Norbert Wiener Center Archives for Fall 2013 to Spring 2014
Organization Meeting
When: Tue, September 4, 2012 - 2:00pmWhere: Math 3206
The Jacobi Ensemble and Discrete Uncertainty Principles
When: Tue, September 11, 2012 - 2:00pmWhere: Math 3206
Speaker: Brendan Farrell (Computing and Mathematical Sciences, California Institute of Technology)
Abstract: Our starting point concerns a discrete uncertainty principle: how small can the support sets of a vector and its discrete Fourier transform be? By taking a probabilistic and geometric approach we relate this question to the third ensemble of random matrix theory, the Jacobi ensemble. We present the limiting empirical spectral distribution of a random matrix arising in the discrete Fourier setting and the first universality result for the Jacobi ensemble. We discuss the relationship between these two types of random matrices, as well an unexpected instance of universality. This talk is partially based on joint work with Laszlo Erdos.
Almost Sure Convergence of the Kaczmarz Algorithm with Random Measurements
When: Tue, September 18, 2012 - 2:00pmWhere: Math 3206
Speaker: Xuemei Chen (UMCP)
Abstract: The Kaczmarz algorithm is an iterative method for reconstructing a signal $x \in R^d$ from an overcomplete collection of linear measurements $y_n = < x, phi_n >$, $n \geq 1$. We prove quantitative bounds on the rate of almost sure exponential convergence in the Kaczmarz algorithm for suitable classes of random measurement vectors $\{\phi_n\} \subset R^d$. Refined convergence results are given for the special case when each $\phi_n$ has i.i.d. Gaussian entries and, more generally, when each $\phi_n / ||\phi_n||$ is uniformly distributed on $S^{d-1}$. Interestingly, the work on finding the best convergence rate among all the probability measures on the sphere is closely related to logarithmic potential theory. The conclusion is that the best convergence rate occurs when $\phi_n / ||\phi_n||$ is uniformly distributed on $S^{d-1}$.
Applications of Wiener-Hopf Theory in Math Finance
When: Tue, September 25, 2012 - 2:00pmWhere: Math 3206
Speaker: Bryant Angelos (UMCP)
Abstract: Levy processes are a class of stochastic process which are useful in several areas of mathematics, especially insurance and financial mathematics. One area of specific interest is in functionals of the path of a Levy process, such as extrema, first passage time, and the last time an extrema was achieved. Wiener-Hopf factorization has proven to be a powerful tool in answering questions of this nature.
In this presentation, I will outline the general theory of Levy processes, and then describe Wiener-Hopf factorization theory in this context. I will give several examples, including the recent work by Alexey Kuznetsov in this area. An application to barrier option pricing will conclude the talk.
Diffusion Maps for Changing Data
When: Tue, October 2, 2012 - 2:00pmWhere: Math 3206
Speaker: Matthew Hirn (Yale University)
Abstract: Much of the data collected today is massive and high dimensional, yet hidden within is a low dimensional structure that is key to understanding it. As such, recently there has been a large class of research that utilizes nonlinear mappings into low dimensional spaces in order to organize high dimensional data according to its intrinsic geometry. Examples include, but are not limited to, locally linear embedding (LLE), ISOMAP, Hessian LLE, Laplacian eigenmaps, and diffusion maps. The type of question we shall ask in this talk is the following: if my data is in some way dynamic, either evolving over time or changing depending on some set of input parameters, how do these low dimensional embeddings behave? Is there a way to go between the embeddings, or better still, track the evolution of the data in its intrinsic geometry? Can we understand the global behavior of the data in a concise way? Focusing on the diffusion maps framework, we shall address these questions and a few others. We will begin with a review the original work on diffusion maps by Coifman and Lafon, and then present some current theoretical results. Various synthetic and real world examples will be presented to illustrate these ideas in practice, including examples taken from image analysis and dynamical systems. Parts of this talk are based on joint work with Ronald Coifman, Simon Adar, Yoel Shkolnisky, Eyal Ben Dor, and Roy Lederman.
Wavelets with Composite Dilations
When: Tue, October 9, 2012 - 2:00pmWhere: Math 3206
Speaker: Glenn Easley (Senior Scientist, System Planning Corporation)
Abstract: It is widely recognized that the performance of many image processing algorithms can be significantly improved by applying multiscale image representations with the ability to handle very efficiently directional and other geometric features. Wavelets with composite dilations offer a flexible and especially effective framework for the construction of such representations. Unlike traditional wavelets, this approach enables the construction of waveforms ranging not only over various scales and locations but also over various orientations and other orthogonal transformations. Several useful constructions are derived from this approach, including the well-known shearlet representation and new ones. In this talk, we shall introduce and apply a novel multiscale image decomposition algorithm for the efficient digital implementation of wavelets with composite dilations. Due to its ability to handle geometric features efficiently, our new image processing algorithms provide consistent improvements upon competing state-of-the-art methods.
