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Found 28 papers, 6 papers with code
An Optimal Transport Approach for Computing Adversarial Training Lower Bounds in Multiclass Classification
Recent works have developed a connection between AT in the multiclass classification setting and multimarginal optimal transport (MOT), unlocking a new set of tools to study this problem.
A New Perspective On Denoising Based On Optimal Transport
We then prove that, under appropriate identifiability assumptions on the model, our OT-based denoiser can be recovered solely from information of the marginal distribution of $Z$ and the posterior mean of the model, after solving a linear relaxation problem over a suitable space of couplings that is reminiscent of a standard multimarginal OT (MOT) problem.
Spectral Neural Networks: Approximation Theory and Optimization Landscape
There is a large variety of machine learning methodologies that are based on the extraction of spectral geometric information from data.
Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds
Let $\mathcal{M} \subseteq \mathbb{R}^d$ denote a low-dimensional manifold and let $\mathcal{X}= \{ x_1, \dots, x_n \}$ be a collection of points uniformly sampled from $\mathcal{M}$.
FedCBO: Reaching Group Consensus in Clustered Federated Learning through Consensus-based Optimization
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 local data set, in a way that is sensitive to data privacy and to communication loss constraints.
On the existence of solutions to adversarial training in multiclass classification
We study three models of the problem of adversarial training in multiclass classification designed to construct robust classifiers against adversarial perturbations of data in the agnostic-classifier setting.
On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it
These interacting particle dynamics are shown to converge toward appropriate mean-field limit equations in certain large number of particles regimes.
Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective
To prove our results we provide a comprehensive landscape analysis of a matrix factorization problem with a least squares objective, which serves as a critical bridge.
The Multimarginal Optimal Transport Formulation of Adversarial Multiclass Classification
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 paper and 2) a family of multimarginal optimal transport problems where the number of marginals is equal to the number of classes in the original classification problem.
On the regularized risk of distributionally robust learning over deep neural networks
In this paper we explore the relation between distributionally robust learning and different forms of regularization to enforce robustness of deep neural networks.
Large sample spectral analysis of graph-based multi-manifold clustering
We investigate sufficient conditions that similarity graphs on data sets must satisfy in order for their corresponding graph Laplacians to capture the right geometric information to solve the MMC problem.
Adversarial Classification: Necessary conditions and geometric flows
Using the necessary conditions, we derive a geometric evolution equation which can be used to track the change in classification boundaries as $\varepsilon$ varies.
Lipschitz regularity of graph Laplacians on random data clouds
As a byproduct of our general regularity results, we obtain high probability $L^\infty$ and approximate $\mathcal{C}^{0, 1}$ convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators.
Traditional and accelerated gradient descent for neural architecture search
In this paper we introduce two algorithms for neural architecture search (NASGD and NASAGD) following the theoretical work by two of the authors [5] which used the geometric structure of optimal transport to introduce the conceptual basis for new notions of traditional and accelerated gradient descent algorithms for the optimization of a function on a semi-discrete space.
Semi-discrete optimization through semi-discrete optimal transport: a framework for neural architecture search
With this aim in mind, we discuss the geometric and theoretical motivation for new techniques for neural architecture search (in a companion paper we show that algorithms inspired by our framework are competitive with contemporaneous methods).
From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data clouds
In this work we study statistical properties of graph-based clustering algorithms that rely on the optimization of balanced graph cuts, the main example being the optimization of Cheeger cuts.
Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs
In this paper we improve the spectral convergence rates for graph-based approximations of Laplace-Beltrami operators constructed from random data.
Local Regularization of Noisy Point Clouds: Improved Global Geometric Estimates and Data Analysis
Several data analysis techniques employ similarity relationships between data points to uncover the intrinsic dimension and geometric structure of the underlying data-generating mechanism.
Geometric structure of graph Laplacian embeddings
More precisely, we assume that the data is sampled from a mixture model supported on a manifold $\mathcal{M}$ embedded in $\mathbb{R}^d$, and pick a connectivity length-scale $\varepsilon>0$ to construct a kernelized graph Laplacian.
Variational Characterizations of Local Entropy and Heat Regularization in Deep Learning
The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization.
A maximum principle argument for the uniform convergence of graph Laplacian regressors
This paper investigates the use of methods from partial differential equations and the Calculus of variations to study learning problems that are regularized using graph Laplacians.
Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator
sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero.
On the Consistency of Graph-based Bayesian Learning and the Scalability of Sampling Algorithms
A popular approach to semi-supervised learning proceeds by endowing the input data with a graph structure in order to extract geometric information and incorporate it into a Bayesian framework.
Continuum Limit of Posteriors in Graph Bayesian Inverse Problems
We consider the problem of recovering a function input of a differential equation formulated on an unknown domain $M$.
Gromov-Hausdorff limit of Wasserstein spaces on point clouds
We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ uniformly distributed on the flat torus $\mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d $, and construct a geometric graph on the cloud by connecting points that are within distance $\varepsilon$ of each other.
Variational limits of k-NN graph based functionals on data clouds
This paper studies the large sample asymptotics of data analysis procedures based on the optimization of functionals defined on $k$-NN graphs on point clouds.
A new analytical approach to consistency and overfitting in regularized empirical risk minimization
This work considers the problem of binary classification: given training data $x_1, \dots, x_n$ from a certain population, together with associated labels $y_1,\dots, y_n \in \left\{0, 1 \right\}$, determine the best label for an element $x$ not among the training data.
Consistency of Cheeger and Ratio Graph Cuts
We consider point clouds obtained as samples of a ground-truth measure.
