30 citations · 180 across the 40 of their papers we have counts for
47 papers
On the Diverse Dynamical Behaviors Arising in Deep Linear Transformers
Sixu Li, Thomas Jacob Maranzatto, Jan Peszek +5
We study the inference-time behavior of deep linear encoder-only transformers through the lens of interacting particle systems. In this perspective, tokens are modeled as particles…
Wasserstein-Cramér-Rao Theory of Unbiased Estimation
Nicolás García Trillos, Adam Quinn Jaffe, Bodhisattva Sen
The quantity of interest in the classical Cramér-Rao theory of unbiased estimation (e.g., the Cramér-Rao lower bound, its exact attainment for exponential families, and asymptotic…
Lower Bounds on Adversarial Robustness for Multiclass Classification with General Loss Functions
Camilo Andrés García Trillos, Nicolás García Trillos
We consider adversarially robust classification in a multiclass setting under arbitrary loss functions and derive dual and barycentric reformulations of the corresponding learner-a…
The Future of Artificial Intelligence and the Mathematical and Physical Sciences (AI+MPS)
Andrew Ferguson, Marisa LaFleur, Lars Ruthotto +97
This community paper developed out of the NSF Workshop on the Future of Artificial Intelligence (AI) and the Mathematical and Physics Sciences (MPS), which was held in March 2025 w…
Central limit theorems for the eigenvalues of graph Laplacians on data clouds
Chenghui Li, Nicolás García Trillos, Housen Li +1
Given i.i.d.\ samples from a distribution supported on a low dimensional manifold embedded in Eucliden space, we consider the graph Laplacian ope…
Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds
Nicolás García Trillos, Chenghui Li, Raghavendra Venkatraman
We study the problem of estimating eigenpairs of elliptic differential operators from samples of a distribution supported on a manifold . The operators discussed in the pape…
Vector valued optimal transport: from dynamic to static formulations
Katy Craig, Nicolás García Trillos, Đorđe Nikolić
Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, fr…
Defending Against Diverse Attacks in Federated Learning Through Consensus-Based Bi-Level Optimization
Nicolás García Trillos, Aditya Kumar Akash, Sixu Li +2
Adversarial attacks pose significant challenges in many machine learning applications, particularly in the setting of distributed training and federated learning, where malicious a…
CBO: Consensus-Based Bi-Level Optimization
Nicolás García Trillos, Sixu Li, Konstantin Riedl +1
Bi-level optimization problems, where one wishes to find the global minimizer of an upper-level objective function over the globally optimal solution set of a lower-level objective…
An Optimal Transport Approach for Computing Adversarial Training Lower Bounds in Multiclass Classification
Nicolas Garcia Trillos, Matt Jacobs, Jakwang Kim +1
Despite the success of deep learning-based algorithms, it is widely known that neural networks may fail to be robust. A popular paradigm to enforce robustness is adversarial traini…
A New Perspective On Denoising Based On Optimal Transport
Nicolas Garcia Trillos, Bodhisattva Sen
In the standard formulation of the denoising problem, one is given a probabilistic model relating a latent variable and an observation $Z \…
Spectral Neural Networks: Approximation Theory and Optimization Landscape
Chenghui Li, Rishi Sonthalia, Nicolas Garcia Trillos
There is a large variety of machine learning methodologies that are based on the extraction of spectral geometric information from data. However, the implementations of many of the…
Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithms
Nicolás García Trillos, Anna Little, Daniel McKenzie +1
We analyze the convergence properties of Fermat distances, a family of density-driven metrics defined on Riemannian manifolds with an associated probability measure. Fermat distanc…
Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds
Nicolas Garcia Trillos, Melanie Weber
Let denote a low-dimensional manifold embedded in Euclidean space and let be a collection of points uniformly sampled from it. We study the relatio…
FedCBO: Reaching Group Consensus in Clustered Federated Learning through Consensus-based Optimization
Jose A. Carrillo, Nicolas Garcia Trillos, Sixu Li +1
Federated learning is an important framework in modern machine learning that seeks to integrate the training of learning models from multiple users, each user having their own loca…
It begins with a boundary: A geometric view on probabilistically robust learning
Leon Bungert, Nicolás García Trillos, Matt Jacobs +3
Although deep neural networks have achieved super-human performance on many classification tasks, they often exhibit a worrying lack of robustness towards adversarially generated e…
From Optimization to Sampling Through Gradient Flows
N. Garcia Trillos, B. Hosseini, D. Sanz-Alonso
This article overviews how gradient flows, and discretizations thereof, are useful to design and analyze optimization and sampling algorithms. The interplay between optimization, s…
On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it
Camilo Garcia Trillos, Nicolas Garcia Trillos
We propose iterative algorithms to solve adversarial problems in a variety of supervised learning settings of interest. Our algorithms, which can be interpreted as suitable ascent-…
Wasserstein Barycenter-based Model Fusion and Linear Mode Connectivity of Neural Networks
Aditya Kumar Akash, Sixu Li, Nicolás García Trillos
Based on the concepts of Wasserstein barycenter (WB) and Gromov-Wasserstein barycenter (GWB), we propose a unified mathematical framework for neural network (NN) model fusion and u…
Rates of Convergence for Regression with the Graph Poly-Laplacian
Nicolás García Trillos, Ryan Murray, Matthew Thorpe
In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularisation. Higher order regularity can be ob…
Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective
Yuetian Luo, Nicolas Garcia Trillos
We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization and consider a particular Riema…
Mathematical Foundations of Graph-Based Bayesian Semi-Supervised Learning
Nicolas García Trillos, Daniel Sanz-Alonso, Ruiyi Yang
In recent decades, science and engineering have been revolutionized by a momentous growth in the amount of available data. However, despite the unprecedented ease with which data a…
The Multimarginal Optimal Transport Formulation of Adversarial Multiclass Classification
Nicolas Garcia Trillos, Matt Jacobs, Jakwang Kim
We study a family of adversarial multiclass classification problems and provide equivalent reformulations in terms of: 1) a family of generalized barycenter problems introduced in…
The Geometry of Adversarial Training in Binary Classification
Leon Bungert, Nicolás García Trillos, Ryan Murray
We establish an equivalence between a family of adversarial training problems for non-parametric binary classification and a family of regularized risk minimization problems where…