Adaptive acquisition of correct features from massive datasets is at the core of modern data analysis. One particular interest in medicine is the extraction of hidden dynamics from an observed time series composed of multiple oscillatory signals. The mathematical and statistical problems are made challenging by the structure of the signal which consists of non-sinusoidal oscillations with time varying amplitude and time varying frequency, and by the heteroscedastic nature of the noise. In this talk, I will discuss recent progress in solving this kind of problem. Based on the cepstrum-based nonlinear time-frequency analysis and manifold learning technique, a particular solution will be given along with its theoretical properties. I will also discuss the application of this method to two medical problems – (1) the extraction of a fetal ECG signal from a single lead maternal abdominal ECG signal; (2) the simultaneous extraction of the instantaneous heart rate and instantaneous respiratory rate from a PPG signal during exercise. If time permits, an extension to multiple-time series will be discussed.

For a range of physical and biological processes—from dynamics of granular media to biological swarming—the evolution of a large number of interacting agents is modeled according to the competing effects of pairwise attraction and (possibly degenerate) diffusion. In the slow diffusion limit, the degenerate diffusion formally becomes a hard height constraint on the density of the population, as arises in models of pedestrian crown motion. Motivated by these applications, we bring together new results on the Wasserstein gradient flow of nonconvex energies with the theory of free boundaries to study a model of Coulomb interaction with a hard height constraint. Our analysis demonstrates the utility of Wasserstein gradient flow as a tool to construct and approximate solutions, alongside the strength of viscosity solution theory in examining their precise dynamics. By combining these two perspectives, we are able to prove quantitative estimates on convergence to equilibrium, which relates to recent work on asymptotic behavior of the Keller-Segel equation. This is joint work with Inwon Kim and Yao Yao.

Contemporary and ancient demographic structure in human populations has shaped the genomic variation observed in modern humans, and severely affected the distribution of functional and disease related genetic variation. Using next-generation sequencing technologies, researchers gather increasing amounts of genomic sequencing data for large samples in many different human population groups. These datasets present unprecedented opportunities to study genomic variation in complex demographic scenarios, and this area has received a lot of attention in recent years. In this talk, I will present a method for the inference of demographic histories from full-genome sequencing data of multiple individuals developed by me and my collaborators. I will apply this method to a genomic dataset of Native American individuals to unravel the ancient demographic events underlying the peopling of the Americas. Moreover, I will discuss a novel method for demographic inference that has the potential to improve inference especially in the recent past, which is of particular importance in the context of complex genetic diseases in humans.

Sparsity plays a central role in recent developments of many fields such as signal and image processing, compressed sensing, statistics, and optimization. In practice, sparsity is promoted through the additional of an L1 norm (or related quantity) as a constraint or penalty in a variational model. Motivated by the success of sparsity-inducing methods in imaging and information sciences, there is a growing interest in exploiting sparsity in dynamical systems and partial differential equations. In this talk, we will investigate the connections between compressed sensing, sparse optimization, and numerical methods for nonlinear differential equations. In particular, we will discuss about sparse modeling as well as the advantage of sparse optimization in solving various differential equations arising from physical and data sciences.

In the first part of the talk, we will discuss a numerical method for wave propagation in inhomogeneous media. The Trefftz method relies on basis functions that are solution of the homogeneous equation. In the case of variable coefficients, basis functions are designed to solve an approximation of the homogeneous equation. The design process yields high order interpolation properties for solutions of the homogeneous equation. This introduces a consistency error, requiring a specific analysis.
In the second part of the talk, we will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudo-spectral algorithms were developed for inverting the Laplace-Beltrami operator and computing the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, well-conditioned solvers for the Maxwell equations will rely on these algorithms.

This talk focuses on two instances in which scientific fields outside mathematics benefit from incorporating the geometry of the data. In each instance, the application area motivates the need for new mathematical approaches and algorithms and leads to interesting new questions. (1) A method to determine and predict drug treatment effectiveness for patients based off their baseline information. This motivates building a function adapted diffusion operator on high-dimensional data X when the function F can only be evaluated on large subsets of X, and defining a localized filtration of F and estimation values of F at a finer scale than it is reliable naively. (2) The current empirical success of deep learning in imaging and medical applications, in which theory and understanding is lagging far behind. By assuming the data lie near low-dimensional manifolds and building local wavelet frames, we improve on existing theory that breaks down when the ambient dimension is large (the regime in which deep learning has seen the most success).

