Department of Mathematics, University of California Los Angeles
Geometric Scattering on Measure Spaces
Geometric Deep Learning is an emerging field of research that aims to extend the success of convolutional neural networks (CNNs) to data with non-Euclidean geometric structure. Despite being in its relative infancy, this field has already found great success in many applications such as recommender systems, computer graphics, and traffic navigation. In order to improve our understanding of the networks used in this new field, several works have proposed novel versions of the scattering transform, a wavelet-based model of CNNs for graphs, manifolds, and more general measure spaces. In a similar spirit to the original Euclidean scattering transform, these geometric scattering transforms provide a mathematically rigorous framework for understanding the stability and invariance of the networks used in geometric deep learning.Additionally, they also have many interesting applications such as drug discovery, solving combinatorial optimization problems, and predicting patient outcomes from single-cell data. In particular, motivated by these applications to single-cell data, I will also discuss recent work proposing a diffusion maps style algorithm with quantitative convergence guarantees for implementing the manifold scattering transform from finitely many samples of an unknown manifold.
Department of Applied Mathematics, University of Colorado at Boulder
Direct parallel solvers for a variable coefficient Helmholtz equation applied to 3D wave propagation problems.
We present an efficient parallel solver for the linear system that arises from the Hierarchical Poincare-Steklov (HPS)
discretization of three dimensional variable coefficient Helmholtz problems. We tackle problems of 100 wavelengths in
each direction that require more than a billion unknowns to achieve approximately 4 digits of accuracy in less than 20 minutes.
Additionally, we show results on a parallel Direct solver for the same problem and explore the potential of both techniques
for inverse problems. This work was funded by Total Energies and NSF.
Department of Mathematics, University of Colorado Denver
Sandia National Labs
Data-Consistent Inversion: A Collaborative Presentation
(Brief Notes: We utilize Jupyter notebooks to re-create some of our published results and also build a "computational
intuition" for the ideas presented. These materials will be made available after the presentation. The presentation is
collaborative between CSU Alumni Drs. Tim Wildey (Sandia National Laboratories) and Troy Butler (CU Denver). The speakers
have a strong history of collaboration since graduating from CSU. The presentation is organized around an evolving set of
research questions that provide insight into how collaborative research can be sustained over years.)
Models are useful for simulating key processes and generating significant amounts of (simulated) data on quantities of
interest (QoI) extracted from the model solution. This simulated data can be compared directly to observable data to
address many important questions in scientific modeling. However, many key characteristics governing system behavior
described as input parameters in the model remain hidden to direct observation. Thus, scientific inference fundamentally
depends on the formulation and solution of a stochastic inverse problem (SIP) to describe sets of probable model
Statistical Bayesian inference is the most common approach solving the SIP using both data and an assumed error model on
the QoI to construct posterior distributions of model inputs and model discrepancies. This approach has proven effective at
quantifying uncertainties that are often best categorized as epistemic (i.e., reducible) in nature. We have recently
developed an alternative framework based on the measure-theoretic principles to solve the SIP when uncertainties are
categorized as aleatoric. We prove that this approach produces a distribution that is consistent in the sense that its
push-forward through the QoI map will match the distribution on the observable data, i.e., we say that this distribution is
consistent with the model and the data. This method is therefore referred to as data-consistent inversion. Our approach
only requires approximating the push-forward probability density of an initial density, which is fundamentally a forward
propagation of uncertainty. We briefly summarize this approach including existence, uniqueness, and stability of solutions.
A comparison to statistical Bayesian inference is also provided.
Motivated by computationally expensive models, we then discuss the impact of using approximate models to approximate the
QoI on the construction of the push-forward of the initial density. We then outline, at a high-level, the basic theoretical
argument of convergence of the push-forward of the initial density using a generalized version of the Arzela-Ascoli theorem
to prove a converse of Scheffe's theorem and discuss rates of convergence.
Along the way, we also provide brief highlights from some of the other research directions that Drs. Wildey and Butler have
undertaken, both together and with other collaborators, that are rooted in the above joint work.
Department of Mathematics and Statistics, University of Nevado, Reno
Deriving and analyzing ODE models using tools and concepts from stochastic processes via the generalized linear chain trick
ODE models are ubiquitous in scientific applications, and are a cornerstone of the field of dynamical systems. Often, in applications, ODE models are derived using ``rule of thumb" as this is less laborious for practitioners than deriving a mean field deterministic model from an explicitly stochastic model. This can lead to oversimplified model assumptions and model biases. One way modelers have attempted to address this issue is to use classical linear chain trick (LCT), which I will introduce in this talk. I will also present our generalized linear chain trick (GLCT) which allows modelers to efficiently derive systems of ODEs from first principles when framed using continuous time Markov chains. This technique also provides modelers with a framework that offers some additional advantages for interpreting analytical results for such models.
I will illustrate the utility of the GLCT using an example application of this technique to derive, and then find reproduction numbers for, a family of generalized SEIRS models with an arbitrary number of state variables. Reproduction numbers, like the basic reproduction number $R_0$, play an important role in the analysis and application of dynamic models of contagion spread (and parallels exist elsewhere, e.g., in multispecies ecological models). One difficulty in deriving these quantities is that they typically are computed on a model-by-model basis, since it is impractical to obtain and interpret general reproduction number expressions applicable to a family of related models. This is especially true if these models are of different dimensions (i.e., differing numbers of state variables). I will show how to find general reproduction number expressions for such model families using the next generation operator approach in conjunction with the GLCT, and how the GLCT draw insights from these results by leveraging theory and intuition from continuous time Markov chains (CTMCs) and their absorption time distributions (i.e., phase-type probability distributions).
These results highlight the utility of the GLCT for the derivation and analysis of mean field ODE models, especially when used in conjunction with Markov chain theory and associated phase-type distributions.
Department of Mathematics, The Ohio State University
Machine learning algorithms are designed for data in Euclidean space. When naively representing data in a Euclidean space V, there is often a nontrivial group G of isometries such that different members of a common G-orbit represent the same data point. To properly model such data, we want to map the set V/G of orbits into Euclidean space in a way that is bilipschitz in the quotient metric.
In this talk, we have some good news and some bad news. The bad news is G needs to be pretty special for there to exist a polynomial invariant that is bilipschitz, and so we need to move beyond classical invariant theory to solve our problem. The good news is we can take inspiration from an inverse problem called phase retrieval to find a large and flexible class of bilipschitz invariants that we call max filter banks. We discuss how max filter banks perform in theory and in practice, and we conclude with several open problems.
Department of Mathematics, University of Texas at Arlington
Optimal Transport Based Manifold Learning
We will discuss the use of optimal transport in the setting of nonlinear dimensionality reduction and applications to image data. We illustrate the idea with
an algorithm called Wasserstein Isometric Mapping (Wassmap) which works for data that can be viewed as a set of probability measures in Wasserstein space.
The algorithm provides a low-dimensional, approximately isometric embedding. We show that the algorithm is able to exactly recover parameters of some
image manifolds including those generated by translations or dilations of a fixed generating measure. We will discuss computational speedups to the
algorithm such as use of linearized optimal transport or the Nystrom method. Testing of the proposed algorithms on various image data
manifolds show that Wassmap yields good embeddings compared with other global and local techniques.