sequence
converges to zero, as does the projection of the gradient vector onto the subspace spanned by
the D-d eigenvectors corresponding to the D-d largest eigenvalues of the local inverse
covariance matrix.
Furthermore, three new variations of the SCMS algorithm are proposed and the running
times and performance of the resulting algorithms are compared with original SCMS algorithm. We also propose an adaptive version of the SCMS algorithm to consider the effect
of new incoming samples without running the algorithm on the whole data set.
As well, we develop some new potential applications of the MS and SCMS algorithm.
These applications involve ?nding straight lines in digital images; pre-processing data be fore applying locally
linear embedding (LLE) and ISOMAP for dimensionality reduction; noisy source vector quantization where the clean data need to be estimated before the
quanization step; improving the performance of kernel regression in certain situations; and
skeletonization of digitally stored handwritten characters.