Faraway Fourier Talks 20212022
The Norbert Wiener Center Online Seminar on Harmonic Analysis and Applications
 

When:
Mondays at 2 pm EDT.
Where:
Zoom. For access to the Zoom link, please join the mailing list by entering your information here.
Recorded talks will become available on our Youtube channel.
Our speaker for May 9th is Professor Soledad Villar of JHU, presenting on the topic of 'Enforcing exact group equivariances in machine learning'.
Scheduled and completed talks:
Sept. 27th, 2021 
Professor David Walnut (GMU, UMD) 
Exponential bases for partitions of intervals 
For a partition of [0,1] into intervals I_{1},...,I_{n}
we prove the existence of a partition of ℤ into
Λ_{1},..., Λ_{n}
such that the complex exponential functions with frequencies in Λ_{k} form a Riesz basis for L^{2}(I_{k}),
and furthermore, that for any J⊆{1,2,...,n}, the exponential functions with frequencies in
⋃_{j∈J} Λ_{j} form a Riesz basis for L^{2}(I)
for any interval I with length I=Σ_{j∈J} I_{j}.
The construction extends to infinite partitions of [0,1], but with size
limitations on the subsets J⊆ ℤ.
The construction utilizes an interesting assortment of tools from analysis, probability, and number theory.
This is joint work with Shauna Revay (GMU and Novetta), and
Goetz Pfander (Catholic University of EichstaettIngolstadt).

Concluded  Recording 
Slides 
Oct. 4th, 2021 
Professor Anne Gelb (Dartmouth) 
Empirical Bayesian inference using joint sparsity

We develop a new empirical Bayesian inference algorithm for solving a linear inverse problem given multiple measurement vectors (MMV) of undersampled and noisy observable data. Specifically, by exploiting the joint sparsity across the multiple measurements in the sparse domain of the underlying signal or image, we construct a new support informed sparsity promoting prior. While a variety of applications can be modeled using this framework, our prototypical example comes from synthetic aperture radar (SAR) data, from which data are acquired from neighboring aperture windows. Hence a good test case is to consider the observations modeled as noisy Fourier samples. Our numerical experiments demonstrate that using the support informed sparse prior not only improves accuracy of the recovery, but also reduces the uncertainty in the posterior when compared to standard sparsity producing priors.
This is joint work with Theresa Scarnati formerly of the Air Force Research Lab Wright Patterson and now working at Qualis Corporation in Huntsville, AL, and Jack Zhang, recent bachelor degree recipient at Dartmouth College and now enrolled at University of Minnesota’s PhD program in mathematics.

Concluded  Recording  
Oct 11th, 2021 
Professor Zuowei Shen (National University of Singapore) 
Deep Approximation via Deep Learning 
The primary task of many applications is approximating/estimating a function through samples drawn from a probability distribution on the input space. The deep approximation is to approximate a function by compositions of many layers of simple functions, that can be viewed as a series of nested feature extractors. The key idea of deep learning network is to convert layers of compositions to layers of tuneable parameters that can be adjusted through a learning process, so that it achieves a good approximation with respect to the input data. In this talk, we shall discuss mathematical theory behind this new approach and approximation rate of deep network; we will also how this new approach differs from the classic approximation theory, and how this new theory can be used to understand and design deep learning network.

Concluded  Recording  
Oct 18th, 2021 
Professor Amit Singer (Princeton) 
Wilson Statistics: Derivation, Generalization, and Applications to CryoEM 
The power spectrum of proteins at high frequencies is remarkably well described by the flat Wilson statistics. Wilson statistics therefore plays a significant role in Xray crystallography and more recently in cryoEM. Specifically, modern computational methods for threedimensional map sharpening and atomic modeling of macromolecules by single particle cryoEM are based on Wilson statistics. In this talk we use certain results about the decay rate of the Fourier transform to provide the first rigorous mathematical derivation of Wilson statistics. The derivation pinpoints the regime of validity of Wilson statistics in terms of the size of the macromolecule. Moreover, the analysis naturally leads to generalizations of the statistics to covariance and higher order spectra. These in turn provide theoretical foundation for assumptions underlying the widespread Bayesian inference framework for threedimensional refinement and for explaining the limitations of autocorrelation based methods in cryoEM.

