Workshop on Dynamical Sampling, Frame Theory, Harmonic Analysis, and Applications

Location: Department of mathematics Stevenson Center 1

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Special Event

Party at Akram's house on Saturday at 7:30pm.

Address: 2712 Brightwood Ave., Nashville, 27212

Cell Phone: 615-260-5766

Titles and Abstracts

Uncertainty Principle via Short-time Fourier Transform

Pu-Ting Yu, GA Tech

A triangular inversion method for frame reconstruction

Dave Larson, Texas A M University

Abstract: We introduce a Triangular Inversion Method as a technique for obtaining frame reconstruction (synthesis) without inverting the standard frame operator. The problem of inverting the standard frame operator can be replaced with the problem of inverting a "cross-frame operator" which can be taken to be triangular when certain hypotheses are attained, and hence is amenable to a standard inversion technique for infinite dimensional triangular operators. There are applications to sampling theory.

Declipping and the recovery of vectors from saturated measurements

Dorsa Ghoreishi, Saint Louis University

Abstract: Frames for a Hilbert space allow for a linear and stable reconstruction of a vector from linear measurements. In many real-world applications, sensors are set up such that any measurement above and below a certain threshold would be clipped as the sensor gets saturated. We study the recovery of a vector from such measurements which is called declipping or saturation recovery. We use a frame theoretic approach to saturation recovery and we characterize when saturation recovery of all vectors in the unit ball is possible. We will be discussing fundamental concepts, proposing some open problems, and providing additional resources. This is a joint work with Wedad Alharbi, Daniel Freeman, Brody Johnson and N. Randrianarivony

A Mathematical Model of CT Scans for Lung Cancer Diagnosis

Glenn Webb, Vanderbilt University

Abstract: A diffusive logistic partial differential equation is developed to provide a dynamic model corresponding to the histograms of Hounsfield units obtained from the ground glass opacity measurements of lung CT scans. The model simulations provide agreement with the five phases of lung cancer tumor development as profiled by the histograms. The model quantifies the tumor growth dynamics and can be used to schedule additional CT scans or medical procedures. The model is compared to patient data at Vanderbilt University Veterans Administration Hospital.

Two Reconstruction Algorithms for Irregular Sampling

Sam Scholze, University of Wisconsin Eau Claire

Abstract: In this talk, I will discuss two techniques to reconstruct a band-limited signal (with band π) from irregularly spaced sample values. The first algorithm aims to reconstruct a signal from sampled values at the points {pj+εj :0<p≤1,j∈Z} where εj is a temporal noise? term which is equal to zero for all but finitely many j ∈ Z. This algorithm reconstructs from finitely many irregular samples by inverting a finite dimensional matrix. The algorithm is stable provided you sample at least four times the Nyquist rate (p < 1/4) and supj∈Z |εj| is sufficiently small. The second algorithm, known as triangular bridging, reconstructs from irregular samples by iteratively preconditioning linear combinations of the irregularly sampled values, and constructing a dual frame to the preconditioned dataset by inverting a finite upper triangular matrix at each step. Based on Kadec’s 1/4 Theorem, we conjecture that this triangular operator is bounded and invertible provided that the sampled values do not differ too much from an evenly spaced lattice.

Recovery of Structured Signals and Its Implications to Frames

Xuemei Chen, University of North Carolina Wilmington

Abstract: In many applications we seek to recover signals from linear measurements far fewer than the ambient dimension, given they have exploitable structures such as sparse vectors or low rank matrices. In this talk we work in a general setting where signals are approximately sparse in an so called atomic set. We provide general recovery results stating that a convex programming can stably and robustly recover signals if the null space of the sensing map satisfies certain properties. Some new results for recovering signals sparse in a frame, and recovering low rank matrices are derived as a consequence.

Dynamical Dual Frames

Rocio Martin Diaz, Vanderbilt, University

Abstract: We use annihilating polynomials to derive characterizations of dynamical frames and "dynamical dual frames" in finite-dimensional vector spaces. We use this approach to prove that every redundant finite frame has infinitely many distinct dynamical dual frames that are generated by a dynamic operator with spectral radius strictly less than 1. This allows a better control of the error bounds for the recent dynamical quantization algorithm presented by J. Ashbrock and A. Powell. The talk is based on a joint work with J. Ashbrock, B. Johnson, I. Medri, and A. Powell

Robust Estimation of Smooth Graph Signals from Randomized Space-time Samples

Longxiu Huang, Michigan State University

Abstract: Heat diffusion processes have found wide applications in modeling dynamical system over graphs. In this talk, I will talk about the recovery of a k-bandlimited graph signal that is an initial signal of a heat diffusion process from its space-time samples. In this work, we have proposed three random space-time sampling regimes, termed dynamical sampling techniques, that consist in selecting a small subset of space-time nodes at random according to some probability distribution. We show that the number of space-time samples required to ensure stable recovery for each regime depends on a parameter called the spectral graph weighted coherence, that depends on the interplay between the dynamics over the graphs and sampling probability distributions. Then, we propose a computationally efficient method to reconstruct k-bandlimited signals from their space-time samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we test dynamical sampling techniques on a wide variety of graphs. The numerical results on support our theoretical findings and demonstrate the efficiency.

Barron space for Graph Convolution Neural Networks

Qiyu Sun, University of Central Florida

Abstract: In this talk, we introduce a Barron space of functions on a compact domain of graph signals, discuss its various properties, such as reproducing kernel Banach space property and universal approximation property. We will also discuss well approximation property of functions in the Barron space by outputs of some graph convolution neural networks, and learnability of functions in the Barron space from their random samples.