Generalizations of Generating Functions for Hypergeometric Orthogonal Polynomials
When: Tue, October 16, 2012 - 2:00pmWhere: Math 3206
Speaker: Howard Cohl (NIST)
Abstract: We generalize generating functions for hypergeometric orthogonal polynomials, namely Wilson, Laguerre, Jacobi, Gegenbauer, Chebyshev, and Legendre polynomials. These generalizations of generating functions are accomplished through series rearrangement using connection relations for these orthogonal polynomials.
Probabilistic Frames and Optimal Transport
When: Fri, November 2, 2012 - 2:15pmWhere: Math 1310
Speaker: Clare Wickman (UMCP)
Abstract: Probabilistic frames are an extension of frames for Euclidean space to probability measures with finite second moment on that space . While many of the familiar results from finite frame theory are retrievable in this setting, new tools also become available. In particular, one can impose a metric, the 2-Wasserstein distance, on this space. The construction of this metric and of geodesics in this space relies on characterization of solutions to the Monge-Kantorovich optimal transport problem. In this talk, we shall describe probabilistic frames in detail, explore the nature of solutions to the Monge-Kantorovich problem in the context of probabilistic frames, and present some initial results on constructions of probabilistic frames using optimal transport.
The Paulsen Prooblem
When: Tue, November 6, 2012 - 2:00pmWhere: Math 3206
Speaker: Jameson Cahill (University of Missouri)
Abstract: The Paulsen Problem asks if a frame is nearly tight and nearly equal norm, how close is it to being simultaneously tight and equal norm. This problem has been actively worked on for over ten years now, and still no satisfactory answer has been given. We will give a precise formulation of this problem, and review what is known. We will conclude by giving some recent equivalent reformulations which will hopefully shed some light.
Shearlet Ginzburg Landau energy: Gamma-convergence and applications
When: Tue, November 20, 2012 - 2:00pmWhere: Math 3206
Speaker: Julia Dobrosotskaya (UMCP)
Abstract: We design a new class of shearlet-based functionals resembling the
classical Ginzburg-Landau energy, and prove the variational
convergence of those to the weighted TV functionals as the diffuse
interface parameter approaches zero.
We use the essential features of the differential operator
representations in the Fourier domain to create a new class of
anisotropic diffusive operators based on sparse representations.
While preserving the convenient diffusive features, new operators
bring in the combined advantages of sparsity and
non-locality.
The anisotropic shearlet energies and associated operators are
(by design) highly tunable and very effective in the signal recovery:
we illustrate it with examples of directional-sensitive image
inpainting.
The Posterior Concentration Phenomenon in Bayesian Compressed Sensing
When: Tue, November 27, 2012 - 2:00pmWhere: Math 3206
Speaker: Nathaniel Strawn (Duke University)
Abstract: The decoding step in Compressed Sensing may be thought of as a maximum a posteriori (MAP) estimate where sparsity of the data is enforced through a highly structured prior. Since the full posterior encodes uncertainty about this estimate, a good prior should result in a posterior that is sufficiently concentrated around the MAP estimate if the data is representative of a sample from the prior.
By leveraging the properties of the Dantzig estimator and the principles behind Schwartz's theorem, we exhibit a universal finite-sample posterior concentration bound, which is the first theoretical evidence of this concentration phenomenon. Due to the imprecision of the estimates associated with Schwartz's theorem, this bound is suboptimal, but we may still utilize it to exhibit reasonable posterior concentration for some common priors. In certain cases, we demonstrate that a much stronger bound can be acquired through brute force.
In this talk, we shall discuss these results in depth and we shall also discuss ongoing work on sharpening these results.
A New Alternative for Constructing Multivariate Wavelets
When: Tue, December 4, 2012 - 2:00pmWhere: Math 3206
Speaker: Youngmi Hur (Johns Hopkins University)
Abstract: Wavelets are useful for many applications including signal
and image processing. Tensor product has been a predominant method for
constructing multivariate wavelets. In this talk, I will briefly
review the limitations and benefits of the tensor product
construction. Then I will introduce a new alternative to the tensor
product, to which we refer as coset sum. We will see that many
benefits of the tensor product are shared by coset sum, while some of
its limitations are overcome by coset sum. Highlights of the coset sum
approach include the following: the coset sum works for any spatial
dimension and for a wide range of lowpass filters, and it can be
associated with wavelets that have algorithms faster than the tensor
product ones. Some experimental results that compare the performance
of the two methods will be presented.
