arXiv subjects

Keaton Hamm

Publications and source records attributed to Keaton Hamm.

At least 19 recordsLinked to original sources

arXiv2026

Structured Approximations of Measures

We study the approximation of probability measures in the Wasserstein-$p$ distance by structured classes of approximators, motivated by applications in imaging, machine learning, and physical measurement under sensor constraints. We obtain three sets of results. First, for measures with densities bounded away from zero on a bounded Lipschitz domain $Ω$, we prove that any approximation scheme for functions in $\mathrm{L}_p(Ω)$ transfers, with linear rate, to a corresponding approximation scheme for measures in $\mathrm{W}_p(Ω)$. The argument applies a theorem of Bogovskii on regularity of solutions to the continuity equation in the Benamou-Brenier formulation of optimal transport. We exhibit concrete approximation schemes (polynomials, shift-invariant spaces, cardinal interpolation with radial basis functions, kernel density estimators, and piecewise approximations on nonuniform Voronoi partitions) that fit the framework. As a matter of independent interest, we prove a negative Sobolev lower bound that generalizes existing bounds from $p=2$ to all $p\in(1,\infty)$. We also consider deterministic bounds for discrete approximations to arbitrary measures in terms of the mesh norm of a quasi-uniform set of points. We specialize these bounds to show that compactly supported measures admit a deterministic $N$-term approximation $μ_N$ such that $\mathrm{W}_p(μ,μ_N) = O(N^{-\frac{1}{d}})$ for all $d\geq 1$, which matches the asymptotic optimal quantizer rate. We also extend these results to non-compactly supported measures with appropriate tail decay.

stat.ML↗

arXiv2026

LOTFormer: Doubly-Stochastic Linear Attention via Low-Rank Optimal Transport

Transformers have proven highly effective across modalities, but standard softmax attention scales quadratically with sequence length, limiting long context modeling. Linear attention mitigates this by approximating attention with kernel feature maps, yet most attention mechanisms remain row normalized and can over concentrate mass on a few tokens, harming robustness and information flow. Doubly stochastic attention counteracts this by balancing token participation across both rows and columns, but existing approaches often add significant overhead. We propose LOTFormer, a linear time doubly stochastic attention mechanism derived from an optimal transport view of attention as a coupling between query and key measures. LOTFormer enforces a low rank transport plan by conditioning on a learnable pivot measure with small support. We solve two entropic transport problems, queries to pivot and pivot to keys, and compose them into a conditional coupling that is provably doubly stochastic, has rank at most $r \ll n$, and applies to values in $O(nr)$ time without forming the full $n \times n$ matrix. The pivot locations and masses are learned end-to-end. Across vision and text benchmarks, LOTFormer delivers strong accuracy efficiency tradeoffs when plugged into standard backbones including Swin, DeiT, and BERT.

cs.LG↗

arXiv2025

Neighbor Embeddings Using Unbalanced Optimal Transport Metrics

This paper proposes the use of the Hellinger--Kantorovich metric from unbalanced optimal transport (UOT) in a dimensionality reduction and learning (supervised and unsupervised) pipeline. The performance of UOT is compared to that of regular OT and Euclidean-based dimensionality reduction methods on several benchmark datasets including MedMNIST. The experimental results demonstrate that, on average, UOT shows improvement over both Euclidean and OT-based methods as verified by statistical hypothesis tests. In particular, on the MedMNIST datasets, UOT outperforms OT in classification 81\% of the time. For clustering MedMNIST, UOT outperforms OT 83\% of the time and outperforms both other metrics 58\% of the time.

stat.ML↗

arXiv2025

Recovering Wasserstein Distance Matrices from Few Measurements

This paper proposes two algorithms for estimating square Wasserstein distance matrices from a small number of entries. These matrices are used to compute manifold learning embeddings like multidimensional scaling (MDS) or Isomap, but contrary to Euclidean distance matrices, are extremely costly to compute. We analyze matrix completion from upper triangular samples and Nyström completion in which $\mathcal{O}(d\log(d))$ columns of the distance matrices are computed where $d$ is the desired embedding dimension, prove stability of MDS under Nyström completion, and show that it can outperform matrix completion for a fixed budget of sample distances. Finally, we show that classification of the OrganCMNIST dataset from the MedMNIST benchmark is stable on data embedded from the Nyström estimation of the distance matrix even when only 10\% of the columns are computed.