Concluded  Recording 
Slides 
Oct 25th, 2021 
Professor John Klauder 
Expanding Quantum Field Theory Using Affine Quantization 
Quantum field theory uses canonical quantization (CQ), and often fails, e.g., φ^{4}_{4}, etc. Affine quantization (AQ)  which will be introduced  can solve a variety of problems that CQ cannot. AQ can even be used to solve certain models regarded as nonrenormalizable. The specific procedures of AQ lead to a novel Fourier transformation that illustrates how AQ can create a generous contribution to quantum field theory.

Concluded  Recording 
Slides 
Nov. 1st, 2021 
Professor Thomas Strohmer (UC Davis) 
Fighting Surveillance Capitalism with Mathematics 
'Sharing is Caring', we are taught. However, in the Age of Surveillance
Capitalism, a new economic system that pushes
for relentless data capture and analysis, we better think twice what we
share. As data sharing is increasingly locking horns with dataprivacy
concerns, synthetic data are gaining traction as a potential solution to
the aporetic conflict between privacy and utility.
The goal of synthetic data is to create an asrealisticaspossible
dataset, one that not only maintains the nuances
of the original data, but does so without risk of exposing sensitive
information. As such, synthetic data can be instrumental in
reestablishing the balance between the need of data that drives AI
advances and the fundamental right to data protection for citizens and
consumers. However, the road to privacy is paved with NPhard problems!
In this talk I will present three recent mathematical breakthroughs in
the NPhard challenge of computationally
efficiently creating synthetic data that come with provable privacy and
utility guarantees. We draw from a wide range of mathematical concepts,
including Boolean Fourier analysis, duality, empirical processes, and
microaggregation. For instance, we will see some surprising connections
between theoretical probability and
anonymization. I will also present the first noisefree method to achieve
differential privacy and discuss applications of our approach for data
analysis tasks arising in the Intensive Care Unit.
This is joint work with March Boedihardjo and Roman Vershynin.

Concluded  Recording 

Nov. 8th, 2021 
Professor Robert Calderbank (Duke) 
Climbing the Diagonal Clifford Hierarchy 
Quantum computers are moving out of physics labs and becoming generally programmable. In this talk, we start from quantum algorithms like magic state distillation and Shor factoring that make essential use of diagonal logical gates. The difficulty of reliably implementing these gates in some quantum error correcting code (QECC) is measured by their level in the Clifford hierarchy, a mathematical framework that was defined by Gottesman and Chuang when introducing the teleportation model of quantum computation. We describe a method of working backwards from a target logical diagonal gate at some level in the Clifford hierarchy to a quantum error correcting code (CSS code) in which the target logical can be implemented reliably.
This talk describes joint work with my graduate students Jingzhen Hu and Qingzhong Liang.

Concluded  Recording 
Slides 
Nov. 15th, 2021 
Professor Hans Feichtinger (Vienna) 
Conceptual Harmonic Analysis: Tools and Goals The Ubiquitous Role of BUPUs 
Since almost 14 years the speaker tries to promote the idea of
``CONCEPTUAL HARMONIC ANALYIS'' as a way to combine or rather reconcile
Abstract Harmonic Analysis (AHA) with Computational Harmonic Analysis (CHA)
and much more. In particular, the long history Fourier Analysis (by now
200 years!) has contributed to a diversification of methods and standards.
This has led to the unpleasant situation that mathematicians, engineers or
physicists have their own notations, their own settings and habits, and
numerical work is often only seen as a way to illustrate the continuous
theory, or to simulate a problem in order to improve the heuristic basis
for the proper development of a mathematical theory.
Going back to Andre Weil and Hans Reiter one can say that the natural domain
for Fourier Analysis are LCA groups. The same is true for timefrequency
analysis and Gabor Analysis. But in the world of AHA we can discuss the
analogy between different groups G. Once the dual group G^ has been identified
we can define the forward and inverse Fourier transform, define timefrequency
shifts and the STFT and discuss the reconstruction from samples (for
bandlimited functions or from the STFT).
Obviously one expects that the FFT should be useful in computing at least
approximately the Fourier transform of a nice function, or the convolution
of two functions, or perhaps even measures. We should motivate the approaches
and ideally provide a guarantee (in the spirit of numerical integration methods)
for computations to deliver good quantitative results. Ideally to approach
should avoid unnecessary technicalities (such as Lebesgue integration or Frechet
spaces such as S(R)), at least for the problems relevant for digital signal
processing. Of course, suitable function spaces are required in order to express
properly that computations deliver a good approximation of a given signal.
In the talk the speaker will report on attempts to rebuild Fourier Analysis
over LCA groups (including R^d) from scratch. First convolution of bounded
measures is introduced via translation invariant systems and then the
Fourier Stieltjes transform is introduced, up to the convolution theorem.
BUPUs (bounded uniform partitions play an important role here).
As an intermediate goal the space S_0(G) is introduced, and finally
the Banach Gelfand Triple (S_0,L_2,S_0*). Most spaces relevant for classical
Fourier Analysis are then sandwiched between S_0 and S_0* and are isometrically
invariant under the timefrequency shifts.
Overall, the focus of the talk will be on alternative ways to provide a proper
foundation for AHA, it will talk about nonstandard function spaces (avoiding
Lebesgue spaces as a starting point) and suggest an interpretation of signals
as ``mild distributions'' (members of S_0*), having a bounded STFT. On the
other hand we need computational tools plus quantitative and constructive
approximations of guaranteed quality.

