2022 Program for Women and Mathematics: The Mathematics of Machine Learning

Mirjeta Pasha, Arizona State University
7:00 pm - 7:20 pm
Title: On Deterministic and Statistical Methods for Large-scale Dynamic Inverse Problems
Abstract: Inverse problems are ubiquitous in many fields of science such as engineering, biology, medical imaging, atmospheric science, and geophysics. Three emerging challenges on obtaining relevant solutions to large-scale and data-intensive inverse problems are ill-posedness of the problem, large dimensionality of the parameters, and the complexity of the model constraints. In this talk we discuss efficient methods for computing solutions to dynamic inverse problems, where both the quantities of interest and the forward operator may change at different time instances. We consider large-scale ill-posed problems that are made more challenging by their dynamic nature and, possibly, by the limited amount of available data per measurement step. In the first part of the talk, to remedy these difficulties, we apply efficient regularization methods that enforce simultaneous regularization in space and time (such as edge enhancement at each time instant and proximity at consecutive time instants) and achieve this with low computational cost and enhanced accuracy [2]. In the remainder of the talk, we focus on designing spatial-temporal Bayesian models for estimating the parameters of linear and nonlinear dynamical inverse problems [1]. Numerical examples from a wide range of applications, such as tomographic reconstruction, image deblurring, and chaotic dynamical systems are used to illustrate the effectiveness of the described approaches.

1. S. Lan, S. Li, and M. Pasha. Bayesian spatiotemporal modeling for inverse problems. in preperation, 2022.

2. M. Pasha, A. K. Saibaba, S. Gazzola, M. I. Espanol, and E. de Sturler. Efficient edge-preserving methods for dynamic inverse problems. arXiv preprint arXiv:2107.05727, 2021.

Kara Ycoubou Djima, Amherst College
7:20 pm - 7:40 pm
Title: Extracting Autism's Biomarkers in Placenta using Multiscale Methods
Abstract: The placenta is the essential organ of maternal-fetal interactions, where nutrient, oxygen, and waste exchange occur. In recent studies, differences in the morphology of the placental chorionic surface vascular network (PCSVN) have been associated with developmental disorders such as autism. This suggests that the PCSVN could potentially serve as a biomarker for the early diagnosis and treatment of autism. Studying PCSVN features in large cohorts requires a reliable and automated mechanism to extract the vascular networks. In this talk, we present a method for PCSVN extraction. Our algorithm builds upon a directional multiscale mathematical framework based on a combination of shearlets and Laplacian eigenmaps and can isolate vessels with high success in high-contrast images such as those produced in CT scans. 

Jamie Haddock, Harvey Mudd College
7:40 pm - 8:00 pm
Title: Tensor Models, Methods, and Medicine
Abstract: There is currently an unprecedented demand for efficient, quantitative, and interpretable methods to study large-scale (often multi-modal) data. One key area of interest is that of tensor decomposition, which seeks to automatically learn latent trends or topics of complex data sets, providing practitioners a view of what is “going on” inside their data. This talk will survey tools for topic modeling on matrix and tensor data.  These tools are of interest across the many fields and industries producing, capturing, and analyzing big data, but are of particular interest in applications where expert supervision is available and often essential (e.g., medicine).  We will describe an application of these methods to medical data; an ongoing application to cardiovascular imaging data.