Norbert Wiener Center Archives for Academic Year 2016

Recent developments in operator sampling

When: Mon, August 24, 2015 - 2:00pm
Where: MATH 1311
Speaker: Prof. Götz Pfander (Jacobs University of Bremen) -

Methods for Spectral Analysis of Stochastic Networks

When: Tue, September 15, 2015 - 2:00pm
Where: Math 1311
Speaker: Prof. Maria Cameron (UMD) -
Abstract: Stochastic networks (continuous-time Markov chains) with pairwise transition rates containing a small parameter arise in modeling natural processes. Time-reversible processes include the dynamics of clusters of interacting particles, conformal changes in molecules, and protein folding, while time-irreversible ones are exemplified e.g. by walks of molecular motors.
The spectral decomposition of the generator matrix of the stochastic network gives a key to understanding its dynamics, the extraction of quasi-invariant sets, and building coarse-grained models. However, its direct calculation is exceedingly difficult if the matrix is large and its entries vary by tens of orders of magnitude. I will introduce a single-sweep algorithm for computing the asymptotic spectral decomposition
and discuss continuation techniques. This approach allows us to easily interpret the results and largely avoid the issues associated with the floating point arithmetic. An application to the Lennard-Jones cluster of 75 atoms will be discussed.

Non-Asymptotic Bounds for Geometric Multiresolution Analysis

When: Tue, September 22, 2015 - 2:00pm
Where: Math 1311
Speaker: Nate Strawn (Georgetown University) -
Abstract: Geometric Multiresolution Analysis (GMRA) is an efficient procedure which combines two primitives to create an approximation to a dataset: partitioning and Principal Component Analysis. In this talk, we exhibit nearly minimax optimal finite sample probabilistic bounds for the mean square error (MSE) of the GMRA procedure of Allard, Chen, and Maggioni for ``noisy manifold'' models. To do so, we independently show that ''good'' partitions with respect to a base probability distribution produce GMRAs satisfying an MSE bound, and then show that partitions constructed from running the cover trees algorithm of Beygelzimer et al. are ``good'' with high probability if the base probability distribution is a "noisy manifold." This is joint work with Stanislav Minsker of UC Southern California and Mauro Maggioni of Duke University.

Structured sparse recovery with sparse sampling matrices

When: Tue, September 29, 2015 - 2:00pm
Where: Math 1311
Speaker: Bubacarr Bah (University of Texas) -
Abstract: Compressed sensing seeks to exploit the simplicity (sparsity) of a signal to under sample the signal significantly. Sparsity is a first order prior information on the signal. In many applications signals exhibit an additional structure beyond sparsity. Exploiting this second order prior information about the signal not only enables further sub-sampling but also improves accuracy of reconstruction. On the other hand, a lot of the sampling matrices, for which we are able to prove optimal recovery guarantees, are dense and hence do not scale well with the dimension of the signal. Sparse matrices scale better than their dense counterparts but they are more difficult to give provable guarantees on. The sparse sampling operators we consider are adjacency matrices of lossless expander graphs. They are non-mean zero and they reflect more some of the applications of compressed sensing like the single pixel camera. We also propose a non-convex algorithm that converges linearly with the signal dimension and a convex algorithm that is comparable and sometimes outperforms existing popular algorithms. We also derived sharp sample complexity bounds. This talk will be about these results.


When: Tue, October 6, 2015 - 2:00pm
Where: Math 1311
Speaker: Thomas Goldstein (University of Maryland) -
Abstract: TBA

ShapeFit: Exact location recovery from corrupted pairwise directions.

When: Tue, October 13, 2015 - 2:00pm
Where: Math 1311
Speaker: Paul Hand (Rice University) -
Abstract: We consider the problem of recovering a set of locations given observations of the direction between pairs of these locations. This recovery task arises from the Structure from Motion problem, in which a three-dimensional structure is sought from a collection of two-dimensional images. In this context, the locations of cameras and structure points are to be found from epipolar geometry and point correspondences among images. These correspondences are often incorrect because of lighting, shadows, and the effects of perspective. Hence, the resulting observations of relative directions contain significant corruptions. To solve the location recovery problem in the presence of corrupted relative directions, we introduce a tractable convex program called ShapeFit. Empirically, ShapeFit can succeed on synthetic data with over 50% corruption. Rigorously, we prove that ShapeFit can recover a set of locations exactly when a fraction of the measurements are adversarially corrupted and when the data model is random. This and subsequent work was done in collaboration with Choongbum Lee, Vladislav Voroninski, and Tom Goldstein.

