| Abstracts will be posted as they come available All talks will be held in Science Hall 107 unless otherwise noted. Title: Phase retrieval in a shift-invariant space
Abstract: Phase retrieval arises in various fields of science and engineering.
In this talk, we consider an infinite-dimensional phase retrieval problem to reconstruct
real-valued signals living in a shift-invariant space from their phaseless samples taken either
on the whole line or on a discrete set with finite sampling rate. We find an equivalence
between nonseparability of signals in a shift-invariant space and their phase retrievability
with phaseless samples taken on the whole line. For spline signals of order N, we show that they can be well approximated, up to a sign,
from their noisy phaseless samples taken on a set with sampling rate
2N-1. We also propose a robust algorithm to reconstruct nonseparable signals in a shift-invariant space from their phaseless samples
corrupted by bounded noises.Title: Nonlinear Kaczmarz-type recovery algorithm for fast phase retrieval.
Abstract: Signal recovery from phaseless measurements is a fundamental
nonlinear problem that arises in many sensing and imaging application where recording of
phase information is impossible due to physical device constraint. In recent years the
PhaseLift algorithm and its variants have received a lot of attention for their theoretical
guarantees based on the convex geometry tools. However this lifting techinique is not efficient for
very high dimensional problems due to its quadratic expansion of the signal dimensionality.
Therefore iterative recovery methods that avoid lifting have been of interest very recently,
especially with the help of new developments such as the Wirtinger flow.
One of iterative methods of this type is a certain nonlinear Kaczmarz recovery algorithm.
In this talk, I will present a revised version of this algorithm and show its exponential (linear)
convergence using tools from stochastic process analysis,
which exploits its negative drift property to bound the hitting time of residual error evolution.
This talk is based on joint work with my advisor Sinan Gunturk. Title: Associating vectors in C^{n} with rank 2 projections in R^{n}
Abstract: We will see that vectors in C^{n} have natural analogs as rank 2 projections in R^{n}.
We will use this association to prove the complex version of a theorem of Edidin in phase retrieval.
In particular we will show that a set of complex subspaces in C^{n} does phase retrieval if and only if it has real codimension 1.
We will also prove a similar theorem about norm retrieval in the process. Title: Multi-dimensional Sublinear Sparse Fourier Algorithm
Abstract: We introduce the development of a sublinear sparse Fourier algorithm for high-dimensional
data. In "Adaptive Sublinear Time Fourier Algorithm" by D. Lawlor, Y. Wang and A. Christlieb (2013) ,
an efficient algorithm with empirically O(klogk) runtime and O(k) sampling complexity for the one-dimensional sparse FFT
was developed for signals of bandwidth
N, where k is the number of significant modes such that k< < N.
In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals,
extending some of the ideas in the paper mentioned above. Note
a higher dimensional signal can always be unwrapped into a one dimensional signal,
but when the dimension gets larger unwrapping a higher dimensional signal into a
one dimensional array is far too expensive to be realistic.
Our approach here introduces two
new concepts: a "partial unwrapping'' and a "tilting.''
These two ideas allow us to efficiently compute the sparse FFT of higher dimensional signals.
Moreover, a multiscale method iteratively fixes approximation for noisy signals.Title: Universal spatiotemporal sampling sets for discrete spatially invariant evolution processes
Abstract: We consider the spatiotemporal sampling problem in evolution processes in which we seek to recover evolving signals from coarse samples taken
at varying time instances instead of a finer sampling taken at the earliest time. In this talk, we propose an universal spatiotemporal sampling scheme
for spatially invariant evolution processes and show how it is related to the sparse signal processing theory. Title: Bypassing the limits of L^{1} regularization: Convex non-convex optimization for signal processing
Abstract: Sparsity has become the basis of some important signal processing
methods over the last ten years. Many signal processing problems (e.g., denoising, deconvolution,
non-linear component analysis) can be expressed as inverse problems. Sparsity is invoked through
the formulation of an inverse problem with suitably designed regularization terms.
Often, the L^{1} norm is used to induce sparsity, so much so that L^{1} regularization is considered
to be 'modern least-squares.' Convex regularization via the L^{1} norm, however, tends to under-estimate
the non-zero values of sparse signals. In order to estimate the non-zero values more accurately,
non-convex regularization is often favored over convex regularization. However, non-convex
regularization generally leads to non-convex optimization, which suffers from numerous issues such
as sensitivity to initialization, multiple local minima, etc.
This talk will discuss parameterized non-convex penalty functions that are designed so as to ensure that
convexity of the objective function and their application to several problems of interest. Title: Some applications of Hilbert-Schmidt frames
Abstract: In this talk, we will first discuss some characterizations of Hilbert-Schmidt frames in separable Hilbert spaces.
Then we will discuss the Parseval type identities and new inequalities for Hilbert-Schmidt frames.
These results generalize and improve the remarkable results which have been obtained by Balan et al. and Gavruta.Title: On the Strong Convergence of Some Proximal Point
Algorithms
Abstract: We consider a proximal point algorithm with errors for a maximal monotone
operator in a real Hilbert space, previously studied by Boikanyo and Morosanu,
where they assumed that the zero set of the operator is nonempty and the error
sequence is bounded. In this paper, by using our own approach, we significantly
improve the previous results by giving a necessary and sufficient condition for the
zero set of the operator to be nonempty, and by showing that in this case, this
iterative sequence converges strongly to the metric projection of some point onto
the zero set of the operator, without assuming the boundedness of the error
sequence. We study also in a similar way, the strong convergence of a new
proximal point algorithm, and present some applications of our results to
optimization and variational inequalities..Title: A Posteriori Error Bounds for Two Point Boundary Value Problem with Uncertain ParametersAbstract: Estimating errors of the solution of differential equations is a very important task in
many areas of science and enerinering. In this presentation approximation errors and uncertainty
will be taken into account simultaneously. In this work the authors assumed that only
upper and lower bounds of parameters are given. To goal is to find upper and
lower bound of the solution with high accuracy. The problem can be solved by using variational formulation
and the Finite Element Method. Extreme values of the solution can be found by using special optimization methods.
In order to increase the accuracy of the solution a posteriori error estimation was applied.
The grid points can be found by using special adaptive algorithm.
Presented methodology can be applied to large scale problems and solved by using parallel computing.
The method was applied for the solution of sample two point boundary problem with the uncertain parameters. |