Concluded  Recording 

Nov. 22nd, 2021 
Professor Rama Chellappa (Johns Hopkins) 
Design of Unbiased, Adaptive and Robust AI Systems 
Over the last decade, algorithms and systems based on deep learning and other datadriven methods have contributed to the reemergence of Artificial Intelligencebased systems with applications in national security, defense, medicine, intelligent transportation, and many other domains. However, another AI winter may be lurking around the corner if challenges due to bias, domain shift and lack of robustness to adversarial attacks are not considered while designing the AI systems. In this talk, I will present our approach to bias mitigation and designing AI systems that are robust to domain shift and a variety of adversarial attacks.

Concluded  Recording 
Slides 
Dec. 6th, 2021 
Professor Dustin Mixon (OSU) 
Three proofs of the BenedettoFickus theorem 
In 2003, Benedetto and Fickus introduced a vivid intuition for an objective function called the frame potential, whose global minimizers are fundamental objects known today as unit norm tight frames. Their main result was that the frame potential exhibits no spurious local minimizers, suggesting local optimization as an approach to construct these objects. Local optimization has since become the workhorse of cuttingedge signal processing and machine learning, and accordingly, the community has identified a variety of techniques to study optimization landscapes. This talk applies some of these techniques to obtain three modern proofs of the BenedettoFickus theorem. Joint work with Tom Needham, Clayton Shonkwiler, and Soledad Villar.

Concluded  Recording 

Dec. 13th, 2021 
Professor Hrushikesh Mhaskar (Claremont Graduate University) 
Learning without training 
The fundamental problem of machine learning is often formulated as the problem of function approximation as follows. Starting with a data of the form {(x_i, y_i)} sampled from an unknown joint distribution t, approximate f(x) = E_t (yx). Since t is unknown, it is impossible to give a constructive method to find the minimizer of the generalization error defined as the deviation of the model from the target function f in L^2(t). Instead, the estimation of this error is made independently of the actual construction, known as training, which is based on the minimization of another objective function. In this talk, we will point out some pitfalls of this paradigm, and describe our efforts to bypass this procedure and construct a “good” approximation to f directly from the data. While our construction is universal in the sense that it does not involve any assumptions on the target function, we will obtain probabilistic estimates on the pointwise deviation between our approximation and f under some smoothness assumption on f. The talk is mostly theoretical, but some proofofconcept applications are discussed.

Concluded  Recording 

Dec. 20th, 2021 
Professor Weilin Li (Courant Institute) 
Function approximation with onebit Bernstein and neural networks 
The celebrated universal approximation theorems for neural networks typically state that every sufficiently nice function can be arbitrarily well approximated by a neural network with carefully chosen parameters. Motivated by recent questions regarding compression and overparameterization, we ask whether it is possible to represent any reasonable function with a neural network whose parameters are restricted to a small set of values, with the extreme case being onebit {+1,1} neural networks? We answer this question in the affirmative. One of our main innovations is a novel approximation result for linear combinations of multivariate Bernstein polynomials, with only +1 and 1 coefficients. Joint work with Sinan Gunturk.

