Seminars and Colloquia by Series

The Connections Between Discrete Geometric Mechanics, Information Geometry and Machine Learning

Series
School of Mathematics Colloquium
Time
Thursday, March 5, 2020 - 11:00 for
Location
Speaker
Melvin LeokUCSD

Please Note: Melvin Leok is a professor in the Department of Mathematics at the University of California, San Diego. His research interests are in computational geometric mechanics, computational geometric control theory, discrete geometry, and structure-preserving numerical schemes, and particularly how these subjects relate to systems with symmetry. He received his Ph.D. in 2004 from the California Institute of Technology in Control and Dynamical Systems under the direction of Jerrold Marsden. He is a three-time NAS Kavli Frontiers of Science Fellow, and has received the NSF Faculty Early Career Development (CAREER) award, the SciCADE New Talent Prize, the SIAM Student Paper Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis. He has given plenary talks at the Society for Natural Philosophy, Foundations of Computational Mathematics, NUMDIFF, and the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control. He serves on the editorial boards of the Journal of Nonlinear Science, the Journal of Geometric Mechanics, and the Journal of Computational Dynamics, and has served on the editorial boards of the SIAM Journal on Control and Optimization, and the LMS Journal of Computation and Mathematics.

Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-$h$ flow map of a Hamiltonian system on the space of probability distributions.

The A. D. Aleksandrov problem of existence of convex hypersurfaces in Space with given Integral Gaussian curvature and optimal transport on the sphere

Series
High Dimensional Seminar
Time
Wednesday, March 4, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Vladimir OlikerEmory University

In his book Convex Polyhedra, ch. 7 (end of subsection 2) A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of convex polytopes with given geometric data. As an example of a geometric problem in which variational approach was successfully applied, Aleksandrov quotes the Minkowski problem. He also mentions the Weyl problem of isometric embedding for which a variational approach was proposed (but not fully developed and not completed) by W. Blashke and G. Herglotz. The first goal of this talk is to give a variational formulation and solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gaussian curvature (also posed by Aleksandrov who solved it using topological methods). The second goal of this talk is to show that in variational form the Aleksandrov problem is closely connected to the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.

Stable phase retrieval for infinite dimensional subspaces of L_2(R)

Series
Analysis Seminar
Time
Wednesday, March 4, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel FreemanSt. Louis University

 The problem of phase retrieval for a set of functions $H$ can be thought of as being able to identify a function $f\in H$ or $-f\in H$ from the absolute value $|f|$.  Phase retrieval for a set of functions is called stable if when $|f|$ and $|g|$ are close then $f$ is proportionally close to $g$ or $-g$.  That is, we say that a set $H\subseteq L_2({\mathbb R})$ does stable phase retrieval if there exists a constant $C>0$ so that
$$\min\big(\big\|f-g\big\|_{L_2({\mathbb R})},\big\|f+g\big\|_{L_2({\mathbb R})}\big)\leq C \big\| |f|-|g| \big\|_{L_2({\mathbb R})} \qquad\textrm{ for all }f,g\in H.
$$
 It is known that phase retrieval for finite dimensional spaces is always stable.  On the other hand, phase retrieval for infinite dimensional spaces using a frame or a continuous frame is always unstable.  We prove that there exist infinite dimensional subspaces of $L_2({\mathbb R})$ which do stable phase retrieval.  This is joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman.

Optimization over the Diffeomorphism Group Using Riemannian BFGS with Application

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, March 4, 2020 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Darshan Bryner Naval Surface Warfare Center, Panama City Division

Please Note: This is a part of IEEE Signal Processing Society Lecture Series, organized by Dr. Alessio Medda (alessiomedda@ieee.org). PLEASE RSVP to https://events.vtools.ieee.org/m/222947

The set of diffeomorphisms of the unit interval, or “warping functions,” plays an important role in many in functional data analysis applications. Most prominently, the problem of registering, or aligning, pairs of functions depends on solving for an element of the diffeomorphism group that, when applied to one function, optimally aligns it to the other.
The registration problem is posed as the unconstrained minimization of a cost function over the infinite dimensional diffeomorphism function space. We make use of its well-known Riemannian geometry to implement an efficient, limited memory Riemannian BFGS optimization scheme. We compare performance and results to the benchmark algorithm, Dynamic Programming, on several functional datasets. Additionally, we apply our methodology to the problem of non-parametric density estimation and compare to the benchmark performance of MATLAB’s built-in kernel density estimator ‘ksdensity’.

Modeling Phytoplankton Blooms with a Reaction-Diffusion Predator-Prey Model

Series
Mathematical Biology Seminar
Time
Wednesday, March 4, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Seth CowallMercer University

Phytoplankton are the base of the marine food web. They are also responsible for much of the oxygen we breathe, and they remove carbon dioxide from the atmosphere. The mechanisms that govern the timing of seasonal phytoplankton blooms is one of the most debated topics in oceanography. Here, we present a macroscale plankton ecology model consisting of coupled, nonlinear reaction-diffusion equations with spatially and temporally changing coefficients to offer insight into the causes of phytoplankton blooms. This model simulates biological interactions between nutrients, phytoplankton and zooplankton. It also incorporates seasonally varying solar radiation, diffusion and depth of the ocean’s upper mixed layer because of their impact on phytoplankton growth. The model is analyzed using seasonal oceanic data with the goals of understanding the model’s dependence on its parameters and of understanding seasonal changes in plankton biomass. A study of varying parameter values and the resulting effects on the solutions, the stability of the steady-states, and the timing of phytoplankton blooms is carried out. The model’s simulated blooms result from a temporary attraction to one of the model’s steady-states.

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