We run weekly seminars to present our research work and/or teach our peers interesting (applied) math and stats!
September 15th
Drew Henrichsen
Title:
How much data do you need? An approach to uncertainty quantification using ordinary classifiers.
Abstract:
Class-membership probability is a central object for uncertainty quantification for classification. Many approaches begin by constructing a probabilistic classifier and using its scores to estimate or bound this quantity. This paper studies whether informative confidence intervals can instead be obtained from a finite family of ordinary classifiers by reasoning on positive-probability domains of the feature space. We show that weighted ordinary classifiers induce domains of the feature space that naturally separate regions of higher and lower class-membership probability. For weighted Bayes classifiers, these domains yield almost-sure deterministic bounds on the class-membership probability. For classifiers whose classification error is close to the Bayes-optimal error, i.e. classifiers with small Bayes excess risk, we derive finite-sample and asymptotic confidence intervals whose width depends explicitly on the Bayes excess risk. These results provide a framework that links uncertainty quantification directly to Bayes excess risk and, consequently, to the quantity of available training data.
September 22th
Title: TBD
Abstract: TBD
September 29th
Title: TBD
Abstract: TBD
October 6th
Yue Wu
Title:
A parameter-free restart scheme
Abstract: TBD
October 13th
Title:
A complete characterization of optimal methods for convex Lipschitz minimization
Abstract: TBD
October 20th
Anaïs Després
Title: TBD
Abstract: TBD
October 27th
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
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
Abdel Ghani Labassi
Title: TBD
Abstract: TBD
Chia-Cheng (Allen) Hao
Title: TBD
Abstract: TBD
November 24th
Thanksgiving Break
December 1st
Barbara Fiedorowicz
Title: TBD
Abstract: TBD
Dilum Fernando
Title: TBD
Abstract: TBD
December 8th
Title: TBD
Abstract: TBD
Organizer(s): Chia-Cheng (Allen) Hao, Sehee Park