Dynamical Sampling for unmanned aerial vehicles

Ursula Molter, Universidad de Buenos Aires and CONICET

Abstract: In this talk I will show an application of dynamical sampling to unmanned aerial vehicles.

Operator Estimation in Dynamical Sampling

Sui Tang, UC Santa Barbara

Abstract: Dynamical sampling explores the recovery of dynamical systems using limited space-time samples. This presentation will focus on recent theoretical and algorithmic advancements in reconstructing evolution operators from these sparse space-time samples.

Spherical Sliced Optimal Transport for Earth Sciences

Soheil Kolouri, Vanderbilt University

Abstract: Many geoscientific phenomena can be represented through spherical distributions, making the comparison of these distributions of high interest in these domains. Traditionally, these distributions have been approached through directional statistics, also known as circular/spherical statistics. This specialized field is dedicated to the statistical analysis of directions, orientations, and rotations. More recently, the growing application of optimal transport theory, attributed to its favorable statistical, geometrical, and topological properties, has sparked increased interest in using optimal transport for comparing spherical distributions. This talk aims to establish computationally robust transport-based distances for comparing spherical distributions. We will introduce a family of spherical sliced Wasserstein distances and discuss their application in geoscience.

Computational Physics and Machine Learning Approaches for Monitoring and Modeling Ice Sheet Evolution

Ravindra Duddu, Vanderbilt University

Abstract: Antarctic and Greenland ice sheets are dynamic components of the Earth system and interact with the ocean and the atmosphere to affect global and regional climate. Additionally, these ice sheets contain enormous quantities of frozen fresh water with a combined sea level equivalent of 67 meters. The dynamic mass loss from ice sheets directly into oceans is the greatest source of uncertainty in predicting sea level rise, which has vast socio-economic implications for coastal regions around the world. Ice flow is modulated by mechanical processes such as sliding, yielding, creep, and fracture in addition to thermodynamic processes melting. In the past two decades, advances in computational physics and mathematics approaches have created Stokes-flow-based formulations to model ice sheet evolution, considering mass, momentum, and energy conservation. More recently, machine learning approaches have been utilized for processing remotely sensed data to monitor ice sheet evolution and other relevant surface processes. In this presentation, I will provide an overview of the computational physics models currently used to model Antarctic and Greenland ice sheets, and briefly introduce the corresponding partial differential equations (PDEs). I will then discuss a few emerging machines learning approaches, involving convolutional neural networks (CNNs) and physics-informed neural networks. I will end my talk with a discussion of the computational challenges persisting with ice sheet modeling.

Wavelet-based change point detection

Darrin Speegle, University of Saintb Louis

A Bayesian wavelet change point detection algorithm is presented. This algorithm can be combined with dimension reduction techniques to find discontinuities in smooth series of millions of variables. We present a (toy) application to computer network data, where we detect change points in traffic patterns in the computer network. This talk combines projects with Rob Steward, Flavio Esposito, Mary Silverglate, Troy Hofstrand (UG), and Addie Wisniewski (UG).

Restriction Theory, Uncertainty Principle, and Signal Recovery

Alex Iosevich, University of Rochester

The classical problem, explored by Donoho and Stark, is to consider a signal (function from {\mathbb Z}_N \to {\mathbb C} with missing frequencies, i.e {\{\widehat{f}(m)\}}_{m \in S} and ask whether the original signal can be recovered EXACTLY. They introduced the uncertainty principle as a key tool in this endeavor. We are going to describe several results where the classical restriction theory of the Fourier transform and some ideas from additive combinatorics are used to obtain a stronger uncertainty principle, and, as a result, a less stringent exact signal recovery condition. This is joint work with Azita Mayeli.

WOVEN FRAMES IN FINITE DIMENSION

Carlos Cabrelli, UUNIVERSITY OF BUENOS AIRES AND IMAS-CONICET

In this talk we present some recent results related to woven finite frames. We provide a characterization and then use it to construct some examples. These results extend to infinite dimensions. Finally we consider the case of Riesz bases of translations. This is motivated by applications to distributed processing.

Restriction Theory, Uncertainty Principle, and Signal Recovery

Alex Iosevich, University of Rochester

The classical problem, explored by Donoho and Stark, is to consider a signal (function from {\mathbb Z}_N \to {\mathbb C} with missing frequencies, i.e {\{\widehat{f}(m)\}}_{m \in S} and ask whether the original signal can be recovered EXACTLY. They introduced the uncertainty principle as a key tool in this endeavor. We are going to describe several results where the classical restriction theory of the Fourier transform and some ideas from additive combinatorics are used to obtain a stronger uncertainty principle, and, as a result, a less stringent exact signal recovery condition. This is joint work with Azita Mayeli.

Quantifying Eigenvalues of Multidimensional Space-Spectral Limiting Operators: Insights and Applications in Signal Processing

Azita Mayeli, CUNY

TIn this presentation, we will explore space-spectral limiting operators (SSLO), their eigenvalue distribution, and their crucial role in signal processing and imaging. A significant focus is placed on the distribution of eigenvalues in higher-dimensional spaces. By connecting theoretical frameworks with practical applications, this talk highlights the transformative potential of SSLOs in advancing signal manipulation and imaging techniques. This is a joint work with Arie Israel.