Department of Mathematics, Colorado State University
Title
On the notion of "information" in inverse problems
Abstract
Inverse problems are ones where one would like to reconstruct a
spatially variable function from measurements of a system in which
this function appears as a coefficient or right hand side. Examples
are biomedical imaging and seismic imaging of the earth.
In many inverse problems, practitioners have an intuitive notion of
how much one "knows" about the coefficient in different parts of the
domain -- that is, that there is a spatially variable "amount of
information". For example, in seismic imaging, we can only know about
those parts of the earth that are traversed by seismic waves on their
way from earthquake to receiving seismometer, whereas we can not know
about places not on these raypaths.
Despite the fact that this concept of "information" is intuitive to
understand, there are no accepted quantitative measures of information
in inverse problems. I will present an approach to define such a
spatially variable "information density" and illustrate both how it
can be computed practically, as well as used in applications. The
approach is based on techniques borrowed from Bayesian inverse
problems, and an approximation of the covariance matrix using the
Cramer-Rao bound.
Department of Biomedical Informatics, University of Colorado Denver/Anschutz
Title
Novel Technologies for Fast MRI
Abstract
Magnetic Resonance Imaging (MRI) displays significant contrast between soft tissues and images without any
ionizing radiation. This is especially appealing when imaging fetuses and neonates who are most at
risk for the dangers of ionizing radiation. However, MRI is a slow imaging modality; with conventional
MRI, the scan for a 3D volume would require approximately 10 minutes. Since MRI requires the patient
to remain still during the scan, this limits its applicability to these vulnerable patients.
In this talk, we will discuss several technologies to accelerate MRI including non-rectangular field-of-views,
structured compressed sensing, and deep learning with enforced data consistency. When combined,
these technologies will reduce the scan time for 3D MRI to under 10 seconds, limiting the amount
of motion possible during the scan. This will enable new applications in infant and fetal imaging.
Abstract
The optimal transport (OT) problem seeks the most efficient way to transfer a distribution of mass
from one configuration to another while minimizing an associated transportation cost. This framework
has found diverse applications in machine learning due to its ability to define meaningful distances between
probability distributions, known as Wasserstein distances. However, classical Wasserstein distances can
be computationally expensive, particularly in high-dimensional settings. In this talk, we will
introduce new OT-based metrics designed to mitigate these computational challenges. These metrics retain,
to some extent, desirable properties and qualitative behaviors of the classical OT-distances while
offering significant improvements in efficiency, making them more suitable for large-scale applications.
Title
Interpretable, Explainable, and Adversarial AI: Data Science Buzzwords and You (Mathematicians)
Abstract
Many state-of-the-art methods in machine learning are black boxes which do not allow humans to understand
how decisions are made. In a number of applications, like medicine and atmospheric science, researchers
do not trust such black boxes. Explainable AI can be thought of as attempts to open the black box of
neural networks, while interpretable AI focuses on creating clear boxes. Adversarial attacks are small
perturbations of data that cause a neural network to misclassify the data or act in other undesirable
ways. Such attacks are potentially very dangerous when applied to technology like self-driving cars.
The goal of this talk is to introduce mathematicians to problems they can attack using their favorite
mathematical tools. The mathematical structure of transformers, the powerhouse behind
large language models like ChatGPT, will also be explained.
Abstract
It is often important in data science to identify data that is semantically the same. For example,
if one imaged an object in different orientations, one would typically want a machine learning algorithm
to classify the object in each of the images as being the same.
Motivated by such problems in data science, we study the following questions: (1) Given a Hilbert space
V and a group G of linear isometries, does there exist a bilipschitz embedding of the quotient metric
space V/G into a Hilbert space? (2) What are necessary and sufficient conditions for such embeddings?
(3) Which embeddings minimally distort the metric?
We answer these questions in a variety of settings, and we conclude with several open problems.
Title
Topological deep learning for COVID-19 variants prediction, RNA motif design, and DockQ prediction.
Abstract
In this talk, I will introduce an integration of computational topology and artificial intelligence (AI),
showcasing a novel topological deep-learning approach to forecast future COVID-19 variants, predict possible
RNA motif structures, and enhance peptide-protein complex structures. Specifically, I will first
introduce the capacities of persistent spectral graphs (PSGs), a mathematical approach in Topological
Data Analysis (TDA), for analyzing the intricate topological changes and homotopy shape evolution of
high-dimensional biological data. Next, I will demonstrate how PSG-based representations are integrated
into AI and applied to diverse biological datasets, such as RNA motifs and protein interaction networks, to
accurately predict RNA motif structures, enhance peptide-protein complex prediction, and forecast
COVID-19 variant trends.
