Seminar Abstracts (spring 2009)


Laurent Younes
Smoothing in the dual space; Applications to vector fields and frames
Abstract: We discuss a new variational paradigm to measure the smoothness of data like unit vectors fields, or fields of rotation matrices over spatial domains, and related denoising method for such datasets. Our point of view is to crepreset such fields as linear forms acting on suitable reproducing kernel Hilbert spaces and work in the dual. Experimental results are based on synthetic data and diffusion tensor–magnetic resonance imaging datasets.

Melvin Leok
Lie group and homogeneous variational integrators and their applications to geometric optimal control theory
Abstract: The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.
We will discuss the application of geometric structure-preserving numerical schemes to the optimal control of mechanical systems. In particular, we consider Lie group variational integrators, which are based on a discretization of Hamilton's principle that preserves the Lie group structure of the configuration space. In contrast to traditional Lie group integrators, issues of equivariance and order-of-accuracy are independent of the choice of retraction in the variational formulation. The importance of simultaneously preserving the symplectic and Lie group properties is also demonstrated.
Recent extensions to homogeneous spaces yield intrinsic methods for Hamiltonian flows on the sphere, and have potential applications to the simulation of geometrically exact rods, structures and mechanisms. Extensions to Hamiltonian PDEs and uncertainty propagation on Lie groups using noncommutative harmonic analysis techniques will also be discussed.
We will place recent work in the context of progress towards a coherent theory of computational geometric mechanics and computational geometric control theory, which is concerned with developing a self-consistent discrete theory of differential geometry, mechanics, and control.
This research is partially supported by NSF grants DMS-0714223, DMS-0726263, and DMS-0747659.

Dan Naiman
Multivariate Records
Abstract: Given a vector-valued time series, a multivariate record is said to occur at time t if no previous observation dominates it in every coordinate. This notion of a record generalizes the usual notion in one dimension, and gives rise to some interesting phenomena, some of which will be presented. This talk will describe an efficient algorithm for sampling the multivariate records process, as well as a method, based on the theory of abstract tubes, for calculating the probability of a new multivariate record at time t, conditional on the past up to time t. (Joint work with Fred Torcaso.)

Anant Godbole
Omnibus Sequences
Abstract: Consider locating words of length $k$ over an alphabet of size $a$ as subsequences (but not necessarily substrings) of a random length $n$ string. An $n$ string with the property that each of the $a^k$ words is present as a subsequence is called an {\it Omnibus Sequence.} We derive necessary and sufficient conditions for a string to be omnibus and give conditions under which a random string is almost always or almost never omnibus. An analysis of the number of missing words provides another way of looking at this problem. Several applications are presented. For example, we demonstrate how Tolstoy's ``War and Peace" contains this abstract, or any other ``abstract" of this length as a subsequence. Parallels are drawn with the fundamental results of de Bruijn, and Chung, Diaconis, & Graham, on Universal Cycles . This is joint work with Sunil Abraham (Oxford), Greg Brockman (Harvard) and Stephanie Sapp (JHU-AMS).

Jay Rosen
The $L^2$ modulus of continuity of local times of Brownian motion
Abstract: Let $\{L^x_t; (x,t)\in\mathbb R^1\times \mathbb R^1_+\}$ denote the local time of Brownian motion. We refer to $\int(L_t^{x+h}-L_t^x)^2dx$ as the $L^2$ modulus of continuity. We explain the basic ideas behind the law of large numbers $$ \lim_{h\downarrow 0}\int \frac{(L_t^{x+h}-L_t^x)^2}{h} dx = 4t \mathrm{\ a. s.} $$ and central limit theorem $$ \lim_{h\downarrow 0}\int \frac{(L_t^{x+h}-L_t^x)^2-4ht}{h^{3/2}} dx \stackrel{\mathcal L}{\Longrightarrow} c\sqrt{\alpha_t}W_1 $$ for this modulus of continuity. Here $$ \alpha_t = \int (L_t^x)^2 dx $$ and $\{W_t; t\geq 0\}$ is a Brownian motion independent of $\alpha_t$.
We also discuss analogous results for the local times of stable and other Lévy processes.
Based on joint work with Xia Chen, Wenbo Li, and Michael B. Marcus

Lisa Fleischer
Submodular Approximation: Sampling-based Algorithms and lower Bounds
Abstract: We introduce several natural optimization problems using submodular functions and give asymptotically tight or close upper and lower bounds on approximation guarantees achievable using polynomial number of queries to a function-value oracle. The optimization problems we consider are submodular load balancing, submodular sparsest cut, submodular balanced cut, submodular knapsack.
We also give a new lower bound for approximately learning a monotone submodular function; and show that much tighter lower bounds will require a different approach.