This talk is about representation learning with a nontrivial geometry of variables. A convolutional neural network can be viewed as a statistical machine to detect and count features in an image progressively through a multi-scale system. The constructed features are insensitive to nuance variations in the input, while sufficiently discriminative to predict labels. We introduce the Haar scattering transform as a model of such a system for unsupervised learning. Employing Haar wavelets makes it applicable to data lying on graphs that are not necessarily pixel grids. When the underlying graph is unknown, an adaptive version of the algorithm infers the geometry of variables by optimizing the construction of the Haar basis so as to minimize data variation. Given time, I will also mention an undergoing project of flow cytometry data analysis, where histogram-like features are used for comparing empirical distributions. After "binning" samples on a mesh in space, the problem can be closely related to feature learning when a variable geometry is present.

At the beginning of this talk, We will introduce the background of tensor network states (TNS) in various areas such as quantum physics, quantum chemistry and numerical partial differential equations. Famous TNS including tensor trains (TT), matrix product states (MPS), projected entangled pair states (PEPS) and multi-scale entanglement renormalization ansatz (MERA). Then We will explain how to define TNS by graphs and we will define tensor network ranks which can be used to measure the complexity of TNS. We will see that the notion of tensor network ranks is an analogue of tensor rank and multilinear rank. We will discuss basic properties of tensor network ranks and the comparison among tensor network ranks, tensors rank and multilinear rank. If time permits, we will also discuss the dimension of tensor networks and the geometry of TNS. This talk is based on papers joined with Lek-Heng Lim.

The derivation of nonlinear dispersive PDE, such as the nonlinear Schr\"{o}dinger (NLS) from many particle quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE that describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE. We will discuss the process of going from a quantum many particle system of bosons to the NLS via the GP. The most involved part in such a derivation of NLS consists in establishing uniqueness of solutions to the GP. In the talk we will focus on approaches to the uniqueness step that are motivated by the perspective coming from nonlinear dispersive PDE, including the approach that we developed with Chen, Hainzl and Seiringer based on the quantum de Finetti's theorem. Also we will look into what else the nonlinear PDE such as the NLS can tell us about the GP hierarchy, and will present recent results on infinitely many conserved quantities for the GP hierarchy that are obtained with Mendelson, Nahmod and Staffilani.

Obtaining predictive dynamical equations from data lies at the heart of science and engineering modeling, and is the linchpin of our technology. In mathematical modeling one typically progresses from observations of the world (and some serious thinking!) first to selection of variables, then to equations for a model, and finally to the analysis of the model to make predictions. Good mathematical models give good predictions (and inaccurate ones do not) --- but the computational tools for analyzing them are the same: algorithms that are typically operating on closed form equations.
While the skeleton of the process remains the same, today we witness the development of mathematical techniques that operate directly on observations --- data, and appear to circumvent the serious thinking that goes into selecting variables and parameters and deriving accurate equations. The process then may appear to the user a little like making predictions by "looking into a crystal ball". Yet the "serious thinking" is still there and uses the same --- and some new --- mathematics: it goes into building algorithms that "jump directly" from data to the analysis of the model (which is now not available in closed form) so as to make predictions. Our work here presents a couple of efforts that illustrate this "new" path from data to predictions. It really is the same old path, but it is traveled by new means.

Guided waves arise in a variety of applications like underwater acoustics, optics, the design of musical instruments, and so on. We present an analysis of wave propagation and reflection in an acoustic waveguide with random sound soft boundary and a turning point. The waveguide has slowly bending axis and variable cross section. The variation consists of a slow and monotone change of the width of the waveguide and small and rapid fluctuations of the boundary, on the scale of the wavelength. These fluctuations are modeled as random. The turning point is many wavelengths away from the source, which emits a pulse that propagates toward the turning point, where it is reflected. We consider a regime where scattering at the random boundary has a significant effect on the reflected pulse. We determine from first principles when this effects amounts to a deterministic pulse deformation. This is known as a pulse stabilization result. The reflected pulse shape is not the same as the emitted one. It is damped, due to scattering at the boundary, and is deformed by dispersion in the waveguide. An example of an application of this result is in inverse problems, where the travel time of reflected pulses at the turning points can be used to determine the geometry of the waveguide.

