The optimal transportation problem defines a geometry of probability measures which leads to a definition for weighted averages (barycenters) of measures, finding application in the machine learning and computer vision communities as a signal processing tool. Here, we implement a barycentric coding model for measures which are supported on a graph, a context in which the classical optimal transport geometry becomes degenerate, by leveraging a Riemannian structure on the simplex induced by a dynamic formulation of the optimal transport problem. We approximate the exponential mapping associated to the Riemannian structure, as well as its inverse, by utilizing past approaches which compute action minimizing curves in order to numerically approximate transport distances for measures supported on discrete spaces. Intrinsic gradient descent is then used to synthesize barycenters, wherein gradients of a variance functional are computed by approximating geodesic curves between the current iterate and the reference measures; iterates are then pushed forward via a discretization of the continuity equation. Analysis of measures with respect to given dictionary of references is performed by solving a quadratic program formed by computing geodesics between target and reference measures. We compare our novel approach to one based on entropic regularization of the static formulation of the optimal transport problem where the graph structure is encoded via graph distance functions, we present numerical experiments validating our approach, and we conclude that intrinsic gradient descent on the probability simplex provides a coherent framework for the synthesis and analysis of measures supported on graphs.

Two active learning algorithms for hyperspectral image (HSI) classification are proposed that combine density-aware Fermat distances with Poisson-reweighted harmonic label propagation. Our methods actively query points using an uncertainty-based acquisition function, extending Poisson ReWeighted Laplace Learning (PWLL). Our first algorithm, Fermat Active Laplace Learning (FALL), builds an affinity matrix using Fermat distances between all data points. Then, PWLL is run with a diagonal perturbation using the minimum-norm acquisition function. In contrast, Approximate FALL (A-FALL) computes Fermat distances between each data point and landmark pixels selected via farthest-point sampling and constructs the affinity matrix using landmark multidimensional scaling. After several query rounds, A-FALL selects the Fermat exponent p using a leave-one-out cross-validation variant. FALL and A-FALL leverage Fermat distances and subsequent harmonic label propagation to provide a density-aware estimation of the data manifold, improving labeling accuracy. Experiments on Salinas A and Pavia show the effectiveness of FALL and the scalability of A-FALL to large HSI scenes.

An unsupervised framework for hyperspectral image (HSI) clustering is proposed that incorporates masked deep representation learning with diffusion-based clustering, extending the Spatially-Regularized Superpixel-based Diffusion Learning (S^2DL) algorithm. Initially, a denoised latent representation of the original HSI is learned via an unsupervised masked autoencoder (UMAE) model with a Vision Transformer backbone. The UMAE takes spatial context and long-range spectral correlations into account and incorporates an efficient pretraining process via masking that utilizes only a small subset of training pixels. In the next stage, the entropy rate superpixel (ERS) algorithm is used to segment the image into superpixels, and a spatially regularized diffusion graph is constructed using Euclidean and diffusion distances within the compressed latent space instead of the HSI space. The proposed algorithm, Deep Spatially-Regularized Superpixel-based Diffusion Learning (DS^2DL), leverages more faithful diffusion distances and subsequent diffusion graph construction that better reflect the intrinsic geometry of the underlying data manifold, improving labeling accuracy and clustering quality. Experiments on Botswana and KSC datasets demonstrate the efficacy of DS^2DL.

This paper considers the problem of estimating a matrix that encodes pairwise distances in a finite metric space (or, more generally, the edge weight matrix of a network) under the barycentric coding model (BCM) with respect to the Gromov-Wasserstein (GW) distance function. We frame this task as estimating the unknown barycentric coordinates with respect to the GW distance, assuming that the target matrix (or kernel) belongs to the set of GW barycenters of a finite collection of known templates. In the language of harmonic analysis, if computing GW barycenters can be viewed as a synthesis problem, this paper aims to solve the corresponding analysis problem. We propose two methods: one utilizing fixed-point iteration for computing GW barycenters, and another employing a differentiation-based approach to the GW structure using a blow-up technique. Finally, we demonstrate the application of the proposed GW analysis approach in a series of numerical experiments and applications to machine learning.

Hyperspectral images capture vast amounts of high-dimensional spectral information about a scene, making labeling an intensive task that is resistant to out-of-the-box statistical methods. Unsupervised learning of clusters allows for automated segmentation of the scene, enabling a more rapid understanding of the image. Partitioning the spectral information contained within the data via dictionary learning in Wasserstein space has proven an effective method for unsupervised clustering. However, this approach requires balancing the spectral profiles of the data, blurring the classes, and sacrificing robustness to outliers and noise. In this paper, we suggest improving this approach by utilizing unbalanced Wasserstein barycenters to learn a lower-dimensional representation of the underlying data. The deployment of spectral clustering on the learned representation results in an effective approach for the unsupervised learning of labels.