Trac Tran
Novel Compressed Sensing Applications in Distributed Video Sensing and Error-Resilient Video Transmission
Abstract: This talk presents several practical frameworks and algorithms for video sensing and communications via the viewpoint of the recently emerging compressed sensing (CS) theory.
We will first start with a novel framework called Distributed Compressed Video Sensing (DisCoS) where video sequences are sampled intra-frame and reconstructed inter-frame via exploitation of temporal correlation. The key observation here is that a pixel block in the current frame can be sparsely represented by a linear combination of few neighboring blocks in previously reconstructed frame(s), enabling it to be predicted from its CS measurements by soling the L1 minimization problem. Simulation results show that DisCoS significantly outperforms the intra-frame-sensing and intra-frame-sparse recovery scheme by 8-10 dB. Unlike all other previous distributed video coding schemes, DisCoS can perform encoding operations in the analog/optical domain with very low-complexity, making it a promising candidate for applications where the sampling process is very expensive, e.g., in Terahertz imaging.
The second part of the talk discusses a framework named layered compressed sensing (LaCoS) for robust video transmission over lossy packet-switched channels. In the proposed transmission system, the encoder consists of a base layer and an enhancement layer. The base layer is a conventionally encoded bitstream and transmitted without any error protection. The additional enhancement layer is a stream of compressed-sensing measurements taken across slices of video signals for error-resilience. By exploiting the side information (base layer) at the decoder, a sparse recovery algorithm can not only recover the lost packets but the enhancement layer is also required to transmit a minimal amount of compressed measurements (only proportional to the packet-loss percentage). Simulation results show that both compression efficiency and error-resilience capacity of the proposed LaCos framework are competitive with those of other state-of-the-art robust transmission methods, in which Wyner-Ziv coders are often used to generate the enhancement layer.

Jeffrey Leek
A general framework for multiple testing dependence
Abstract: I will present a general framework for performing large-scale significance testing in the presence of arbitrarily strong dependence. We have derived a low-dimensional set of random vectors, called a dependence kernel, that fully captures the dependence structure in an observed high-dimensional dataset. This result is a surprising reversal of the "curse of dimensionality" in the high-dimensional hypothesis testing setting. We have shown theoretically that conditioning on a dependence kernel is sufficient to render statistical tests independent regardless of the level of dependence in the observed data. This framework for multiple testing dependence has implications in a variety of common multiple testing problems, such as in gene expression studies, brain imaging, and spatial epidemiology. I will illustrate our approach on a large-scale clinical genomics study of trauma.

Matt Feiszli
Weil-Peterson metric on plane curves
Abstract: The Weil-Peterson Riemannian metric arises in Teichmuller theory, where it measures deformation of conformal structures on Riemann surfaces. We will discuss its use as a metric on shapes, where it measures changes in the conformal structure of the interior and exterior of a simple closed plane curve. We will examine the geometric properties that the WP metric is measuring, and present some estimates relating WP path lengths to the geometry of the region being deformed. There is a close connection between the WP metric and deformations of the medial axes of both the interior and the exterior of the curve.

Youngmi Hur
Design of nonseparable $n$-D biorthogonal wavelet filter banks
Abstract: We consider the design of nonseparable $n$-D biorthogonal wavelet filter banks for signals with different correlation along different directions. Other properties (such as fast algorithms) of the new wavelet filters, as well as the details of the design will be discussed. We illustrate the use of our wavelet filter banks with a $2$-D example. If time permits, we will discuss the properties (such as smoothness) of the corresponding biorthogonal wavelet functions.

Jonathan Taylor
Abstract: This talk describes a "prototypical" application of random field theory (RFT) to neuroimaging data. The data we are interested in represent anatomical differences in the brain between controls and patients who have suffered non-missile trauma. To model the difference between patients and control, we use a multivariate linear model at each location in space (by varying spatial location we arrive at a random field model). The test we use to compare patients and controls at each point is a Hotelling's $T^2$ to detect differences between cases and controls. RFT is used in the final stage: detecting regions of activation in the Hotelling's $T^2$ data, which we do by thresholding the image of $T^2$ statistics.

Jonathan Taylor
Abstract: In various scientific fields from astrophysics to neuroimaging, researchers observe entire images or functions rather than single observations. In my first talk, I describe a particular application in which these random functions, or fields, are used to detect differences between populations in a neuroimaging study. The integral geometric properties, notably the Euler characteristic of the level/excursion sets of these random functions, typically modelled as Gaussian random fields figure in a crucial way in these applications of random fields. In this talk, I will describe some of the integral geometric properties of these random sets, particularly their average Lipschitz-Killing curvature measures. I will focus on describing the results for a class of non-Gaussian random fields (built up of Gaussians) which highlights the relation between their Lipschitz-Killing curvature measures and the classical Kinematic Fundamental Formulae of integral geometry.

Robert McCann
Extremal Doubly Stochastic Measures and Optimal Transportation
Abstract: Imagine some commodity being produced at various locations and consumed at others. Given the cost per unit mass transported, the optimal transportation problem is to pair consumers with producers so as to minimize total transportation costs. Despite much study, surprisingly little is understood about this problem when the producers and consumers are continuously distributed over smooth manifolds, and optimality is measured against a cost function encoding some geometry of the product space. This talk will be an introduction to the optimal transportation, its relation to Birkhoff's problem of characterizing of extremality among doubly stochastic measures, and recent progress linking the two. It culminates in the presentation of a criterion for uniqueness of solutions which subsumes all previous criteria, yet which is among the very first to apply to smooth costs on compact manifolds, and only then when the topology is simple.

Robert McCann
Continuity, curvature, and the general covariance of optimal transportation
Abstract: In this talk, I describe my surprising discovery with Young-Heon Kim (University of Toronto) that the regularity theory of Ma, Trudinger, Wang and Loeper for optimal maps is based on a hidden pseudo-Riemannian structure induced by the cost function on the product space. Non-negativity of certain sectional curvatures of this metric give a necessary and sufficient condition for such maps to be continuous. This gives a simple direct proof of a key result in the theory, leads to new examples, and opens several new research directions, including links to maximal Lagrangian submanifolds of the product space and para-Kähler geometry.

Michael Kazhdan


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