stat.ML↗

arXiv2025

Manifold learning in Wasserstein space

This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures $\mathcal{P}_{\mathrm{a.c.}}(Ω)$ with $Ω$ a compact and convex subset of $\mathbb{R}^d$, metrized with the Wasserstein-2 distance $\mathbb{W}$. We begin by introducing a construction of submanifolds $Λ$ in $\mathcal{P}_{\mathrm{a.c.}}(Ω)$ equipped with metric $\mathbb{W}_Λ$, the geodesic restriction of $\mathbb{W}$ to $Λ$. In contrast to other constructions, these submanifolds are not necessarily flat, but still allow for local linearizations in a similar fashion to Riemannian submanifolds of $\mathbb{R}^d$. We then show how the latent manifold structure of $(Λ,\mathbb{W}_Λ)$ can be learned from samples $\{λ_i\}_{i=1}^N$ of $Λ$ and pairwise extrinsic Wasserstein distances $\mathbb{W}$ on $\mathcal{P}_{\mathrm{a.c.}}(Ω)$ only. In particular, we show that the metric space $(Λ,\mathbb{W}_Λ)$ can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes $\{λ_i\}_{i=1}^N$ and edge weights $W(λ_i,λ_j)$. In addition, we demonstrate how the tangent space at a sample $λ$ can be asymptotically recovered via spectral analysis of a suitable ``covariance operator'' using optimal transport maps from $λ$ to sufficiently close and diverse samples $\{λ_i\}_{i=1}^N$. The paper closes with some explicit constructions of submanifolds $Λ$ and numerical examples on the recovery of tangent spaces through spectral analysis.

stat.ML↗

arXiv2024

Persistent Classification: A New Approach to Stability of Data and Adversarial Examples

There are a number of hypotheses underlying the existence of adversarial examples for classification problems. These include the high-dimensionality of the data, high codimension in the ambient space of the data manifolds of interest, and that the structure of machine learning models may encourage classifiers to develop decision boundaries close to data points. This article proposes a new framework for studying adversarial examples that does not depend directly on the distance to the decision boundary. Similarly to the smoothed classifier literature, we define a (natural or adversarial) data point to be $(γ,σ)$-stable if the probability of the same classification is at least $γ$ for points sampled in a Gaussian neighborhood of the point with a given standard deviation $σ$. We focus on studying the differences between persistence metrics along interpolants of natural and adversarial points. We show that adversarial examples have significantly lower persistence than natural examples for large neural networks in the context of the MNIST and ImageNet datasets. We connect this lack of persistence with decision boundary geometry by measuring angles of interpolants with respect to decision boundaries. Finally, we connect this approach with robustness by developing a manifold alignment gradient metric and demonstrating the increase in robustness that can be achieved when training with the addition of this metric.

cs.LG↗

arXiv2024

On Wasserstein distances for affine transformations of random vectors

We expound on some known lower bounds of the quadratic Wasserstein distance between random vectors in $\mathbb{R}^n$ with an emphasis on affine transformations that have been used in manifold learning of data in Wasserstein space. In particular, we give concrete lower bounds for rotated copies of random vectors in $\mathbb{R}^2$ by computing the Bures metric between the covariance matrices. We also derive upper bounds for compositions of affine maps which yield a fruitful variety of diffeomorphisms applied to an initial data measure. We apply these bounds to various distributions including those lying on a 1-dimensional manifold in $\mathbb{R}^2$ and illustrate the quality of the bounds. Finally, we give a framework for mimicking handwritten digit or alphabet datasets that can be applied in a manifold learning framework.

stat.ML↗

arXiv2023

Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning

In this paper, we propose Wasserstein Isometric Mapping (Wassmap), a nonlinear dimensionality reduction technique that provides solutions to some drawbacks in existing global nonlinear dimensionality reduction algorithms in imaging applications. Wassmap represents images via probability measures in Wasserstein space, then uses pairwise Wasserstein distances between the associated measures to produce a low-dimensional, approximately isometric embedding. We show that the algorithm is able to exactly recover parameters of some image manifolds including those generated by translations or dilations of a fixed generating measure. Additionally, we show that a discrete version of the algorithm retrieves parameters from manifolds generated from discrete measures by providing a theoretical bridge to transfer recovery results from functional data to discrete data. Testing of the proposed algorithms on various image data manifolds show that Wassmap yields good embeddings compared with other global and local techniques.

