Thesis records
https://feeds.library.caltech.edu/people/Chandrasekaran-V/thesis.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 15 Apr 2024 13:27:14 +0000Convex Optimization Methods for Graphs and Statistical Modeling
https://resolver.caltech.edu/CaltechAUTHORS:20121008-130644748
Authors: {'items': [{'id': 'Chandrasekaran-V', 'name': {'family': 'Chandrasekaran', 'given': 'Venkat'}}]}
Year: 2012
An outstanding challenge in many problems throughout science and engineering is
to succinctly characterize the relationships among a large number of interacting entities.
Models based on graphs form one major thrust in this thesis, as graphs often
provide a concise representation of the interactions among a large set of variables. A
second major emphasis of this thesis are classes of structured models that satisfy certain
algebraic constraints. The common theme underlying these approaches is the development
of computational methods based on convex optimization, which are in turn useful
in a broad array of problems in signal processing and machine learning. The specific
contributions are as follows:
We propose a convex optimization method for decomposing the sum of a sparse
matrix and a low-rank matrix into the individual components. Based on new
rank-sparsity uncertainty principles, we give conditions under which the convex
program exactly recovers the underlying components.
Building on the previous point, we describe a convex optimization approach to
latent variable Gaussian graphical model selection. We provide theoretical guarantees
of the statistical consistency of this convex program in the high-dimensional
scaling regime in which the number of latent/observed variables grows with the
number of samples of the observed variables. The algebraic varieties of sparse and
low-rank matrices play a prominent role in this analysis.
We present a general convex optimization formulation for linear inverse problems,
in which we have limited measurements in the form of linear functionals of a signal
or model of interest. When these underlying models have algebraic structure, the resulting convex programs can be solved exactly or approximately via semidefinite
programming. We provide sharp estimates (based on computing certain Gaussian
statistics related to the underlying model geometry) of the number of generic
linear measurements required for exact and robust recovery in a variety of settings.
We present convex graph invariants, which are invariants of a graph that are convex
functions of the underlying adjacency matrix. Graph invariants characterize
structural properties of a graph that do not depend on the labeling of the nodes;
convex graph invariants constitute an important subclass, and they provide a systematic
and unified computational framework based on convex optimization for
solving a number of interesting graph problems.
We emphasize a unified view of the underlying convex geometry common to these
different frameworks. We describe applications of these methods to problems in financial
modeling and network analysis, and conclude with a discussion of directions for future
research.https://authors.library.caltech.edu/records/m5g1p-xyp93