I will discuss variational problems arising in machine learning and their limits as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying "ground-truth" measure. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. Many machine learning tasks, such as clustering and classification, can be posed as minimizing functionals on such graphs. We consider functionals involving graph cuts and graph laplacians and their limits as the number of data points goes to infinity. In particular we establish for what graph constructions the minimizers of discrete problems converge to a minimizer of a functional defined in the continuum setting. The talk is primarily based on joint work with Nicolas Garcia Trillos, as well as on works with Xavier Bresson, Moritz Gerlach, Matthias Hein, Thomas Laurent, James von Brecht and Matt Thorpe.

Interpolatory matrix factorizations provide alternatives to the singular value decomposition (SVD) for obtaining low-rank approximations; this class includes the CUR factorization, where the C and R matrices are subsets of columns and rows of the target matrix. While interpolatory approximations lack the SVD's optimality, their ingredients are easier to interpret than singular vectors: since they are copied from the matrix itself, they inherit the data's key properties (e.g., nonnegative/integer values, sparsity, etc.). We shall provide an overview of these approximate factorizations, describe how they can be analyzed using interpolatory projectors, and introduce a new method for their construction based on the Discrete Empirical Interpolation Method (DEIM). (This talk describes joint work with Dan Sorensen (Rice).)

A common question arising in the study of graphs is how to partition nodes into well-connected clusters. One standard methodology is known as spectral clustering and utilizes an eigenvector embedding as a staring point for clustering the nodes. Given that embedding, we present a new algorithm for spectral clustering based on a column-pivoted QR factorization. Our method is simple to implement, direct, scalable, and requires no initial guess. We also provide theoretical justification for our algorithm and experimentally demonstrate that its performance tracks recent information theoretic bounds for exact recovery in the stochastic block model. Algorithmically, the origins of this work are in methods for building a localized basis for Kohn-Sham orbitals, and we briefly discuss those connections.

The incompressible Euler equation of fluid mechanics has been derived in 1755. It is one of the central equations of applied analysis, yet due to its nonlinearity and non-locality many fundamental properties of solutions remain poorly understood. In particular, the global regularity vs finite time blow up question for incompressible three dimensional Euler equation remains open. In two dimensions, it has been known since 1930s that solutions to Euler equation with smooth initial data are globally regular. The best available upper bound on the growth of derivatives of the solution has been double exponential in time. I will describe a construction showing that such fast generation of small scales can actually happen, so that the double exponential bound is qualitatively sharp. This work has been motivated by numerical experiments due to Hou and Luo who propose a new scenario for singularity formation in solutions of 3D Euler equation. The scenario is axi-symmetric. The geometry of the scenario is related to the geometry of 2D Euler double exponential growth example and involves hyperbolic points of the flow located at the boundary of the domain. If time permits, I will discuss some recent attempts to gain insight into the three-dimensional fluid behavior in this scenario.

Exploring data structures (e.g, periodicity, sparsity, low-rankness) is a universal method in designing fast algorithms in scientific computing. In the first part of this talk, I will show how this idea is applied to the analysis of oscillatory data in applied harmonic analysis. These fast algorithms have been applied to data analysis ranging from materials science, medicine, and art. In the second part, I will discuss how this idea works in some basic numerical linear algebra routines like matrix multiplications and decompositions, with an emphasis in electronic structure calculation.

I review some works on modeling and the mathematical theory for dilute suspensions of rigid rods. Such problems appear in modeling sedimentation of suspensions of particles. Similar in spirit models are also used for modeling swimming micro-organisms. Here, we focus on a class of models introduced by Doi and describing suspensions of rod¨Clike molecules in a solvent fluid. They couple a microscopic Fokker-Planck type equation for the probability distribution of rod orientations to a macroscopic Stokes flow. One objective is to compare such models with traditional models used in macoscopic viscoelasticity as the well known Oldroyd model. In particular: For the problem of sedimenting rods under the influence of gravity we discuss the instability of the quiescent flow and the derivation of effective equations describing the collective response. We derive two such effective theories: (i) One ammounts to a classical diffusive limit and produces a Keller-Segel type of model. (ii) A second approach involves the derivation of a moment closure theory and the approximation of moments via a quasi-dynamic approximation. This produces a model that belongs to the class of flux-limited Keller-Segel systems. The two theories are compared numerically with the kinetic equation. (joint work with Christiane Helzel, Univ. Duesseldorf).