An Introduction to Quantum Stochastics

When: Tue, October 20, 2015 - 2:00pm
Where: Math 1311
Speaker: Dr. Radhakrishnan Balu (ARL) -
Abstract: Quantum probability (QP) is a generalization of axiomatic probability theory and quantum mechanics. The classical ideas of random variables and measures are extended to operators (Hermitian) and trace operations enabling the rich theory of operators to be exploited. In QP probability amplitudes play a prominent role producing counter intuitive interference effects not possible in classical situations. Quantum analogues of stochastic processes can be defined as operator processes that can be used to model noise in quantum systems interacting with an environment. We will start with a review of notions in classical probability space and define the corresponding notions in QP. To make the ideas concrete we will consider a discrete time quantum processes to model classical Markov chains to express a bias removal algorithm. We will also discuss examples of continuous time versions in modeling quantum networks. As a second application we will discuss a quantum mechanics based logic programming language to express probability distributions and statistical correlations, by extending the notion of propositions to operators. We will discuss examples from quantum communication protocols in this new language and outline how to prove properties about them. Finally, we will discuss issues in designing efficient algorithms in solving the stochastic partial differential equations derived as part of modeling quantum noises.

Examining sonar targets with topological invariants

When: Tue, October 27, 2015 - 2:00pm
Where: Math 1311
Speaker: Prof. Michael Robinson (American University) -
Abstract: For various reasons, synthetic aperture sonar (SAS) target classification in various clutter contexts is usually done using a data-driven, machine learning approach. Unfortunately, the resulting feature set can be rather inscrutable -- what features is it really using? If you train on simulated data, there is a substantial risk of overfitting or worse. Maybe you've trained on artifacts that won't really happen in actual data! On the other hand, experimental data may contain un-leveraged symmetries that are not adequately captured by simulation. I will describe a principled, foundational analysis of target echo structure through the lens of topological signal processing. I will describe how this can be used to extract of actionable, physics-aware features for classification from simulated and experimental data.

Nonuniform sampling and multiscale computation

When: Tue, November 3, 2015 - 2:00pm
Where: Math 1311
Speaker: Dr. Christina Frederick (Georgia Tech.) -
Abstract: The Heterogeneous Multiscale Method (HMM) is a framework for efficiently solving problems that couple physical models on different scales, for example, determining the effective properties of composite materials. Simulations often involve discretizations by means of sampling. We discuss an interesting connection between HMM for numerical homogenization problems and nonuniform sampling strategies for bandlimited functions. The sampling and reconstruction problem is formulated by viewing typical functions in homogenization theory as structured multiband signals. Our main result is a stable reconstruction algorithm for this class of signals using $d-$dimensional periodic nonuniform sampling. The proposed sampling sets are of optimal rate according to the minimal sampling requirements of Landau.

Semi-Supervised Graph-Based Methods for Image Segmentation and Classification

When: Tue, November 10, 2015 - 2:00pm
Where: Math 1311
Speaker: Prof. Nathan Cahill (RIT) -
Abstract: Laplacian Eigenmaps (LE) is a powerful graph-based technique for manifold learning and recovery. It solves the same generalized eigenvector problem that arises in the continuous relaxation of Normalized Cut (NCut)-based graph partitioning. Schroedinger Eigenmaps (SE) has emerged as a powerful semi-supervised extension of LE that uses barrier and/or cluster potentials to encode expert/labeled information provided at a subset of graph vertices. In this talk, we first explore how SE can be used for hyperspectral image analysis, including in the fusion of spatial/spectral data for classification, the exploitation of user-provided "paintbrush" strokes for segmentation, and the inclusion of target spectra for target detection. Then, we show how the idea of cluster potentials in SE can be used to construct an analogous Semi-Supervised version of the Normalized Cuts algorithm, which we call SSNCuts, and we show how SSNCuts improves RGB image segmentation of complicated scenes.

Scattering transform and its applications

When: Tue, November 17, 2015 - 2:00pm
Where: Math 1311
Speaker: Yiran Li (UMD) -

Quantization of Phaseless Measurements

When: Tue, November 24, 2015 - 2:00pm
Where: Math 1311
Speaker: Thang Huynh (CIMS) -
Abstract: In this talk, I will discuss how the distributed noise-shaping method of Chou and Gunturk can be extended to the quantization problem of phaseless measurements and will show that a suitably modified recovery algorithm based on PhaseLift guarantees near-optimal error performance. Joint work with Gunturk and Jeong.

Single image superresolution through directional representations

When: Tue, December 1, 2015 - 2:00pm
Where: Math 1311
Speaker: Dr. James Murphy (Duke) -

Minimal scalings and scalable frames

When: Tue, December 8, 2015 - 2:00pm
Where: Math 1311
Speaker: Professor Yeonhyang Kim (Central Michigan U) -
Abstract: It is known that the set of all scalings of a scalable frame F is a convex polytope with vertices corresponding minimal scalings. In this talk, we provide a method to find a subset of contact points which provides a decomposition of the identity, and an estimate of the number of minimal scalings of a scalable frame. We provide a characterization of when minimal scalings are affinely dependent. Using this characterization, we can conclude that all strict scalings of F have the same structural property. We also present the uniqueness of orthogonal partitioning property of any set of minimal scalings, which provides all possible tight subframes of a given scaled frame.

