Abstract

Weil-Petersson curves are a class of rectifiable closed curves in the plane, defined as the closure of the smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2006. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, geometric measure theory, probability theory, knot theory, computer vision, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than twenty equivalent conditions. One involves inscribed polygons and can be explained to a calculus student. Another is a strengthening of Peter Jones's traveling salesman condition characterizing rectifiable curves. A third says a curve is Weil-Petersson iff it bounds a minimal surface in hyperbolic 3-space that has finite total curvature. I will discuss these and several other characterizations and sketch why they are all equivalent to each other.

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Abstract

In this talk, we introduce two new aspects of inverse problems formulated as PDE-constrained optimization. Firstly, while current approaches assume deterministic parameters, many real-world problems exhibit stochastic behavior. We present a novel approach that treats the PDE solver as a push-forward map to recover the full distribution of unknown random parameters. We introduce a gradient-flow equation to estimate the ground-truth parameter probability distribution. Secondly, as problem dimensions increase, Monte Carlo methods regain relevance. However, directly applying them to gradient-based PDE-constrained optimization poses challenges due to the product of forward and adjoint solutions involving Dirac deltas. We propose strategies to rescue Monte Carlo methods and make them compatible with gradient-based optimization.

Abstract

Quantum Signal Processing (QSP) is a revolutionary technique that uses a product of unitary matrices to represent polynomials, with numerous applications in quantum computing. In this talk, I will introduce QSP in a fashion that does not require prior knowledge on quantum computing. We introduce optimization-based algorithms that can efficiently find the ""phase factors"" used to represent a given polynomial. We also identify a surprising connection between the smoothness of the target function and the decay properties of a specific branch of the phase factors. Y. Dong, L. Lin, H. Ni, J. Wang, Infinite quantum signal processing [arXiv:2209.10162] J. Wang, Y. Dong, L. Lin, On the energy landscape of symmetric quantum signal processing, Quantum 6, 850, 2022 Y. Dong, X. Meng, K. B. Whaley, L. Lin, Efficient phase factor evaluation in quantum signal processing, Phys. Rev. A 103, 042419, 2021

Abstract

Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.

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Abstract

Wigner-Smith (WS) time delay concepts have been used extensively in quantum mechanics to characterize delays experienced by particles interacting with a potential well. This presentation will formally extend WS time delay theory to Maxwells equations and explores its potential applications in electromagnetics. The WS time delay matrix relates a lossless and reciprocal systems scattering matrix to its frequency derivative and allows for the construction of modes that experience well-defined group delays when interacting with the system. The matrix entries for guiding, scattering, and radiating systems are energy-like overlap integrals of the electric and/or magnetic fields that arise upon excitation of the system via its ports. Numerous applications in electromagnetics will be highlighted, including the characterization of group delays in multiport systems, the description of electromagnetic fields in terms of elementary scattering processes, and the characterization of frequency sensitivities of fields and multiport antenna impedance matrices. Extensions of WS methods towards lossy and dispersive systems will be analyzed as well, and avenues for leveraging WS concepts in computational electromagnetics will be discussed.

Abstract

There are many real-world processes exhibiting fractal growing shapes - from mineral deposition and coral growth to lightning strikes, and in many of them growth is related to diffusion properties. We will discuss two seminal models: Diffusion Limited Aggregation was introduced by Witten and Sanders in 1981 and was generalized to Dielectric Breakdown Model by Niemayer et al shortly afterwards. Numerically they approximate very well a wide range of physical phenomena. However, despite a very simple definition (DLA cluster grows by attaching particles undergoing Brownian motion when they hit the aggregate), very little is understood today, and even less is known rigorously - essentially, only the famous Harry Kesten upper bound on the DLA growth. We will try to show the flavor of these models and present some new results. Based on a joint preprint with Ilya Losev and some further work.

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A chord-arc curve is a locally rectifiable curve satisfying the property that the length of the shortest arc on the curve joining any two points is comparable to the distance between these two points. In this talk, we will introduce the open problem of the connectivity of the manifold of chord arc curves, already mentioned by G. David in his thesis in 1981, and present some recent results that allow us to transfer the connectivity problem to a problem involving the spectrum of a Beurling- type operator on a particular weighted space.

