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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 $[\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n] = \mathbf{X} \in \mathbb{R}^{d \times n}$ and a vector $\mathbf{y} \in \mathbb{R}^d$, the goal is to find coefficients $\mathbf{w} \in \mathbb{R}^n$ so that $\mathbf{X} \mathbf{w} \approx \mathbf{y}$, subject to some desired structure on $\mathbf{w}$. In this work we seek $\mathbf{w}$ that forms a local reconstruction of $\mathbf{y}$ by solving a regularized least squares regression problem. We obtain local solutions through a locality function that promotes the use of columns of $\mathbf{X}$ that are close to $\mathbf{y}$ when used as a regularization term. We prove that, for all levels of regularization and under a mild condition that the columns of $\mathbf{X}$ 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 $d \ll n$. Under the same condition we also show that for any $\mathbf{y}$ contained in the convex hull of $\mathbf{X}$ there exists a regime of regularization parameter such that the optimal coefficients are supported on the vertices of the Delaunay simplex containing $\mathbf{y}$. This provides an interpretation of the sparsity as having structure obtained implicitly from the Delaunay triangulation of $\mathbf{X}$. We demonstrate that our locality regularized problem can be solved in comparable time to other methods that identify the containing Delaunay simplex., Comment: 26 pages, 8 figures

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., Comment: 27 pages, 9 figures, and 2 tables

This paper addresses the problem of estimating entropy-regularized optimal transport (EOT) maps with squared-Euclidean cost between source and target measures that are subGaussian. In the case that the target measure is compactly supported or strongly log-concave, we show that for a recently proposed in-sample estimator, the expected squared $L^2$-error decays at least as fast as $O(n^{-1/3})$ where $n$ is the sample size. For the general subGaussian case we show that the expected $L^1$-error decays at least as fast as $O(n^{-1/6})$, and in both cases we have polynomial dependence on the regularization parameter. While these results are suboptimal compared to known results in the case of compactness of both the source and target measures (squared $L^2$-error converging at a rate $O(n^{-1})$) and for when the source is subGaussian while the target is compactly supported (squared $L^2$-error converging at a rate $O(n^{-1/2})$), their importance lie in eliminating the compact support requirements. The proof technique makes use of a bias-variance decomposition where the variance is controlled using standard concentration of measure results and the bias is handled by T1-transport inequalities along with sample complexity results in estimation of EOT cost under subGaussian assumptions. Our experimental results point to a looseness in controlling the variance terms and we conclude by posing several open problems., Comment: 30 pages, 7 figures

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.

This paper considers the use of recently proposed optimal transport-based multivariate test statistics, namely rank energy and its variant the soft rank energy derived from entropically regularized optimal transport, for the unsupervised nonparametric change point detection (CPD) problem. 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 theoretical analyses remove the need for resampling and out-of-sample extensions previously required to obtain such rates. In contrast 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., Comment: 36 pages, 5 figures

Purpose: To assess the fiducial motion in abdominal stereotactic body radiotherapy (SBRT) using the cone-beam computed tomography (CBCT) projections acquired for pre-treatment patient set-up. Materials and Methods: Pre-treatment CBCT projections and anterior-posterior (AP) and lateral (LAT) pair of fluoroscopic sequences of 7 pancreatic and 6 liver SBRT patients with implanted fiducials were analyzed for 49 treatment fractions retrospectively. A tracking algorithm based on template matching and sequential stereo triangulation algorithms was used to track the fiducials in the CBCT projections and the fluoro sequence pairs. We predicted the clinical couch adjustment from CBCT tracking and compared it with the clinical couch decision made during the patient's treatment. Results: In 3D coordinate, the fiducial motion ranges for pancreas cases were 9.90+/-3.52 mm, 10.65+/-5.91 mm, and 10.74+/-6.24 mm for CBCT, AP, and LAT fluoro, respectively, while in the liver, they were 13.93+/-3.39 mm, 11.17+/-3.75 mm, and 11.52+/-4.33 mm, respectively. Prediction of couch adjustment in LAT, SI, and AP coordinates from CBCT tracking agrees with the actual clinical couch correction within 0.92+/-0.74 mm, 1.37+/-1.26 mm, and 0.68+/-0.56 mm for pancreas cases and within 1.12+/-0.96 mm, 1.15+/-0.92 mm and 0.90+/-0.86 mm for liver cases, respectively. Conclusion: Tracking pre-treatment CBCT projections using template matching and sequential stereo triangulation is suitable for assessing fiducial motion and adjusting the patient setup for abdominal SBRT. CBCT can be used for motion modeling, potentially eliminating the need for additional fluoroscopic pair acquisition and thus reducing the imaging dose to the patient and the total treatment time.