We apply entropically regularized Wasserstein geometry to the setting of a discrete metric space, where the cost functions are given by different choices of graph metric, which capture varying levels of connectivity information. We find that despite the degeneracy of the Monge-Kantorovich geometry in the case of discrete metric spaces, regularized transport leads to geometric objects (e.g., geodesics and more generally barycenters) which convincingly resemble what one would expect from a Wasserstein-type geometry on the space of probability distributions defined for a network. We consider both synthesis (combining measures with respect to the regularized Wasserstein geometry) and analysis (decomposing signals with respect to fixed reference measures) on real geography data, demonstrating the utility of our approach. Our code is available on GitHub1.

Linear representation learning is widely studied due to its conceptual simplicity and empirical utility in tasks such as compression, classification, and feature extraction. Given a set of points [𝐱_1, 𝐱_2, …, 𝐱_n] = 𝐗∈ℝ^d × n and a vector 𝐲∈ℝ^d, the goal is to find coefficients 𝐰∈ℝ^n so that 𝐗𝐰≈𝐲, subject to some desired structure on 𝐰. In this work we seek 𝐰 that forms a local reconstruction of 𝐲 by solving a regularized least squares regression problem. We obtain local solutions through a locality function that promotes the use of columns of 𝐗 that are close to 𝐲 when used as a regularization term. We prove that, for all levels of regularization and under a mild condition that the columns of 𝐗 have a unique Delaunay triangulation, the optimal coefficients' number of non-zero entries is upper bounded by d+1, thereby providing local sparse solutions when d ≪ n. Under the same condition we also show that for any 𝐲 contained in the convex hull of 𝐗 there exists a regime of regularization parameter such that the optimal coefficients are supported on the vertices of the Delaunay simplex containing 𝐲. This provides an interpretation of the sparsity as having structure obtained implicitly from the Delaunay triangulation of 𝐗. We demonstrate that our locality regularized problem can be solved in comparable time to other methods that identify the containing Delaunay simplex.

In this paper, we propose the OTSeg score, a new quantitative metric that utilizes optimal transport to measure the degree of segregation on graphs. We characterize maximizers and minimizers for the metric and consider its relationship to classical measures of segregation including those based on spectral graph theory. We validate the effectiveness of OTSeg through experiments on synthetic and real datasets derived from American political geography, highlighting its potential for applications in computational social science.

We consider synthesis and analysis of probability measures using the entropy-regularized Wasserstein-2 cost and its unbiased version, the Sinkhorn divergence. The synthesis problem consists of computing the barycenter, with respect to these costs, of reference measures given a set of coefficients belonging to the simplex. The analysis problem consists of finding the coefficients for the closest barycenter in the Wasserstein-2 distance to a given measure. Under the weakest assumptions on the measures thus far in the literature, we compute the derivative of the entropy-regularized Wasserstein-2 cost. We leverage this to establish a characterization of barycenters with respect to the entropy-regularized Wasserstein-2 cost as solutions that correspond to a fixed point of an average of the entropy-regularized displacement maps. This characterization yields a finite-dimensional, convex, quadratic program for solving the analysis problem when the measure being analyzed is a barycenter with respect to the entropy-regularized Wasserstein-2 cost. We show that these coefficients, as well as the value of the barycenter functional, can be estimated from samples with dimensionindependent rates of convergence, and that barycentric coefficients are stable with respect to perturbations in the Wasserstein-2 metric. We employ the barycentric coefficients as features for classification of corrupted point cloud data, and show that compared to neural network baselines, our approach is more efficient in small training data regimes.

We propose the linear barycentric coding model (LBCM) that utilizes the linear optimal transport (LOT) metric for analysis and synthesis of probability measures. We provide a closed-form solution to the variational problem characterizing the probability measures in the LBCM and establish equivalence of the LBCM to the set of Wasserstein-2 barycenters in the special case of compatible measures. Computational methods for synthesizing and analyzing measures in the LBCM are developed with finite sample guarantees. One of our main theoretical contributions is to identify an LBCM, expressed in terms of a simple family, which is sufficient to express all probability measures on the interval [0,1]. We show that a natural analogous construction of an LBCM in ℝ^2 fails, and we leave it as an open problem to identify the proper extension in more than one dimension. We conclude by demonstrating the utility of LBCM for covariance estimation and data imputation.

We present a novel active learning method for hyperspectral images based on representation learning in Wasserstein space. We perform regularized Wasserstein dictionary learning in the space of hyperspectral pixels, then leverage the learned barycentric coefficients to embed the high-dimensional spectra into a low-dimensional space. Sampling in the low-dimensional space leads to high-quality labels that propagate accurately to the remaining pixels in the data. Our method achieves a high level of accuracy with very few training labels, suggesting its utility for hyperspectral image classification in the active labeling setting.

