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1. Riesz energy, L2$L^2$ discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus

   https://doi.org/10.1112/mtk.12245  

   Borda, Bence; Grabner, Peter; Matzke, Ryan W. (March 2024, Mathematika)

   Abstract Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so‐called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere and the flat torus , and the so‐called spherical ensemble on , which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega‐Cerdà on the Riesz ‐energy of the harmonic ensemble to the nonsingular regime , and as a corollary find the expected value of the spherical cap discrepancy via the Stolarsky invariance principle. We find the expected value of the discrepancy with respect to axis‐parallel boxes and Euclidean balls of the harmonic ensemble on . We also show that the spherical ensemble and the harmonic ensemble on and with points attain the optimal rate in expectation in the Wasserstein metric , in contrast to independent and identically distributed random points, which are known to lose a factor of . 

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2. The parallel-transported (quasi)-diabatic basis

   https://doi.org/10.1063/5.0122781  

   Littlejohn, Robert; Rawlinson, Jonathan; Subotnik, Joseph (November 2022, The Journal of Chemical Physics)

   This article concerns the use of parallel transport to create a diabatic basis. The advantages of the parallel-transported basis include the facility with which Taylor series expansions can be carried out in the neighborhood of a point or a manifold such as a seam (the locus of degeneracies of the electronic Hamiltonian), and the close relationship between the derivative couplings and the curvature in this basis. These are important for analytic treatments of the nuclear Schrödinger equation in the neighborhood of degeneracies. The parallel-transported basis bears a close relationship to the singular-value basis; in this article, both are expanded in power series about a reference point and are shown to agree through second order but not beyond. Taylor series expansions are effected through the projection operator, whose expansion does not involve energy denominators or any type of singularity and in terms of which both the singular-value basis and the parallel-transported basis can be expressed. The parallel-transported basis is a version of Poincaré gauge, well known in electromagnetism, which provides a relationship between the derivative couplings and the curvature and which, along with a formula due to Mead, affords an efficient method for calculating Taylor series of the basis states and the derivative couplings. The case in which fine structure effects are included in the electronic Hamiltonian is covered. 

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3. The breakdown of weak null singularities inside black holes

   https://doi.org/10.1215/00127094-2022-0096  

   Van_de_Moortel, Maxime (October 2023, Duke Mathematical Journal)

   It is widely expected that generic black holes have a nonempty but weakly singular Cauchy horizon, due to mass inflation. Indeed this has been proven by the author in the spherical collapse of a charged scalar field, under decay assumptions of the field in the black exterior which are conjectured to be generic. A natural question then arises: can this weakly singular Cauchy horizon close off the space-time, or does the weak null singularity necessarily “break down,” giving way to a different type of singularity? The main result of this paper is to prove that the Cauchy horizon cannot ever “close off” the space-time. As a consequence, the weak null singularity breaks down and transitions to a stronger singularity for which the area-radius r extends to 0. 

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4. The manifold scattering transform for high-dimensional point cloud data

   Chew, J.; Steach, H.; Viswanath, S.; Wu, H. T.; Hirn, M.; Needell, D.; Krishnaswamy S.; Perlmutter, M (November 2022, Proceedings of Machine Learning Research)

   The manifold scattering transform is a deep feature extractor for data defined on a Riemannian manifold. It is one of the first examples of extending convolutional neural network-like operators to general manifolds. The initial work on this model focused primarily on its theoretical stability and invariance properties but did not provide methods for its numerical implementation except in the case of two-dimensional surfaces with predefined meshes. In this work, we present practical schemes, based on the theory of diffusion maps, for implementing the manifold scattering transform to datasets arising in naturalistic systems, such as single cell genetics, where the data is a high-dimensional point cloud modeled as lying on a low-dimensional manifold. We show that our methods are effective for signal classification and manifold classification tasks. 

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5. Communication Efficient Parallel Algorithms for Optimization on Manifolds

   Saparbayeva, Bayan; Zhang, Michael; Lin, Lizhen (January 2018, Advances in Neural Information Processing Systems 31)

   The last decade has witnessed an explosion in the development of models, theory and computational algorithms for big data'' analysis. In particular, distributed inference has served as a natural and dominating paradigm for statistical inference. However, the existing literature on parallel inference almost exclusively focuses on Euclidean data and parameters. While this assumption is valid for many applications, it is increasingly more common to encounter problems where the data or the parameters lie on a non-Euclidean space, like a manifold for example. Our work aims to fill a critical gap in the literature by generalizing parallel inference algorithms to optimization on manifolds. We show that our proposed algorithm is both communication efficient and carries theoretical convergence guarantees. In addition, we demonstrate the performance of our algorithm to the estimation of Fr\'echet means on simulated spherical data and the low-rank matrix completion problem over Grassmann manifolds applied to the Netflix prize data set. 

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