I will present work from these two papers: 1) Regulating Greed Over Time. Stefano Traca and Cynthia Rudin. 2015 Finalist for 2015 IBM Service Science Best Student Paper Award 2) Prediction Uncertainty and Optimal Experimental Design for Learning Dynamical Systems. Chaos, 2016. Benjamin Letham, Portia A. Letham, Cynthia Rudin, and Edward Browne.
There is an important aspect of practical recommender systems that we noticed while competing in the ICML Exploration-Exploitation 3 data mining competition. The goal of the competition was to build a better recommender system for Yahoo!'s Front Page, which provides personalized new article recommendations. The main strategy we used was to carefully control the balance between exploiting good articles and exploring new ones in the multi-armed bandit setting. This strategy was based on our observation that there were clear trends over time in the click-through-rates of the articles. At certain times, we should explore new articles more often, and at certain times, we should reduce exploration and just show the best articles available. This led to dramatic performance improvements.
As it turns out, the observation we made in the Yahoo! data is in fact pervasive in settings where recommender systems are currently used. This observation is simply that certain times are more important than others for correct recommendations to be made. This affects the way exploration and exploitation (greed) should change in our algorithms over time. We thus formalize a setting where regulating greed over time can be provably beneficial. This is captured through regret bounds and leads to principled algorithms. The end result is a framework for bandit-style recommender systems in which certain times are more important than others for making a correct decision.
If time permits I will discuss work on measuring uncertainty in parameter estimation for dynamical systems. I will present "prediction deviation," a new metric of uncertainty that determines the extent to which observed data have constrained the model's predictions. This is accomplished by solving an optimization problem that searches for a pair of models that each provide a good fit for the observed data, yet have maximally different predictions. We develop a method for estimating a priori the impact that additional experiments would have on the prediction deviation, allowing the experimenter to design a set of experiments that would most reduce uncertainty.

The relatively recent introduction of viscosity solutions and the Barles-Souganidis convergence framework have allowed for considerable progress in the numerical solution of fully nonlinear elliptic equations. Convergent, wide-stencil finite difference methods now exist for a variety of problems. However, these schemes are defined only on uniform Cartesian meshes over a rectangular domain. We describe a framework for constructing convergent meshfree finite difference approximations for a class of nonlinear elliptic operators. These approximations are defined on unstructured point clouds, which allows for computation on non-uniform meshes and complicated geometries. Because the schemes are monotone, they fit within the Barles-Souganidis convergence framework and can serve as a foundation for higher-order filtered methods. We present computational results for several examples including problems posed on random point clouds, computation of convex envelopes, obstacle problems, non-continuous surfaces of prescribed Gaussian curvature, and Monge-Ampere equations arising in optimal transportation.

Epitaxial growth is a process in which a thin film is grown above a much thicker substrate. In the simplest case, no deposition is considered, and all the interactions are assumed to be purely elastic. However, since the film may potentially have different rigidity constant from the substate, such growth leads to a nonuniform film thickness. The equations governing epitaxial growth are high order (generally fourth order), nonlocal, and highly nonlinear. In this talk I will present some recent results about the regularity of solutions to several equations arising from epitaxial growth. Joint work with I. Fonseca and G.Leoni.

The beautiful work of Borgs, Chayes, Lovasz, Sos, Szegedy, Vesztergombi, and many others on dense graph limits has received quite a bit of attention in pure math as well as statistics and machine learning. In this talk we will review some of the previous work on dense graph limits and then present recent work on providing a more robust mathematical framework for proving the statistical consistency of graph partitioning algorithms such as spectral clustering. A striking feature of our approach is that it is model-free, compared to the popular iid paradigm. Our results are thus broadly applicable in real-world settings, where it is notoriously difficult to obtain relevant models for network data, and observations are not independent. At the end, I will discuss implications for how mathematical foundations can be developed for other modern data analysis techniques. This is joint work with Dominique Guillot, Apoorva Khare, and Bala Rajaratnam. Preprint available at https://arxiv.org/abs/1608.03860.