Contouring is an indispensable step in Radiotherapy (RT) treatment planning. However, today's contouring software is constrained to only work with a 2D display, which is less intuitive and requires high task loads. Virtual Reality (VR) has shown great potential in various specialties of healthcare and health sciences education due to the unique advantages of intuitive and natural interactions in immersive spaces. VR-based radiation oncology integration has also been advocated as a target healthcare application, allowing providers to directly interact with 3D medical structures. We present VRContour and investigate how to effectively bring contouring for radiation oncology into VR. Through an autobiographical iterative design, we defined three design spaces focused on contouring in VR with the support of a tracked tablet and VR stylus, and investigating dimensionality for information consumption and input (either 2D or 2D + 3D). Through a within-subject study (n = 8), we found that visualizations of 3D medical structures significantly increase precision, and reduce mental load, frustration, as well as overall contouring effort. Participants also agreed with the benefits of using such metaphors for learning purposes., Comment: C. Chen, M. Yarmand, V. Singh, M.V. Sherer, J.D. Murphy, Y. Zhang and N. Weibel, "VRContour: Bringing Contour Delineations of Medical Structures Into Virtual Reality", 2022 IEEE International Symposium on Mixed and Augmented Reality (ISMAR), 2022, pp. 1-10, doi: 10.1109/ISMAR55827.2022.00020

Wasserstein dictionary learning is an unsupervised approach to learning a collection of probability distributions that generate observed distributions as Wasserstein barycentric combinations. Existing methods for Wasserstein dictionary learning optimize an objective that seeks a dictionary with sufficient representation capacity via barycentric interpolation to approximate the observed training data, but without imposing additional structural properties on the coefficients associated to the dictionary. This leads to dictionaries that densely represent the observed data, which makes interpretation of the coefficients challenging and may also lead to poor empirical performance when using the learned coefficients in downstream tasks. In contrast and motivated by sparse dictionary learning in Euclidean spaces, we propose a geometrically sparse regularizer for Wasserstein space that promotes representations of a data point using only nearby dictionary elements. We show this approach leads to sparse representations in Wasserstein space and addresses the problem of non-uniqueness of barycentric representation. Moreover, when data is generated as Wasserstein barycenters of fixed distributions, this regularizer facilitates the recovery of the generating distributions in cases that are ill-posed for unregularized Wasserstein dictionary learning. Through experimentation on synthetic and real data, we show that our geometrically regularized approach yields sparser and more interpretable dictionaries in Wasserstein space, which perform better in downstream applications., Comment: 24 pages

Hyperspectral images, which store a hundred or more spectral bands of reflectance, have become an important data source in natural and social sciences. Hyperspectral images are often generated in large quantities at a relatively coarse spatial resolution. As such, unsupervised machine learning algorithms incorporating known structure in hyperspectral imagery are needed to analyze these images automatically. This work introduces the Spatial-Spectral Image Reconstruction and Clustering with Diffusion Geometry (DSIRC) algorithm for partitioning highly mixed hyperspectral images. DSIRC reduces measurement noise through a shape-adaptive reconstruction procedure. In particular, for each pixel, DSIRC locates spectrally correlated pixels within a data-adaptive spatial neighborhood and reconstructs that pixel's spectral signature using those of its neighbors. DSIRC then locates high-density, high-purity pixels far in diffusion distance (a data-dependent distance metric) from other high-density, high-purity pixels and treats these as cluster exemplars, giving each a unique label. Non-modal pixels are assigned the label of their diffusion distance-nearest neighbor of higher density and purity that is already labeled. Strong numerical results indicate that incorporating spatial information through image reconstruction substantially improves the performance of pixel-wise clustering., Comment: 7 pages, 1 figure