We analyze the convergence properties of Fermat distances, a family of density-driven metrics defined on Riemannian manifolds with an associated probability measure. Fermat distances may be defined either on discrete samples from the underlying measure, in which case they are random, or in the continuum setting, in which they are induced by geodesics under a density-distorted Riemannian metric. We prove that discrete, sample-based Fermat distances converge to their continuum analogues in small neighborhoods with a precise rate that depends on the intrinsic dimensionality of the data and the parameter governing the extent of density weighting in Fermat distances. This is done by leveraging novel geometric and statistical arguments in percolation theory that allow for non-uniform densities and curved domains. Our results are then used to prove that discrete graph Laplacians based on discrete, sample-driven Fermat distances converge to corresponding continuum operators. In particular, we show the discrete eigenvalues and eigenvectors converge to their continuum analogues at a dimension-dependent rate, which allows us to interpret the efficacy of discrete spectral clustering using Fermat distances in terms of the resulting continuum limit. The perspective afforded by our discrete-to-continuum Fermat distance analysis leads to new clustering algorithms for data and related insights into efficient computations associated to density-driven spectral clustering. Our theoretical analysis is supported with numerical simulations and experiments on synthetic and real image data.

A method for active learning of hyperspectral images (HSI) is proposed, which leverages time-evolving diffusion processes on graphs to determine query points and propagate their labels to the full data set. Initially, a diffusion process is defined on a graph in which each HSI pixel is a node, with edge connection strength scaling with the distance between pixels in the spectral domain. At each stage of the iterative sampling process, queried labels are used to update the underlying diffusion matrix by weakening edges between points in different classes, thereby infusing the intrinsic representation encoded in the diffusion matrix with revealed label information. The proposed method, Learning by Evolving Nonlinear Diffusion (LEND), combines robust performance with the mathematical tractability of diffusion geometry, leading to superior labeling accuracy with fewer labeled samples compared to a baseline in which the underlying matrix is not iteratively updated. Experiments on the Salinas A HSI demonstrate the effectiveness and efficiency of LEND.

Hyperspectral images consist of large numbers of pixels across hundreds of spectral bands, making statistical analysis computationally challenging. However, these images often exhibit intrinsic structure that can be leveraged for efficient statistical and machine learning. We propose a novel nonlinear method for unmixing hyperspectral images. In contrast to classical methods which consider an additive linear model, we propose to represent hyperspectral spectra as probability distributions in Wasserstein space and characterize pure spectra as those that allow for typical observations to be reconstructed as entropic Wasserstein barycenters. This allows for the analysis and synthesis of hyperspectral spectra in a geometry-preserving fashion. Results on synthetic data and real HSI show important geometric features of hyperspectral spectra are preserved when utilizing our nonlinear Wasserstein unmixing scheme.

Hyperspectral images (HSI) capture rich information of large spatial scenes, yet generating labeled training data can be expensive and time-consuming. Unsupervised clustering of HSI allows for segmentation in the absence of labels and is an important problem in processing rapidly collected HSI. In order to accurately cluster noisy and high-dimensional HSI, meaningful data representations that capture latent intrinsic structure must be developed. We propose to leverage regularized dictionary learning in Wasserstein space to efficiently and accurately cluster HSI by modeling HSI pixels as probability distributions. We characterize pixels as similar if they can be synthesized as entropic Wasserstein barycenters with a common set of learned reference distributions. Our approach learns representations that preserve the geometry of the space of HSI spectra and our barycentric coding spectral clustering algorithm, which leverages these learned features, shows promise on benchmark HSI data.

This paper considers the use of recently proposed optimal transport-based multivariate goodness-of-fit (GoF) test statistics, namely rank energy and its variant the soft rank energy derived from entropy-regularized optimal transport, for unsupervised non-parametric change point detection (CPD) in multivariate time series data. We show that the soft rank energy enjoys both fast rates of statistical convergence and robust continuity properties which lead to strong performance on real datasets. Our analyses remove the need for resampling and out-of-sample extensions previously required to obtain such rates. Our theoretical results show that the rank energy suffers from the curse of dimensionality in statistical estimation and moreover can signal a change point from arbitrarily small perturbations, which leads to a high rate of false alarms in CPD. Additionally, under mild regularity conditions, we quantify the discrepancy between soft rank energy and rank energy in terms of the regularization parameter. Finally, we show our approach performs favorably in numerical experiments compared to several other optimal transport-based methods as well as maximum mean discrepancy (MMD), which is a popular multivariate GoF statistic.