We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X_1, ...,X_n are i.i.d. standard Gaussian random vectors in R^p, and f is a real-valued function. The eigenvalue distribution of these kernel random matrices is studied in the "large p, large n" regime. It is shown that with suitable normalization the spectral density converges weakly, and we identify the limit. Our analysis applies as long as the rescaled kernel function is generic, and particularly, this includes non-smooth functions, e.g. Heaviside step function. The limiting densities "interpolate" between the Marcenko-Pastur density and the semi-circle density.

If you put sand on a metal plate and start inducing vibrations with a violin bow, the sand jumps around and arranges itself in the most beautiful patterns - this used to be a circus trick in the late 18th century: Napoleon was a big fan and put a prize on giving the best mathematical explanation. Today we know that the sand moves to lines where a certain Laplacian eigenfunction vanishes but these remain mysterious. I will show pictures of sand and demonstrate a new approach: the key ingredient is to make the elliptic equation parabolic and then work with two different interpretations of the heat equation at the same time. If time allows, I will sketch another application of this philosophy to localization phenomena for Schroedinger operators.

A neuron receives thousands of synaptic inputs from other neurons and integrates them to process information. Many experimental results demonstrate this integration could be highly nonlinear, yet few theoretical analyses have been performed to obtain a precise quantitative characterization. Based on asymptotic analysis of an idealized cable model, we derive a bilinear spatiotemporal integration rule for a pair of time-dependent synaptic inputs. Note that the above rule is obtained from idealized models. However, we have confirmed this rule both in simulations of a realistic pyramidal neuron model and in electrophysiological experiments of rat hippocampal CA1 neurons. Our results demonstrate that the integration of multiple synaptic inputs can be decomposed into the sum of all possible pairwise integration with each paired integration obeying a bilinear rule.

I will present several results on the boundedness of maximal operators and Hilbert transforms along variable curves and surfaces, in dimension two or higher. Connections to the circular maximal operator, and the polynomial Carleson operator will also be discussed.

I’ll discuss a geometric approach to graph partitioning where the optimality criterion is given by the sum of the first Laplace-Dirichlet eigenvalues of the partition components. This eigenvalue optimization problem can be solved by a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective. This partitioning method compares well to state-of-the-art approaches on a variety of graphs constructed from manifold discretizations, synthetic data, the MNIST handwritten digit dataset, and images. I'll present a consistency result for geometric graphs, stating convergence of graph partitions to an appropriate continuum partition.

Our knowledge of the long-time behavior of 2d inviscid flows is quite limited. There are some appealing conjectures based on ideas in Statistical Mechanics, but they appear to be beyond reach of the current methods. We will discuss some partial results concerning the dynamics, as well as some results for variational problems to which the Statistical Mechanics methods lead.

In this talk we consider the existence of suitable weak solutions for a quantum energy-transport model for semiconductors. The model is formally derived from the quantum hydrodynamic model by J\"{u}ngel and Mili\v{s}i\'{c} (Nonlinear Anal.: Real World Appl., 12(2011), pp. 1033-1046). It consists of a fourth-order nonlinear parabolic equation for the electron density, an elliptic equation for the electron temperature, and the Poisson equation for the electric potential. Our solution is global in the time variable, while the space variables lie in a bounded Lipschitz domain with a mixed boundary condition. The existence proof is based upon a carefully-constructed approximation scheme which generates a sequence of positive approximate solutions. These solutions are shown to be so regular that they can be used to form a variety of test functions , from which we can derive enough a prior estimates to justify passing to the limit in the approximate problems.

We will discuss recent approximations to boundary value problems to non-linear systems of Boltzmann or Landau (for Coulombic interactions) equations coupled to the Poisson equation. The proposed approximation methods involve hybrid schemes of spectral and Galerkin type, were conservation of flow invariants are achieved by a constrain minimization problem. We will discuss some analytical and computational issues related to these approximations.