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Abstract

In this talk we show how to learn the underlying geometry of data using multiscale data diffusion, and then combine this with deep learning for prediction and inference in several different settings. First we look at capturing graphs using multiscale diffusion based geometric scattering within neural frameworks. We show how to make such networks end-to-end differentiable in order to learn rich representations spaces from which to classify and generate graphs. We then show how to extend this type of analysis to manifolds, where point-clouds of data can be similarly featurized using cascades of wavelets on data graphs to create a manifold scattering transform. Next we show how to derive Wasserstein distances between pointclouds of such data using multiscale diffusion distances. FInally we move from static to dynamic optimal transport using neural ODEs in order to learn dynamic trajectories from static snapshot data—a key problem in inference from single cell data. Throughout the talk, we present examples of such techniques being applied to massively high throughput and high dimensional datasets from biology and medicine.

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Abstract

I will discuss avenues in data driven modeling of complex systems (for my group and myself) that my interaction with Raphy Coifman has enabled - from "variable free" latent space dynamics twenty years ago to "learning what to learn" and "backward in time" today.

Abstract

To elucidate the basic laws that govern processes around us, scientists ask questions and, often, collect data that will let them answer these questions. The answers typically rely on the solutions to so-called inverse problems, namely algorithms for mapping data to latent variables the scientist can interpret. Physical/statistical models often constrain the latent variables and how they relate to data we acquire. In recent years, deep learning algorithms, largely myopic to these constraints, have become a popular method for solving inverse problems. I will argue that the cost of collecting data in science, the need for interpretability, and the dynamic nature of scientific data make vanilla artificial neural networks (ANNs) unsuitable, at best, in scientific settings. I will argue for sparsity of latent representations as a mild form of inductive biase for DNN models of scientific data, which can let us enjoy both the interpretability of traditional methods for solving inverse problems, and the expressive power of ANNs. I will demonstrate that ANNs designed in this fashion make powerful interpretable tools for elucidating the principles of neural computation, and for solving a wide range of inverse problems in imaging, Physics, and beyond, particularly in the data-scare/limited regime that characterizes many scientific settings.

Abstract

Recent theoretical work has shown that under certain conditions, massively overparameterized neural networks are equivalent to kernel regressors with a family of kernels called Neural Tangent Kernels (NTKs). My work in this subject aims to better understand the properties of NTK for various network architectures and relate them to the inductive bias of real neural networks. In particular, I will argue that for input data distributed uniformly on the sphere NTK favors low-frequency predictions over high-frequency ones, potentially explaining why overparameterized networks can generalize even when they perfectly fit their training data. I will further discuss the behavior of NTK when data is distributed nonuniformly and show that NTK (with ReLU activation) is tightly related to the classical Laplace kernel, which has a simple closed-form. Finally, I will discuss our analysis of NTK for convolutional networks, which indicates that these networks are biased toward learning low frequency target functions with any higher frequencies concentrated in local regions. Overall, our results suggest that much insight about neural networks can be obtained from the analysis of NTK.

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Abstract

We consider several estimation problems for stochastic dynamical systems from observations of trajectories: Let A be a linear dynamical system on a graph G. A and G are unknown, we observe a small number of entries of A, A^2, …, A^T, and we wish to estimate A. We study when this problem is well-posed, introduce an estimator of A based on matrix completion of a low-rank structured block-Hankel matrix, obtain results that capture some of the trade-offs between sampling in space and time, and finally show that this estimator can be constructed by a fast algorithm that provably locally converges quadratically to A. We verify this numerically on a variety of examples [C. Kuemmerle, MM, S. Tang]. We consider nonlinear dynamical systems modeling interacting agents. The laws of interactions between the agents are often simple, e.g. they depend only on a function of pairwise interactions. Given observations along trajectories of the agents, we construct statistically and computationally efficient estimators for the laws of interactions, in a nonparametric fashion, and give conditions guaranteeing the problem is well-posed [F. Lu, MM, J. Feng, P. Martin, J.Miller, S. Tang and M. Zhong]. We consider model reduction of fast-slow high-dimensional stochastic systems with a low-dimensional slow manifold M. The fast modes are not assumed to be small, nor orthogonal to M. Both the dynamics and M are unknown; given access to a black-box simulator from which short bursts of simulations can be obtained, we estimate of the manifold M, an effective stochastic process on M, and a simulator thereof, adapted to the dimension of M, and with time steps dependent on the regularity of the effective process. The estimation may be performed on-the-fly, for efficient exploration. We demonstrate the simulation of paths of the effective dynamics, and estimation of crucial features, including the stationary distribution, metastable states, residence times and transition rates [MM, X.-F. Ye, S. Yang].