Wavelets and Besov Spaces on Symmetric Cones
When: Tue, December 11, 2012 - 2:00pmWhere: Math 3206
Speaker: Jens Christensen (Tufts University)
Abstract: Wavelet systems are generated by translation and dilation
$\frac{1}{\sqrt{a}}f(x/a-b/a)$ where $a>0$.
In some situations the dilation by $a$ can be replaced by
the action of a larger group, and we will investigate the case
when case when this group is the automorphism group of a symmetric cone.
We will show that the Besov spaces on symmetric cones,
defined by Bekolle, Bonami, Garrigos and Ricci via a Littlewood-Paley
decomposition of the cone, can be described by wavelets. In particular
we give a wavelet characterization via the quasi-regular representation
of the semi-direct product of the automorphism group on the cone
and the ambient vector space.
Signal Processing for Big Data
When: Tue, January 29, 2013 - 2:00pmWhere: Math 3206
Speaker: Aliaksei Sandryhaila (Carnegie Mellon University)
Abstract: Recent years have witnessed an explosion of interest to the problem of Big Data. At its core, Big Data requires innovative techniques for representation, learning, and processing of massive datasets that arise in very different contexts, including social and economic networks, internet, image and video databases, sensor and transportation networks, molecular and gene interactions. It has become common to represent the structure of these datasets, such as similarities and dependencies between data elements or interactions between individuals in social networks, using graphs. Most existing techniques for learning and processing of structured data, however, resort to studying the graphs representing the structure rather that the datasets themself. In this talk, we discuss a framework for the analysis of structured data that is inspired by, and is an extension of, the traditional signal processing theory. We show that many fundamental signal processing concepts, such as filtering, spectrum, and Fourier transform, can be defined for structured datasets and applied to various problems in data learning and analysis.
Optimal Recovery of 3D X-Ray Data via Shearlet Decompositions
When: Tue, February 5, 2013 - 2:00pmWhere: Math 3206
Speaker: Demetrio Labate (University of Houston)
Abstract: I will discuss a new decomposition of the 3D X-ray transform
based on the shearlet representation, a multiscale directional representation which is optimally efficient in handling 3D piecewise smooth data.
This approach yields a highly effective reconstruction algorithm providing
a near-optimal rate of convergence in estimating piecewise smooth objects
from 3D X-ray tomographic data which are corrupted by white Gaussian noise.
This algorithm applies a thresholding scheme to the 3D shearlet transform coefficients of the noisy data. For a given noise level $\epsilon$, the threshold can be set so that the estimator attains an essentially optimal mean square error rate $O(\epsilon^{2/3})$, as \epsilon \to 0. This approach outperforms standard SVD estimation methods as well as methods based on the Wavelet-Vaguelettes decomposition.
Multi-Wilson Systems
When: Tue, February 19, 2013 - 2:00pmWhere: Math 3206
Speaker: Roza Aceska (Vanderbilt University)
Abstract: Frames are over-complete collections, used for stable representations of signals as linear combinations of basic building atoms. It is very useful when we can use locally adapted atoms, which in addition behave as elements of bases locally. We explore the possibility of using localized parts of frames and bases when building a customized frame. After a review on Gabor frames and Wilson bases, we consider the question of combining parts of these collections into a multi-frame set and look at its properties.
Some New Developments of Digital Topology
When: Tue, February 26, 2013 - 2:00pmWhere: Math 3206
Speaker: Li Chen (University of the District of Columbia)
Abstract: Digital topology was developed by A. Rosenfeld in the late 1970s and early1980s for image processing. It can be traced back to Alexandroff and Hopf in 1935. In this talk, we will review some concepts and application algorithms of digital topology. We will first introduce digital connectivity and the Jordan curve theorem. Then, we will specifically discuss digital surfaces and digital manifolds. At the end, we will present the digital version of Gauss-Bonnet Theorem, the hole-counting formula of digital images, and data reconstruction using digital functions.
Consistent Vertex Classification with Applications in Massive MR Brain-Graphs
When: Tue, March 5, 2013 - 2:00pmWhere: Math 3206
Speaker: Daniel Sussman (Johns Hopkins University)
Abstract: In the first part of this talk I will go over some recent results related to consistent vertex classification for a class of latent position graph models. This work relies on a spectral embedding of the adjacency matrix and the use of k-nearest-neighbors classifiers. In the second part of the talk I will discuss applying these techniques to graphs derived from diffusion tensor MRI. Each vertex in these graphs corresponds to a voxel in the original images. Using gyral based regions of interest (ROI) as the class labels, we demonstrate classification error rates indicating that ROI signal is present in the graph. This is some of the first work to consider networks built at the voxel level.