cs.LG↗

arXiv2023

Boosting Nyström Method

The Nyström method is an effective tool to generate low-rank approximations of large matrices, and it is particularly useful for kernel-based learning. To improve the standard Nyström approximation, ensemble Nyström algorithms compute a mixture of Nyström approximations which are generated independently based on column resampling. We propose a new family of algorithms, boosting Nyström, which iteratively generate multiple ``weak'' Nyström approximations (each using a small number of columns) in a sequence adaptively - each approximation aims to compensate for the weaknesses of its predecessor - and then combine them to form one strong approximation. We demonstrate that our boosting Nyström algorithms can yield more efficient and accurate low-rank approximations to kernel matrices. Improvements over the standard and ensemble Nyström methods are illustrated by simulation studies and real-world data analysis.

stat.ML↗

arXiv2023

Linearized Wasserstein dimensionality reduction with approximation guarantees

We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in $\mathbb{R}^n$, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. We provide guarantees on the embedding quality under such approximations, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.

cs.LG↗

arXiv2023

Multi-Priority Graph Sparsification

A \emph{sparsification} of a given graph $G$ is a sparser graph (typically a subgraph) which aims to approximate or preserve some property of $G$. Examples of sparsifications include but are not limited to spanning trees, Steiner trees, spanners, emulators, and distance preservers. Each vertex has the same priority in all of these problems. However, real-world graphs typically assign different ``priorities'' or ``levels'' to different vertices, in which higher-priority vertices require higher-quality connectivity between them. Multi-priority variants of the Steiner tree problem have been studied in prior literature but this generalization is much less studied for other sparsification problems. In this paper, we define a generalized multi-priority problem and present a rounding-up approach that can be used for a variety of graph sparsifications. Our analysis provides a systematic way to compute approximate solutions to multi-priority variants of a wide range of graph sparsification problems given access to a single-priority subroutine.

cs.DS↗

arXiv2022

Generalized Pseudoskeleton Decompositions

We characterize some variations of pseudoskeleton (also called CUR) decompositions for matrices and tensors over arbitrary fields. These characterizations extend previous results to arbitrary fields and to decompositions which use generalized inverses of the constituent matrices, in contrast to Moore--Penrose pseudoinverses in prior works which are specific to real or complex valued matrices, and are significantly more structured.

math.NA↗

arXiv2022

Riemannian CUR Decompositions for Robust Principal Component Analysis

Robust Principal Component Analysis (PCA) has received massive attention in recent years. It aims to recover a low-rank matrix and a sparse matrix from their sum. This paper proposes a novel nonconvex Robust PCA algorithm, coined Riemannian CUR (RieCUR), which utilizes the ideas of Riemannian optimization and robust CUR decompositions. This algorithm has the same computational complexity as Iterated Robust CUR, which is currently state-of-the-art, but is more robust to outliers. RieCUR is also able to tolerate a significant amount of outliers, and is comparable to Accelerated Alternating Projections, which has high outlier tolerance but worse computational complexity than the proposed method. Thus, the proposed algorithm achieves state-of-the-art performance on Robust PCA both in terms of computational complexity and outlier tolerance.

stat.ML↗

arXiv2021

Computing Steiner Trees using Graph Neural Networks

Graph neural networks have been successful in many learning problems and real-world applications. A recent line of research explores the power of graph neural networks to solve combinatorial and graph algorithmic problems such as subgraph isomorphism, detecting cliques, and the traveling salesman problem. However, many NP-complete problems are as of yet unexplored using this method. In this paper, we tackle the Steiner Tree Problem. We employ four learning frameworks to compute low cost Steiner trees: feed-forward neural networks, graph neural networks, graph convolutional networks, and a graph attention model. We use these frameworks in two fundamentally different ways: 1) to train the models to learn the actual Steiner tree nodes, 2) to train the model to learn good Steiner point candidates to be connected to the constructed tree using a shortest path in a greedy fashion. We illustrate the robustness of our heuristics on several random graph generation models as well as the SteinLib data library. Our finding suggests that the out-of-the-box application of GNN methods does worse than the classic 2-approximation method. However, when combined with a greedy shortest path construction, it even does slightly better than the 2-approximation algorithm. This result sheds light on the fundamental capabilities and limitations of graph learning techniques on classical NP-complete problems.