Concluded  Recording 

Feb. 28th, 2022 
Professor Richard Baraniuk (Rice University) 
A Spline Perspective of Deep Learning 
We study the geometry of deep learning through the lens of approximation theory via spline functions and operators. Our key result is that a large class of deep networks (DNs) can be written as a composition of continuous piecewise affine (CPA) splines, which provide a powerful portal through which to interpret and analyze their inner workings. We explore links to the classical theory of optimal classification via matched filters, the effects of data memorization, vector quantization (VQ), and Kmeans clustering, which open up new geometric avenue to study how DNs organize signals in a hierarchical and multiscale fashion. 
Concluded  Recording 

Mar. 7th, 2022 
Professor Diego Maldonado (Kansas State University) 
A Harnack inequality for certain degenerate/singular elliptic PDEs shaped by convex functions 
Certain convex functions on R^{n} produce quasidistances and Borel measures on R^{n}. Furthermore, their Hessians can shape degenerate/singular elliptic operators. We will describe geometric and measuretheoretic conditions on convex functions that allow to prove a Harnack inequality for nonnegative solutions to their associated elliptic operators. 
Concluded  Recording 

Mar. 14th, 2022 
Professor Margaret Cheney (Colorado State University) 
Passive Source Localization 
This talk introduces the problem of localizing electromagnetic sources from measurements of their radiated fields at two moving sensors.
Two approaches are discussed, the first based on measuring quantities known as “time difference of arrivals“ and “frequency difference of arrivals”. This approach leads to some interesting geometrical problems. The second approach is a syntheticaperture approach that also involves some unsolved problems. 
Concluded  Recording  
Mar. 28th, 2022 
Professor Deanna Needell (UCLA) 
Online nonnegative matrix factorization and applications 
Online Matrix Factorization (OMF) is a fundamental tool for dictionary learning problems, giving an approximate representation of complex data sets in terms of a reduced number of extracted features. Convergence guarantees for most of the OMF algorithms in the literature assume independence between data matrices, and the case of dependent data streams remains largely unexplored. In this talk, we present results showing that a nonconvex generalization of the wellknown OMF algorithm for i.i.d. data converges almost surely to the set of critical points of the expected loss function, even when the data matrices are functions of some underlying Markov chain satisfying a mild mixing condition. As the main application, by combining online nonnegative matrix factorization and a recent MCMC algorithm for sampling motifs from networks, we propose a novel framework of Network Dictionary Learning that extracts `network dictionary patches' from a given network in an online manner that encodes main features of the network. We demonstrate this technique on realworld data and discuss recent extensions and variations. 
Concluded  Recording  
Apr. 4th, 2022 
Professor Alex Cloninger (UCSD) 
Data Representation Learning from a Single Pass of the Data 
In many applications, constructing kernel matrices or pairwise distance matrices can be prohibitively expensive. This can be due to the expense of storing data in a streaming setting, or the expense of accessing multiple large data sets to compute some statistical distance between them. In this talk, I highlight several settings in which we can compute representations of data points (or entire point clouds) on a single pass. This includes Linearized Optimal Transport for computing Wasserstein distances between distributions and performing supervised learning, boosted kernel regression on streaming data, and streaming quantization of translation invariant kernel feature spaces. 
Concluded  Recording  
Apr. 11th, 2022 
Professor Ozgur Yilmaz (UBC) 
Inverting generative models and applications in inverse problems 
Obtaining accurate signal models is a fundamental problem when solving inverse problems. This is because the typical inverse problem, be it denoising, inpainting, compressed sensing, phase retrieval, medical or seismic imaging, is illposed. To find an accurate solution tractably, we need a "good" model of the underlying signal class that can be "imposed" on the solution. Well known, now classical, examples of such models include sparsity with respect to a known basis (compressed sensing), having small total variation (image denoising), and having low rank (matrix and tensor completion). In the Bayesian framework, which provides an alternative approach, signals in a particular class are modeled as realizations of a random vector. The objective is then to model the (prior) distribution of this random vector and use that to identify the posterior distribution, which leads to, for example, the MAP estimator. Recent advances in training deep neural networks have shown that various signal classes can be modeled using generative models leading to generator networks that map a lowdimensional latent space to the highdimensional signal space and provide a straightforward way of sampling the signal space according to the prior distribution. Such models were successfully incorporated in the literature to the Bayesian framework to empirically solve certain inverse problems involving, e.g., handwritten digits and faces.