Ash dieback (Hymenoscyphus fraxineus) is an introduced fungal disease that is causing the widespread death of ash trees across Europe. Remote sensing hyperspectral images encode rich structure that has been exploited for the detection of dieback disease in ash trees using supervised machine learning techniques. However, to understand the state of forest health at landscape-scale, accurate unsupervised approaches are needed. This article investigates the use of the unsupervised Diffusion and VCA-Assisted Image Segmentation (D-VIS) clustering algorithm for the detection of ash dieback disease in a forest site near Cambridge, United Kingdom. The unsupervised clustering presented in this work has high overlap with the supervised classification of previous work on this scene (overall accuracy = 71%). Thus, unsupervised learning may be used for the remote detection of ash dieback disease without the need for expert labeling., Comment: (6 pages, 2 figures). Accepted to Proceedings of IEEE IGARSS 2022

Hyperspectral images encode rich structure that can be exploited for material discrimination by machine learning algorithms. This article introduces the Active Diffusion and VCA-Assisted Image Segmentation (ADVIS) for active material discrimination. ADVIS selects high-purity, high-density pixels that are far in diffusion distance (a data-dependent metric) from other high-purity, high-density pixels in the hyperspectral image. The ground truth labels of these pixels are queried and propagated to the rest of the image. The ADVIS active learning algorithm is shown to strongly outperform its fully unsupervised clustering algorithm counterpart, suggesting that the incorporation of a very small number of carefully-selected ground truth labels can result in substantially superior material discrimination in hyperspectral images., Comment: (6 pages, 2 figures). Accepted to Proceedings of IEEE IGARSS 2022

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 that correspond 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., Comment: 28 pages, 11 figures

Europa's surface composition and evidence for cryovolcanic activity can provide insight into the properties and composition of the subsurface ocean, allowing the evaluation of its potential habitability. One promising avenue for revealing the surface processing and subsurface activity are the relative fractions of crystalline and amorphous water-ice observed on the surface, which are influenced by temperature, charged particle bombardment, vapor deposition, and cryovolcanic activity. The crystallinity observed on Europa's leading hemisphere cannot be reproduced by thermophysical and particle flux modeling alone, indicating that there may be additional processes influencing the surface. We performed a spectral mixture analysis on hyperspectral image cubes from the Galileo Near-Infrared Mapping Spectrometer (NIMS) to identify how surface crystallinity is influenced by physical processing at a high spatial resolution scale. We focus specifically on two image cubes, 15e015 closer to the equator and 17e009 closer to the south pole, both on the leading hemisphere. We performed a nonnegative least-squares spectral mixture analysis to reveal both the non-ice composition and the water-ice crystallinity of the surface. We found that amorphous water-ice dominates the spectrum at the equator and the south pole. We estimated a mean crystallinity of ~35% within the 15e015 NIMS cube and a mean crystallinity of ~15% within the 17e009 NIMS cube, which is consistent with ground-based spectroscopically derived crystallinities. We also identified a correlation of magnesium sulfate, magnesium chloride, and hydrated sulfuric acid with lineae and ridges, which may provide evidence for surface processing by upwelling subsurface material., Comment: Published in the Planetary Science Journal, 17 pages, 17 figures

This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycentric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing., Comment: ICML 2022

In this paper, we use analysis on graphs to study quantitative measures of segregation. We focus on a classical statistic from the geography and urban sociology literature known as Moran's I, which in our language is a score associated to a real-valued function on a graph, computed with respect to a spatial weight matrix such as the adjacency matrix associated to the geographic units that tile a city. Our results characterizing the extremal behavior of I illustrate the important role of the underlying graph structure, especially the degree distribution, in interpreting the score. In addition to the standard spatial weight matrices encoding unit adjacency, we consider the Laplacian L and a doubly-stochastic approximation M. These alternatives allow us to connect I to ideas from Fourier analysis and random walks. We offer illustrations of our theoretical results with a mix of stylized synthetic examples and real geographic/demographic data., Comment: 33 pages, 15 figures, 3 tables

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., Comment: 46 pages, 10 figures. Replacement note: Substantial revision over V2: Title change, first authors contribution change, new improved theoretical results relaxing compactness assumptions