Hyperspectral images (HSIs) provide exceptional spatial and spectral resolution of a scene, crucial for various remote sensing applications. However, the high dimensionality, presence of noise and outliers, and the need for precise labels of HSIs present significant challenges to HSIs analysis, motivating the development of performant HSI clustering algorithms. This paper introduces a novel unsupervised HSI clustering algorithm, Superpixel-based and Spatially-regularized Diffusion Learning (S2DL), which addresses these challenges by incorporating rich spatial information encoded in HSIs into diffusion geometry-based clustering. S2DL employs the Entropy Rate Superpixel (ERS) segmentation technique to partition an image into superpixels, then constructs a spatially-regularized diffusion graph using the most representative high-density pixels. This approach reduces computational burden while preserving accuracy. Cluster modes, serving as exemplars for underlying cluster structure, are identified as the highest-density pixels farthest in diffusion distance from other highest-density pixels. These modes guide the labeling of the remaining representative pixels from ERS superpixels. Finally, majority voting is applied to the labels assigned within each superpixel to propagate labels to the rest of the image. This spatial-spectral approach simultaneously simplifies graph construction, reduces computational cost, and improves clustering performance. S2DL's performance is illustrated with extensive experiments on three publicly available, real-world HSIs: Indian Pines, Salinas, and Salinas A. Additionally, we apply S2DL to landscape-scale, unsupervised mangrove species mapping in the Mai Po Nature Reserve, Hong Kong, using a Gaofen-5 HSI. The success of S2DL in these diverse numerical experiments indicates its efficacy on a wide range of important unsupervised remote sensing analysis tasks.

Molecular dynamics (MD) simulations are a powerful computational tool for elucidation of molecular behavior. These simulations generate an abundance of high-dimensional time series data and parsing these data into a human-interpretable format is nontrivial. Clustering trajectory segments obtained via change point detection has been shown to lower memory complexity and yield improved partitioning resolution of the time series compared to the state of the art. However, accurate change point placement is often inhibited by the presence of gradual changes between long-lived metastable states. The trajectory regions corresponding to these gradual changes are not well-modeled by a single distribution, and therefore are frequently over-segmented. In this work, we model such regions using weighted Wasserstein barycentric interpolations between adjacent metastable states, allowing for gradual changes to be resolved correctly. The improved detection performance of our proposed method is demonstrated on a range of toy and real MD simulation data, showing significant potential for faithfully modeling and compressing complex MD simulations.

The framework of optimal transport has been leveraged to extend the notion of rank to the multivariate setting while preserving desirable properties of the resulting goodness-of-fit (GoF) statistics. In particular, the rank energy (RE) and rank maximum mean discrepancy (RMMD) are distribution-free under the null, exhibit high power in statistical testing, and are robust to outliers. In this paper, we point to and alleviate some of the practical shortcomings of these proposed GoF statistics, namely their high computational cost, high statistical sample complexity, and lack of differentiability with respect to the data. We show that all these practically important issues are addressed by considering entropy-regularized optimal transport maps in place of the rank map, which we refer to as the soft rank. We consequently propose two new statistics, the soft rank energy (sRE) and soft rank maximum mean discrepancy (sRMMD), which exhibit several desirable properties. Given $n$ sample data points, we provide non-asymptotic convergence rates for the sample estimate of the entropic transport map to its population version that are essentially of the order $n^{-1/2}$ when the starting measure is subgaussian and the target measure has compact support. This result is novel compared to existing results which achieve a rate of $n^{-1}$ but crucially rely on both measures having compact support. We leverage this result to demonstrate fast convergence of sample sRE and sRMMD to their population version making them useful for high-dimensional GoF testing. Our statistics are differentiable and amenable to popular machine learning frameworks that rely on gradient methods. We leverage these properties towards showcasing the utility of the proposed statistics for generative modeling on two important problems: image generation and generating valid knockoffs for controlled feature selection.

Hyperspectral images taken from aircraft or satellites contain information from hundreds of spectral bands, within which lie latent lower-dimensional structures that can be exploited for classifying vegetation and other materials. A disadvantage of working with hyperspectral images is that, due to an inherent trade-off between spectral and spatial resolution, they have a relatively coarse spatial scale, meaning that single pixels may correspond to spatial regions containing multiple materials. This article introduces the Diffusion and Volume maximization-based Image Clustering (D-VIC) algorithm for unsupervised material clustering to address this problem. By directly incorporating pixel purity into its labeling procedure, D-VIC gives greater weight to pixels corresponding to a spatial region containing just a single material. D-VIC is shown to outperform comparable state-of-the-art methods in extensive experiments on a range of hyperspectral images, including land-use maps and highly mixed forest health surveys (in the context of ash dieback disease), implying that it is well-equipped for unsupervised material clustering of spectrally-mixed hyperspectral datasets.