September 15th

TBD

September 22th

TBD

September 29th

TBD

October 6th
TBD

October 13th

Aaron Zoll

• Title: 

  • A complete characterization of optimal methods for convex Lipschitz minimization

• Abstract: TBD

October 20th

Anaïs Després

• Title: TBD

• Abstract: TBD
  
  

October 27th

Ian McPherson

• Title: 

  • Part 1 (Conference Talk Practice Generalizing Previous Work): Computing Fr\'{e}chet Means on Unknown Submanifolds of Riemannian Manifolds

  • Part 2 (Fun Math Towards New Applications): Friendly Introduction to Second Order Analysis of Wasserstein and Modern ISOMAP

• Abstract: 

  • Part 1: We study computation of the empirical Fréchet mean of data supported on an unknown compact Riemannian manifold $\mathcal{M}_{1} \subseteq \mathcal{M}_{2}$. Given reference points $\{z_{r}\}_{r=1}^{N}$ from $\mathcal{M}_{1}$, we minimize $f(x) = \frac{1}{2N} \sum_{r=1}^{N} d_{\mathcal{M}_{1}}^{2}(x,z_{i})$ over $\mathcal{M}_{1}$, without explicit access to $\mathcal{M}_{1}$. Assuming access only to local sampler oracles on $\mathcal{M}_{1}$, we propose a graph-based geodesic interpolation method that iteratively refines a local nearest-neighborhood graph and interpolates along graph geodesics. We interpret the algorithm as biased Riemannian SGD, with bias controlled by local metric equivalences between graph and manifold distances. As part of the analysis, we extend biased SGD convergence arguments to the Riemannian setting for structured stochastic gradient estimators. We also extend ISOMAP-type guarantees to embedded submanifolds of general Riemannian ambient spaces. Under geometric assumptions, we obtain an $\mathcal{O}(1/T)$-type expected convergence guarantee where sampling $\widetilde{\mathcal{O}}(T^{4d+3})$ points over an algorithm run suffices with $d = \dim(\mathcal{M}_{1})$. These results provide a framework for provably computing constrained Fréchet means on unknown submanifolds using only graph-based approximations of intrinsic distances. Lastly, we apply this method for problems in molecular dynamics and optimal transport.

  • Part 2: Within this portion of the talk, I will explain basic geometric concepts (second order objects: sectional curvature, Riemann curvature tensor), highlight some basic computations that generalize these to the Wasserstein setting (reducing a geometric calculation to controlling a potential function), and highlight how this is useful in for obtain local metric equivalence bounds between portions of W_{2} and graph constructions (for a submanifold in Wasserstein space). Then, will show how this plugs into a Lemma (or Theorem depending on who you ask) of the previous talk, allowing for more natural notions of submanifolds of Wasserstein than what is currently in the literature.

November 3rd

Merrick Ohata

• Title: TBD

• Abstract: TBD

November 10th

Yuxin Ma

• Title: Any-dimensional learning and size generalization

• Abstract:
  Many modern architectures — including DeepSets, graph neural networks, and transformers — are designed to process inputs of arbitrary size. These models have a desirable feature: they can be trained on small inputs and deployed on larger ones. Yet the theoretical conditions under which such size generalization succeeds or fails remain poorly understood.

In this talk, we reveal a fundamental mechanism behind size generalization: many of these models encode a hidden inductive bias about relationships between objects of different sizes. Specifically, they parameterize functions that are continuous with respect to a pseudometric on the space of inputs across all sizes. When this pseudometric captures equivalences and proximity between objects of different sizes, performance transfers between objects that differ in size but are close under the pseudometric. This perspective yields an approximation and generalization theory in the any-dimensional setting, and lets us determine precisely for which tasks and data distributions a model trained on small inputs will generalize to large ones. It also inspires new architectures, and modifications of existing ones, that achieve size generalization on tasks where standard models fail. If time permits, we will also present applications to weight space learning.

November 17th
TBD

November 24th
TBD

December 1st

Barbara Fiedorowicz

• Title: TBD

• Abstract: TBD

December 8th

TBD