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1. Variational Barycentric Coordinates

   Dodik, Ana; Stein, Oded; Sitzmann, Vincent; Solomon, Justin (December 2023, ACM Transactions on Graphics)

   We propose a variational technique to optimize for generalized barycentric coordinates that offers additional control compared to existing models. Prior work represents barycentric coordinates using meshes or closed-form formulae, limiting the choice of objective function. In contrast, we directly parameterize the continuous function mapping any coordinate in a polytope’s interior to its barycentric coordinates using a neural field. Enabled by our theoretical characterization of barycentric coordinates, we construct neural fields parameterizing valid coordinates. We demonstrate flexibility using various objective functions, validate our algorithm, and present several applications. 

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2. Synthesis and Analysis of Data as Probability Measures With Entropy-Regularized Optimal Transport

   Mallery, Brendan; Murphy, James M; Aeron, Shuchin (January 2025, Open Review)

   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 m reference measures given a set of coefficients belonging to the m-dimensional simplex. The analysis problem consists of finding the coefficients for the closest barycenter in the Wasserstein-2 distance to a given measure μ. 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 regularized barycenters as solutions to a fixed-point equation for the average of the entropic maps from the barycenter to the reference measures. This characterization yields a finite-dimensional, convex, quadratic program for solving the analysis problem when μ is a barycenter. It is shown that these coordinates, as well as the value of the barycenter functional, can be estimated from samples with dimension-independent rates of convergence, a hallmark of entropy-regularized optimal transport, and we verify these rates experimentally. We also establish that barycentric coordinates are stable with respect to perturbations in the Wasserstein-2 metric, suggesting a robustness of these coefficients to corruptions. 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. 

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3. Measure Estimation in the Barycentric Coding Model

   Werenski, M.; Jiang, R.; Tasissa, A; Aeron, S.; Murphy, J.M. (July 2022, Proceedings of the 39th International Conference on Machine Learning)

   Chaudhuri, K.; Stefanie, J.; Song, L.; Szepesvari, C.; Niu, G; Sabato, S. (Ed.)

   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. 

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4. Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes

   https://doi.org/10.1515/cmam-2016-0019  

   Gillette, Andrew; Rand, Alexander; Bajaj, Chandrajit (January 2016, Computational Methods in Applied Mathematics)

   Abstract We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart–Thomas, and Brezzi–Douglas–Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties. 

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5. Sifting through the Static: Moving Object Detection in Difference Images

   https://doi.org/10.3847/1538-3881/ac22ff  

   Smotherman, Hayden; Connolly, Andrew J.; Kalmbach, J. Bryce; Portillo, Stephen K.; Bektesevic, Dino; Eggl, Siegfried; Juric, Mario; Moeyens, Joachim; Whidden, Peter J. (November 2021, The Astronomical Journal)

   Abstract Trans-Neptunian objects provide a window into the history of the solar system, but they can be challenging to observe due to their distance from the Sun and relatively low brightness. Here we report the detection of 75 moving objects that we could not link to any other known objects, the faintest of which has a VR magnitude of 25.02 ± 0.93 using the Kernel-Based Moving Object Detection (KBMOD) platform. We recover an additional 24 sources with previously known orbits. We place constraints on the barycentric distance, inclination, and longitude of ascending node of these objects. The unidentified objects have a median barycentric distance of 41.28 au, placing them in the outer solar system. The observed inclination and magnitude distribution of all detected objects is consistent with previously published KBO distributions. We describe extensions to KBMOD, including a robust percentile-based lightcurve filter, an in-line graphics-processing unit filter, new coadded stamp generation, and a convolutional neural network stamp filter, which allow KBMOD to take advantage of difference images. These enhancements mark a significant improvement in the readiness of KBMOD for deployment on future big data surveys such as LSST. 

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