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:
Structured Self-Supervised Learning with Gaussian Processes: From General Method to Longitudinal Polypharmacy
Abstract:
Many biomedical datasets are high-dimensional, sparsely observed, and governed by meaningful scientific structure, yet outcome labels are often limited or unavailable. Self-supervised learning offers a way to discover useful representations from unlabeled data, but existing approaches frequently depend on artificially constructed positive pairs and provide limited uncertainty quantification. In this talk, I introduce a Gaussian process framework for self-supervised representation learning that uses kernel functions to encode similarity among observations. Gaussian process priors encourage similar observations to have nearby representations, while a generalized Bayesian formulation produces posterior uncertainty that can be propagated to downstream analyses. We discuss connections between the proposed framework, kernel principal component analysis, and neural-network-based self-supervised methods, and demonstrate its performance in classification settings.
We extend this framework to longitudinal medication data, where high dimensionality, irregular observation times, and structured relationships among medications create additional challenges. The resulting two-stage method, LOPEL, first learns visit-level representations using pharmacological relationships encoded by the Anatomical Therapeutic Chemical hierarchy. It then constructs subject-level representations by modeling medication trajectories and using Wasserstein-based similarities to capture both temporal dynamics and uncertainty. Applied to longitudinal cohorts of people with HIV, LOPEL identifies clinically meaningful subgroups that differ in the composition, timing, and progression of medication use. Together, these projects illustrate how scientifically structured probabilistic models can produce informative and uncertainty-aware representations of complex biomedical data.
September 29th
Title:
Spectral Clustering in Very Sparse Graphs
Abstract:
When random graphs have expected degree that is O(log n), existing limit theorems show that their spectral embeddings converge in distribution to a gaussian mixture, providing theoretical support to the use of spectral clustering for tasks on random graphs like community detection. In the sparse case, where expected degrees are fixed as n, the number of vertices, goes to infinity, random graphs require “regularization,” a preprocessing step, to improve the performance of spectral clustering. Existing work provides performance guarantees on regularization when all edge probabilities in a random graph are O(1/n), but these guarantees are not useful for graphs with more heterogeneous edge probability structure. We present one result and a conjecture on the performance of regularization with spectral clustering in sparse graphs.
October 6th
Yue Wu
Title:
A Parameter-Free Restarting Framework without Logarithmic Overheads
Abstract:
It is well-known that first-order methods can offer accelerated convergence rates in the presence of growth structures. Restarting schemes provide a general tool for such speed-ups, but typically require unrealistic problem knowledge, incur logarithmic overhead factors in oracle complexity, and/or have a nontrivial initial burn-in phase. We present a parameter-free framework for restarting any first-order method, avoiding these three drawbacks. Our approach dynamically deploys multiple instances of a given first-order method, associates each with a decrement target, and allows them to operate at different frequencies. Its uniform-frequency specialization reduces the logarithmic overhead of Renegar and Grimmer to $O(\log\log(1/\epsilon))$ for sublinear accelerated rates, while a nonuniform-frequency schedule eliminates this remaining overhead. Our scheme avoids expensive burn-in phases, and the framework itself is independent of the growth parameters; only the optimized target sequence requires a lower bound on the growth exponent.
October 13th
Title:
A Complete Characterization of Optimal Subgradient Methods for Lipschitz Convex Minimization.
Abstract:
We consider the design of optimal fixed-step first-order methods for $M$-Lipschitz convex optimization given $\|x_0-x_\star\|\leq D$. Prior works have identified several distinct fixed-step methods, parameterized by a matrix of stepsizes $W$, with the (information-theoretic) minimax optimal rate $MD/\sqrt{N+1}$ of objective gap convergence. We provide a complete characterization of every optimal fixed-step method. Moreover, we show every optimal fixed-step method can be derived from the constructive approach of Drori and Taylor and provide a polyhedral representation of the set of optimal methods through proof multipliers. From this characterization, we show that no anytime optimal fixed-step subgradient methods exist.
October 20th
Anaïs Després
Title:
Specific Entropic Martingale Optimal Transport: From Duality to Mirror Flow
Abstract:
Classical entropy regularization is not well adapted to continuous-time martingale transport because relative entropy between diffusion laws constrains volatility dynamics. We propose a specific-entropy framework based on Poisson jump approximations of continuous martingales. In the Gaussian-mark setting, this leads to explicit entropy functionals on Wiener space whose limits depend on both the martingale law and the microscopic approximation scheme. We then prove a strong duality result and use it to derive a Sinkhorn-type method based on coupled HJB/FP equations. A mirror-flow formulation reveals exponential contraction of the ideal terminal marginal, while a practical linearized scheme admits local tracking error bounds. Numerical experiments in one and two dimensions illustrate the approach and demonstrate substantial computational gains over gradient flow.
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