In this talk, we present an algorithm, which we call PLUGIn (Partially Linearized Update for Generative Inversion), for recovering an unknown latent (code) vector from (noisy) observations of a signal, under a known generative model with provable guarantees. Furthermore, while requiring an overall expansivity, our convergence guarantees hold for networks with some contractive layers. We will also show that PLUGIn is versatile and can be adopted to various inverse problems such as compressed sensing and reconstruction from quantized measurements.
This is joint work with Babhru Joshi, Xiaowei Li, and Yaniv Plan. 
Concluded  Recording  
Apr. 18th, 2022 
Professor Justin Romberg (Georgia Tech) 
Streaming Solutions for TimeVarying Optimization Problems 
We will discuss streaming optimization problems where the goal is to minimize a sum of functions indexed by time. At each time step, a new function is introduced along with a new set of optimization variables; the function depends the optimization variables at the current time step and the previous one. Problems of this type arise in state estimation problems, including simultaneous localization and mapping (SLAM) in robotics, and in streaming reconstruction problems in signal processing.
We will give structural conditions under which solutions to these problems converge quickly in time. This leads directly to algorithms that have guaranteed performance while using limited memory. We will also show how our framework can be generalized to optimization programs organized on general graphs, where the vertices are associated with functions and the edges are associated with shared variables.
This is joint work with Tomer Hamam. 
Concluded  Recording  
Apr. 25th, 2022 
Professor HauTieng Wu (Duke) 
Nonstationary time series analysis through landmark diffusion with clinical applications 
Compared with the commonly collected health information, longterm and highfrequency physiological time series that are nonstationary in nature contain rich information from the other dimension. However, extracting useful information from these time series for clinical usage is challenging on several fronts. Motivated by clinical unmet needs, physiological knowledge and clinical observations, we introduce a latent diffusion geometry based model to model the dynamics, and an algorithm called Robust and Scalable Embedding via Landmark Diffusion (ROSELAND), which is designed to efficiently handle big and noisy datasets, to extract useful information. Theoretically, we show that the algorithm is robust to colored and heterogeneous noise, and suitable for recovering the latent space under the manifold setup due to the Linfty spectral convergence guarantee. A clinical challenge will be discussed as an application. 
Concluded  Recording  
May 9th, 2022 
Professor Nate Strawn (Georgetown) 
Filament Plots for Data Visualization 
We introduce filament plots for lossless visualization of highdimensional datasets, where each data point in a dataset is mapped isometrically to a 2Dvalued curve in L^{2}, which is then identified with a 3D ``filament" via integration of solutions to a "Bishop frame". The isometric embedding into the space of symmetric curvature functions is chosen to minimize the mean quadratic variation over L^{2} curves, and we characterize the solutions of this minimization problem subject to some constraints (constraints which attempt to remove biases in the visualization). In particular, the solution set recovers a form of 3D Andrews plots. This characterization indicates there are many degrees of freedom for the solution set, and (using recent bounds for quadratic Gauss sums) we demonstrate that there is a particular minimizer which also satisfies an asymptotic projective ``tour" property (which is useful for removing "local" visual biases). Metric comparisons are considered, and we discuss how the finitedimensional analogue of our approach leads to ``hyper frames" which enjoy properties of tight super frames and fusion frames. 
Concluded  Recording  
May 23rd, 2022 
Professor Soledad Villar (JHU) 
Enforcing exact group equivariances in machine learning 
In this talk we will discuss the different approaches people use to implement group equivariances in machine learning. We will focus on ideas based on group equivariant convolutions, irreducible representations, and invariants. We will see applications to image processing, graph neural networks, and particle simulations. Time permitting, we will show different ways to quantify how much better an equivariant model generalizes in comparison to a nonequivariant model. 
Concluded  Recording  
Organizing Committee:
Radu Balan Jacob Bedrossian John Benedetto Maria Cameron Wojciech Czaja Tom Goldstein Vince Lyzinski