cs.LG↗

arXiv2021

Robust CUR Decomposition: Theory and Imaging Applications

This paper considers the use of Robust PCA in a CUR decomposition framework and applications thereof. Our main algorithms produce a robust version of column-row factorizations of matrices $\mathbf{D}=\mathbf{L}+\mathbf{S}$ where $\mathbf{L}$ is low-rank and $\mathbf{S}$ contains sparse outliers. These methods yield interpretable factorizations at low computational cost, and provide new CUR decompositions that are robust to sparse outliers, in contrast to previous methods. We consider two key imaging applications of Robust PCA: video foreground-background separation and face modeling. This paper examines the qualitative behavior of our Robust CUR decompositions on the benchmark videos and face datasets, and find that our method works as well as standard Robust PCA while being significantly faster. Additionally, we consider hybrid randomized and deterministic sampling methods which produce a compact CUR decomposition of a given matrix, and apply this to video sequences to produce canonical frames thereof.

cs.CV↗

arXiv2021

Mode-wise Tensor Decompositions: Multi-dimensional Generalizations of CUR Decompositions

Low rank tensor approximation is a fundamental tool in modern machine learning and data science. In this paper, we study the characterization, perturbation analysis, and an efficient sampling strategy for two primary tensor CUR approximations, namely Chidori and Fiber CUR. We characterize exact tensor CUR decompositions for low multilinear rank tensors. We also present theoretical error bounds of the tensor CUR approximations when (adversarial or Gaussian) noise appears. Moreover, we show that low cost uniform sampling is sufficient for tensor CUR approximations if the tensor has an incoherent structure. Empirical performance evaluations, with both synthetic and real-world datasets, establish the speed advantage of the tensor CUR approximations over other state-of-the-art low multilinear rank tensor approximations.

math.NA↗

arXiv2021

On Matrix Factorizations in Subspace Clustering

This article explores subspace clustering algorithms using CUR decompositions, and examines the effect of various hyperparameters in these algorithms on clustering performance on two real-world benchmark datasets, the Hopkins155 motion segmentation dataset and the Yale face dataset. Extensive experiments are done for a variety of sampling methods and oversampling parameters for these datasets, and some guidelines for parameter choices are given for practical applications.

cs.CV↗

arXiv2021

On additive spanners in weighted graphs with local error

An \emph{additive $+β$ spanner} of a graph $G$ is a subgraph which preserves distances up to an additive $+β$ error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al.\ 2019 and 2020, Ahmed et al.\ 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al.\ 2020] provided constructions of sparse spanners with \emph{global} error $β= cW$, where $W$ is the maximum edge weight in $G$ and $c$ is constant. We improve these to \emph{local} error by giving spanners with additive error $+cW(s,t)$ for each vertex pair $(s,t)$, where $W(s, t)$ is the maximum edge weight along the shortest $s$--$t$ path in $G$. These include pairwise $+(2+\eps)W(\cdot,\cdot)$ and $+(6+\eps) W(\cdot, \cdot)$ spanners over vertex pairs $\Pc \subseteq V \times V$ on $O_{\eps}(n|\Pc|^{1/3})$ and $O_{\eps}(n|\Pc|^{1/4})$ edges for all $\eps > 0$, which extend previously known unweighted results up to $\eps$ dependence, as well as an all-pairs $+4W(\cdot,\cdot)$ spanner on $\widetilde{O}(n^{7/5})$ edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its \emph{lightness}, defined as the total edge weight of the spanner divided by the weight of an MST of $G$. We provide a $+\eps W(\cdot,\cdot)$ spanner with $O_{\eps}(n)$ lightness, and a $+(4+\eps) W(\cdot,\cdot)$ spanner with $O_{\eps}(n^{2/3})$ lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.

cs.DM↗