Fourier Series: Convergence and Quantitative Convergence
When: Tue, March 12, 2013 - 2:00pmWhere: Math 3206
Speaker: Yen Do (Yale University)
Abstract: In this talk I will describe recent joint results with Michael Lacey related to convergence of Fourier series. I will also give a summary of the history of the subject.
Nonparametric Instrumental Variable Regression
When: Tue, March 26, 2013 - 2:00pmWhere: Math 3206
Speaker: Yuan Liao (UMCP)
Abstract: In nonparametric regressions, when the regressor is correlated with the error term, both the estimation and identification of the nonparametric function are ill posed problems. In the econometric literature, people have been using the instrumental variables to solve the problem. But the problem is still very difficult because the identification involves inverting a "Fredholm integration of the first kind", whose inverse either does not exist or is unbounded. I will start by motivating this problem with an application of the effect of education on wage, then explain the concepts of instrumental variables and Fredholm integral equation of the first kind. My proposed Bayesian method does not require the nonparametric function to be identified, so we can never consistently estimate it. Instead, a new consistency concept based on ``partial identification" will be introduced.
Isoperimetric Inequality and Q-curvature (Joint Seminar with Geometric Analysis and PDE Seminars)
When: Tue, April 2, 2013 - 2:00pmWhere: Math 3206
Speaker: Yi Wang (Stanford University)
Abstract: Using the techniques of $A_p$ weights, we study the relationships between
the isoperimetric inequality with the Paneitz Q-curvature. We show
a Fiala-Huber type isoperimetric inequality for higher dimensions in which
the isoperimetric constant depends only on the integrals of the
Q-curvature.
The Hunt Variance Gamma Process with Applications to Option Pricing
When: Tue, April 9, 2013 - 2:00pmWhere: Math 3206
Speaker: Bryant Angelos (UMCP)
Abstract: In this dissertation we develop a spatially inhomogeneous Markov process as a model for financial asset prices. This model is called the Hunt variance gamma process. We define it via its infinitesimal generator, and prove that this generator induces a unique measure on the space of cadlag functions. We next describe a procedure to do computations with this model, by finding a continuous-time Markov chain approximation. This approximation is used to calibrate the model to fit the S&P 500 futures option surface. Next we investigate specific characteristics of the process, showing how it differs from both Levy and Sato processes. We conclude by using the calibrated model to answer questions about properties of the risk-neutral distribution of future stock prices. We observe a more accurate fit to the risk-neutral term structure of volatility, skewness, and kurtosis, and the presence of mean-reversion in conditional probabilities involving large jumps.
Operator Sampling: Recent and Not-So-Recent Developments
When: Tue, April 16, 2013 - 2:00pmWhere: Math 3206
Speaker: David Walnut (George Mason University)
Abstract: This talk reports on joint work with Gotz Pfander of Jacobs University, Bremen. Operator sampling is an outgrowth of pioneering work of T. Kailath and P.A. Bello in the 1950s and 1960s related to finding theoretical limits on the ability to identify a mobile communication channel by sounding it with a single testing signal. The motivation for investigating these questions arose in part from work in the 1950s on spread-spectrum communications. In this talk we will take a brief look at some of the history and motivations behind these investigations, and look at how modern time-frequency tools and techniques have advanced the understanding of these problems considerably. Finally, we will look at some recent developments in the theory including connections to the theory of finite Gabor frames.
Robustness and Stability of Reconstruction from Magnitudes of Frame Coefficients
When: Tue, April 30, 2013 - 2:00pmWhere: Math 3206
Speaker: Radu Balan (UMCP)
Abstract: This paper is concerned with the question of reconstructing a vector in a finite-dimensional real Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We analyze various Lipschitz bounds of the nonlinear analysis map and we establish theoretical performance bounds of any reconstruction algorithm. This is joint work with Yang Wang.
Gabor Analysis: Foundations and Recent Progress
When: Tue, May 7, 2013 - 2:00pmWhere: Math 3206
Speaker: Hans Feichtinger (Institute of Mathematics, University of Vienna)
Abstract: Although the theoretical foundations of Gabor analysis have been established more or less by the end of the last century there is still a lot to be done in Gabor analysis, and various important questions have been settled in the meantime. It is clear that the Banach Gelfand Trip consisting of the Segal Algebra (So,L2,So')(G) is the most approprate setting for many questions in time-frequency analysis.
We will walk a panorama, from the classical setting, the basic facts derived from the specific properties of the Schroedinger representation of the Heisenberg group (resp. phase space) to recent results concernig the robustness of Gabor expansions, the properties of Gabor multipliers, the computation of approximate duals, or the localization of dual Gabor families derived from the Wiener property of certain Banach algebras of infinite matrices.