| Date: | September 16, 2021 |
| Time: | 11:00am |
| Location: | BLOC 302 |
| Speaker: | Keaton Hamm, UT Arlington |
| Title: | Optimal transport methods in nonlinear dimensionality reduction |
| Abstract: | The manifold hypothesis, that high-dimensional data lives on or near a low-dimensional embedded manifold, is ubiquitous in machine learning theory and practice. For imaging data, the data appears as vectors in high-dimensional Euclidean space, where the vectors are acquired from some imaging operator mapping a functional space, such as L2, to Euclidean space. It is unclear that Euclidean distance between the image vectors contains sufficient semantic meaning to understand the structure of the functional data manifold. We consider treatment of the functional data as a set of probability measures, and use pairwise Wasserstein distances to compute similarity. We then utilize these distances in the ISOMAP algorithm for nonlinear dimensionality reduction. We show how the proposed algorithm, WassMap, recovers translational and dilational functional manifolds up to global isometry. We also show further experiments on synthetic data which illustrate the methods success on a variety of other kinds of image manifolds. |
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| Date: | September 24, 2021 |
| Time: | 09:00am |
| Location: | BLOC 302 |
| Speaker: | Audrey Fovelle, Université Bourgogne Franche-Comté |
| Title: | Hamming graphs and concentration properties in Banach spaces |
| Abstract: | In 2008, Kalton and Randrianarivony introduced a concentration property for Lipschitz maps defined on Hamming graphs, that every reflexive asymptotically uniformly smooth Banach space $X$ satisfies. This property, that we will note HFCp,d, provides an obstruction to the coarse Lipschitz embedding of certain spaces into X. Later, Lancien, Raja and Causey proved that this result could be extended to quasi-reflexive spaces, by using a weaker concentration property, that we will call HICp,d.
The goal of this talk is to show that these two properties are stable under lp sums of Banach spaces, in order to obtain a non quasi-reflexive space that satisfies property HICp,d. |
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| Date: | December 3, 2021 |
| Time: | 09:00am |
| Location: | Zoom |
| Speaker: | Kasia Wyczesany, Tel Aviv University |
| Title: | On almost Euclidean and well-complemented subspaces of finite-dimensional normed spaces |
| Abstract: | In this talk I will discuss a version of an old question of Vitali Milman about almost Euclidean and well-complemented subspaces. In particular, I will introduce a notion of ' ε-good points ', which allows for a convenient reformulation of the problem. Let (X,||·||X) be a normed space. It turns out that if a linear subspace Y ⊂ X consists entirely of ε-good points then the restriction of the norm ||·||X to Y must be approximately a multiple of the l2 norm and the operator norm of the orthogonal projection onto Y is close to 1. I will present an example of a normed space X of arbitrarily high dimension, whose Banach-Mazur distance from the l2dim X is at most 2, but such that non of its (even two-dimensional) subspaces consists entirely of ε-good points. The talk is based on joint work with Timothy Gowers. |
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