Upcoming MAMS Colloquium Series

FALL 2022

Friday 10/7— 3:15-4:15pm at Yost 306
Title: Scaling and Scalability: Accelerating Ill-conditioned Low-rank Estimation via Scaled Gradient Descent
Speaker: Yuejie Chi, Carnegie Melon University (Host: Weihong Guo)
Abstract: Many problems encountered in sensing and imaging can be formulated as estimating
a low-rank object from incomplete, and possibly corrupted, linear measurements; prominent
examples include matrix completion and tensor completion. Through the lens of matrix and
tensor factorization, one of the most popular approaches is to employ simple iterative
algorithms such as gradient descent to recover the low-rank factors directly, which allow for
small memory and computation footprints. However, the convergence rate of gradient descent
depends linearly, and sometimes even quadratically, on the condition number of the low-rank
object, and therefore, slows down painstakingly when the problem is ill-conditioned. This talk
introduces a new algorithmic approach, dubbed scaled gradient descent (ScaledGD), that
provably converges linearly at a constant rate independent of the condition number of the low-
rank object, while maintaining the low per-iteration cost of gradient descent. In addition, a
nonsmooth variant of ScaledGD provides further robustness to corruptions by optimizing the
least absolute deviation loss. In total, ScaledGD highlights the power of appropriate
preconditioning in accelerating nonconvex statistical estimation, where the iteration-varying
preconditioners promote desirable invariance properties of the trajectory with respect to the
symmetry in low-rank factorization.

Friday 10/21 — 3:15-4:15pm at Yost 306
Title: What does this pathway do?  Identifying control ensembles for decision-making and other
downstream effects
Speaker: Jonathan Rubin, University of Pittsburgh (Host: Peter Thomas)
Abstract: This talk concerns topics in mathematical neuroscience but will not assume any
specific knowledge of neuroscience.  The basal ganglia (BG) is a brain area that serves as a hub
for the reward signal dopamine and is believed to be involved in decision-making and action
selection.  But the BG circuitry features various feedback pathways and loops, which complicate
efforts to derive a detailed understanding of its function.  In the first section of my talk, I will
introduce this circuitry and will also present joint work with Tim Verstynen, Cati Vich and our
trainees, which applies statistical tools to the dynamics of a BG network model to predict how
different components contribute to the decision-making process.  In the second section of my
talk, I will present work with Ryan Phillips and our neuroscience collaborator Aryn Gittis in
which we model the integration of two signaling pathways by BG output neurons.  Our modeling
explains surprising results seen experimentally and suggests how slow dynamic processes in the
system can contribute to synchronization and oscillations and can affect action selection, in ways
that may be related to Parkinson’s disease.

Friday 11/04— 3:15-4:15pm at Yost 306
Title: Direct and inverse problems for elastic dislocations in geophysics
Speaker: Anna Mazzucato, Penn State (Host: Erkki Somersalo)
Abstract: I will discuss a model for dislocations in an elastic medium, modeling faults in
the Earth’s crust. The direct problem consists in solving a non-standard boundary
value/interface problem for
in-homogeneous, possibly anisotropic linear elasticity with piecewise-Lipschitz
coefficients, for which we prove well-posedness and obtain an integral representation
for the solution. The non-linear inverse problem consists in determining the fault surface
and slip vector from displacement measurements made at the surface. In applications,
these come from GPS arrays and satellite interferometry. We establish uniqueness for
the inverse problem under some geometric conditions, using unique continuation results
for systems. We also derive a shape derivative formula for an iterative reconstruction
algorithm. This is joint work with Andrea Aspri (Milan University, Italy), Elena Beretta
(NYU-Abu Dhabi), and Maarten de Hoop (Rice).