Wavelet transform modulus: phase retrieval and scattering

When: Tue, February 2, 2016 - 2:00pm
Where: MTH3206
Speaker: Dr. Irène Waldspurger (MIT) -
Abstract: We consider the problem of reconstructing a function from the modulus of its wavelet transform. This inverse problem belongs to the class of so-called "phase retrieval problems". From a theoretical point of view, we can show that any function is uniquely determined by its wavelet transform modulus, at least for a specific choice of wavelets. The reconstruction may not be stable to noise in a strong sense; nevertheless, it is locally stable. We also have a reconstruction algorithm that enjoys good empirical performances. After describing these results, we will see which consequences they have for the study of a more sophisticated representation: the scattering transform, introduced by Mallat.

Discrete Directional Gabor Representation

When: Tue, February 9, 2016 - 2:00pm
Where: Math 3206
Speaker: Dr. Benjamin Manning (UMD) -

Sharp Balian-Low Type Theorems for Shift-Invariant Spaces

When: Tue, February 16, 2016 - 2:00pm
Where: Math 3206
Speaker: Michael Northington (Vanderbilt) -
Abstract: Uncertainty principles are results which restrict the localization of a function and its Fourier transform. One class of uncertainty principles studies generators of structured systems of functions, such as wavelets or Gabor systems, under the assumption that these systems form a basis or some generalization of a basis. An example is the Balian-Low Theorem for Gabor systems. In this talk, I will discuss sharp, Balian-Low type, uncertainty principles for finitely generated shift-invariant subspaces of $L^2(\R^d)$. In particular, we give conditions on the localization of the generators and the type of “basis” formed by integer translates of the generators, which prevent these spaces from being invariant under any non-integer shifts.

February Fourier Talks

When: Thu, February 18, 2016 - 9:00am
Where: Math 3206
Speaker: Various Speakers () -

February Fourier Talks

When: Fri, February 19, 2016 - 9:00am
Where: Math 3206
Speaker: Various Speakers () -

Sharp Weighted estimates for Bergman projection in the unit ball

When: Tue, March 1, 2016 - 2:00pm
Where: Math 3206
Speaker: Dr. Edgar Tchoundja (Washington University) -
Abstract: Let Bn be the unit ball of Cn. It is known that the Bekolle-Bonami weight characterize weights w for which the Bergman projection extends to a bounded operator on Lp(w), (p>1). Using modern techniques of dyadic harmonic analysis, we are able to recover this result and prove sharp estimates for the Bergman projection and other related operators in Bn. This talk is based on joint work with R. Rahm and B. Wick.

Phase Retrieval using Structured Measurements

When: Tue, March 8, 2016 - 2:00pm
Where: Math 3206
Speaker: Dr. Yi-Kai Liu (NIST) -
Abstract: Phase retrieval is the task of learning an unknown n-dimensional vector x (over the real or complex numbers) from quadratic measurements. That is, one is given measurements of the squared inner products y_i = |a_i^T x|^2, for i = 1,2,...,m, where the vectors a_i are chosen by the observer. Such measurements arise in a variety of applications, including coherent diffractive imaging, and quantum state tomography. There is a natural approach to solving the phase retrieval problem, by means of a convex relaxation called PhaseLift. Furthermore, PhaseLift is known to perform well (with provable recovery guarantees) when the measurement vectors a_i are chosen independently at random from a Gaussian distribution. Unfortunately, these Gaussian measurements are difficult to implement in real experiments. In this talk, I will show theoretical results on the performance of PhaseLift with different kinds of structured measurements. These include Bernoulli measurements (sampled from the uniform distribution on the hypercube {1,-1}^n), and spherical 2-designs (which are a second-order approximation to the Gaussian distribution). These measurements are easier to implement in experiments, and still ensure successful recovery of most signals, with a few pathological exceptions. (This is joint work with Felix Krahmer at the Technical University of Munich, and Shelby Kimmel at the University of Maryland.)

Sparse Signal Representation in Digital and Biological Systems

When: Tue, April 12, 2016 - 2:00pm
Where: Math 1310
Speaker: Matthew Guay (UMD) -

Low dimensional phase space parametrizations of reproducing formulae and orthonormal bases in space dimensions 1 and 2.

When: Tue, April 26, 2016 - 2:00pm
Where: MTH 3206
Speaker: Prof. Krzysztof Nowak (Drexel University) -

The discrete sign problem: uniqueness, recovery algorithms and phase retrieval applications

When: Tue, May 3, 2016 - 2:00pm
Where: Math 3206
Speaker: Dr. Oren Raz (UMD) -
Abstract: In this talk I will consider the following real-valued finite dimensional sign problem, which is motivated by a particular application of the classical 2-D phase retrieval problem. Let F ? RN be an N-dimensional vector, whose discrete Fourier transform has a compact support. The sign problem is to recover F from its magnitudes |F_i|. First, in contrast to the classical 1-D phase problem which in general has multiple solutions, we prove that with sufficient oversampling, the sign problem admits a unique solution. Next, we show that the sign problem can be viewed as a special case of a more general piecewise constant phase problem. For the former we derive a computationally efficient and robust to noise recovery algorithm. In the noise-free case and with a sufficiently high sampling rate, our algorithm is guaranteed to recover the true sign pattern.