Abstract

Manifold learning algorithms aim to map high-dimensional data into lower dimensions while preserving local and global structure. In this talk, I present Low Distortion Local Eigenmaps (LDLE), a bottom-up manifold learning framework that constructs low-distortion local views of a dataset in lower dimensions and registers them to obtain a global embedding. Motivated by Jones, Maggioni, and Schul (2008), LDLE constructs local views by selecting subsets of the global eigenvectors of the graph Laplacian such that they are locally orthogonal. The global embedding is obtained by rigidly aligning these local views, which is solved iteratively. Our global alignment formulation enables tearing manifolds so as to embed them into their intrinsic dimension, including manifolds without boundary and non-orientable manifolds. We define a strong and weak notion of global distortion to evaluate embeddings in low dimensions. We show that Riemannian Gradient Descent (RGD) converges to an embedding with guaranteed low global distortion. Compared to competing manifold learning and data visualization approaches, we demonstrate that LDLE achieves lowest local and global distortion on real and synthetic datasets.

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Abstract

Curvature is one of the fundamental ingredients in differential geometry. It is interesting to think of combinatorial graphs as manifolds and a number of different notions of curvature have been proposed. I will introduce some of the existing ideas and then propose a new notion based on a simple and completely explicit linear system of equations. This notion satisfies a surprisingly large number of desirable properties -- connections to game theory (especially the von Neumann Minimax Theorem) and potential theory will be sketched. I will also sketch some curious related open problems. No prior knowledge of differential geometry (or graphs) is required.

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Abstract

The success of modern supervised machine learning is predicated on the existence of large labeled data sets, but in many domains, it is cost prohibitive to generate a large number of high quality labels. It is more feasible to first collect a large unlabeled data set, then decide to invest in labeling a subset of this data set. This motivates the consideration of label complexity: for a given supervised learning problem and unlabeled data set, what is the minimal number of labels required so that a model fitted using the labeled subset has almost as much predictive power as would a model fitted on the entire data set if all the labels were available? We present algorithmic results on the label complexity of linear regression: given n data points, how many samples must be labeled to obtain a model with near optimal in-sample prediction error? Existing approaches to reducing the label complexity of linear regression--including various approaches from the randomized numerical linear algebra community such as core-sets and the use of iterative algorithms that touch one data point per iteration-- are applicable when a constant factor approximation is acceptable. New approaches are needed to enter the regime where the approximation factor decreases with the size of the data set. In this setting, we provide a polynomial time algorithm that reduces the label complexity by O(sqrt(n)) additively. The algorithm is based on a tight analysis of the regression error incurred by forming a core-set using backward selection.

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Abstract

The talk will describe how randomized algorithms can effectively, accurately, and reliably solve linear algebraic problems that are omnipresent in scientific computing and in data analysis. We will focus on techniques for low rank approximation, since these methods are particularly simple and powerful. The talk will also briefly survey a number of other randomized algorithms for tasks such as solving linear systems, estimating matrix norms, and computing full matrix factorizations.

Abstract

Many important machine learning problems can be formulated as highly constrained convex optimization problems. One important example is metric constrained problems. In this paper, we show that standard optimization techniques can not be used to solve metric constrained problem. To solve such problems, we provide a general active set framework, called Project and Forget, and several variants thereof that use Bregman projections. Project and Forget is a general purpose method that can be used to solve highly constrained convex problems with many (possibly exponentially) constraints. We provide a theoretical analysis of Project and Forget and prove that our algorithms converge to the global optimal solution and have a linear rate of convergence. We demonstrate that using our method, we can solve large problem instances of general weighted correlation clustering, metric nearness, information theoretic metric learning and quadratically regularized optimal transport; in each case, out-performing the state of the art methods with respect to CPU times and problem sizes. Joint work with Rishi Sonthalia (UCLA)

Abstract

Tensor-network ansatz have long been employed to solve the high-dimensional Schrödinger equation, demonstrating linear complexity scaling with respect to dimensionality. Recently, this ansatz has found applications in various machine learning scenarios, including supervised learning and generative modeling, where the data originates from a random process. In this talk, we present a new perspective on randomized linear algebra, showcasing its usage in estimating a density as a tensor-network from i.i.d. samples of a distribution, without the curse of dimensionality, and without the use of optimization techniques. Moreover, we illustrate how this concept can combine the strengths of particle and tensor-network methods for solving high-dimensional PDEs, resulting in enhanced flexibility for both approaches. (Based on joint works with Yian Chen, Jeremy Hoskins, YoonHaeng Hur, Michael Lindsey, Yifan Peng, Miles Stoudenmire, Xun Tang, and Lexing Ying).