A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The paper gauges AGI's impact on GDP growth using mathematical metrics.
Paper discusses the Fisher metric and differentiability in statistical models.
This is a survey article, based on the author's lectures in the 2015 Current developments in Mathematics meeting; published in "Current developments in Mathematics". Version 2, references corrected and added.
Mathematical approach defines stability conditions for ML models.
New metric for probability measures connects physics and geometry.
Distance metric learning is a branch of machine learning that aims to learn distances from the data, which enhances the performance of similarity-based algorithms. This tutorial provides a theoretical background and foundations on this topic and a comprehensive experimental analysis of the most-known algorithms. We sta…
We correct a mistake in Shen Yibing, Yu Yaoyong, On Projectively Related Randers Metrics, International Journal of Mathematics 19}(2008), no. 5, 503--520, and prove the natural generalization of the projective Lichnerowicz-Obata conjecture for Randers metrics.
This is a continuous work about the nonexistence of some complete metrics on the product of two manifolds studied by Tam-Yu [Asian Journal of Mathematics, 14(2010)]. Motivated by the result of Tossati [Comm.Anal.Geom. 15(2007)]. We generalize the corresponding results of Tam-Yu [Asian Journal of Mathematics, 14(2010)] …
This article is based upon lectures given at the 2013 IAS/Park City Mathematics Institute summer program in geometric analysis.
Introduces Finslerian convolution metrics and their properties.
The paper shows objective derivatives are covariant derivatives on Riemannian metrics.
Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. Th…
Delisle's projection explained by Euler in 18th century.
The b-boundary is a mathematical tool used to attach a topological boundary to incomplete Lorentzian manifolds using a Riemaniann metric called the Schmidt metric on the frame bundle. In this paper, we give the general form of the Schmidt metric in the case of Lorentzian surfaces. Furthermore, we write the Ricci scalar…
Mathematical theory of super fiber bundles and connections developed.
Paper finds previous work on submanifolds incorrect.
A framework evaluates synthetic tabular data quality objectively.
Mathematical framework for understanding attention in neural networks.
We propose a statistical mechanical derivation of Kahler-Einstein metrics, i.e. solutions to Einstein's vacuum field equations in Euclidean signature (with a cosmological constant) on a compact Kahler manifold X. The microscopic theory is given by a canonical free fermion gas on X whose one-particle states are plurican…
In this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion. Such connections have already been classified in the work of Cartan. The maps unde…
We describe four algorithms for neural network training, each adapted to different scalability constraints. These algorithms are mathematically principled and invariant under a number of transformations in data and network representation, from which performance is thus independent. These algorithms are obtained from th…
A new mathematical approach to general covariance using stacks and Lie algebras.
The paper developes a geometrization of a Kronecker -regular vertical fundamental metrical d-tensor on the jet fibre bundle of order one . This geometrization gives a mathematical model for both gravitational and electromagnetic field theory, in a general setting. In this context, the…
We discuss the existence of Killing tensors for certain (physically motivated) stationary and axially symmetric vacuum space-times. We show nonexistence of a nontrivial Killing tensor for a Tomimatsu-Sato metric (up to valence 7), for a C-metric (up to valence 9) and for a Zipoy-Voorhees metric (up to valence 11). The …
Characterizes metrics with finite total Q-curvature and introduces new volume entropy.
This paper investigates the theoretical foundations of metric learning, focused on three key questions that are not fully addressed in prior work: 1) we consider learning general low-dimensional (low-rank) metrics as well as sparse metrics; 2) we develop upper and lower (minimax)bounds on the generalization error; 3) w…
Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound. This partly extends previous a priori estimates of Ye Li (J. Geom. Anal. 17 (20…
Sub-Riemannian geometry connects bike paths to mathematical curves.
Mathematical framework for field theories on Finsler spacetimes.
New metrics on curve spaces improve shape analysis.
This essay, an excerpt of the author's Ph.D. in Philosophy of mathematics (2012) thought of as being a companion to recent discoveries of new explicit Cartan geometry curvatures, analyzes how Gauss, after having devised the isometrically invariant character of curvature, struggled with elimination computations in order…
We improve Riemannian metrics for constrained systems control.
Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop …
The paper explains knowledge distillation by analyzing visual concepts in DNNs.
We consider how microlocal methods developed for tomographic problems can be used to detect singularities of the Lorentzian metric of the Universe using measurements of the Cosmic Microwave Background radiation. The physical model we study is mathematically rigorous but highly idealized.
Newtonian, Lagrangian, and Hamiltonian dynamical systems are well formalized mathematically. They give rise to geometric structures describing motion of a point in smooth manifolds. Riemannian metric is a different geometric structure formalizing concepts of length and angle. The interplay of Riemannian metric and its …
Deep Neural Networks(DNN) have excessively advanced the field of computer vision by achieving state of the art performance in various vision tasks. These results are not limited to the field of vision but can also be seen in speech recognition and machine translation tasks. Recently, DNNs are found to poorly fail when …
We report on some advances made in the problem of singularities in general relativity. First is introduced the singular semi-Riemannian geometry for metrics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like th…
We address the problem of training models with black-box and hard-to-optimize metrics by expressing the metric as a monotonic function of a small number of easy-to-optimize surrogates. We pose the training problem as an optimization over a relaxed surrogate space, which we solve by estimating local gradients for the me…
Topolow embeds dissimilarity data into Euclidean space robustly against non-metricity and sparsity.
The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.
A trivial projective change of a Finsler metric is the Finsler metric . I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is poss…
This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M - the Cheeger-Gromov theory - and extensions thereof to Ricci curvature in place of full curvature. This theory is then applied to study a collection of different issues in mathematical aapects of General Relativi…
The paper broadens a mathematical correspondence to include more balanced metrics.
By exploiting standard facts about and supersymmetric Yang-Mills theory, the Donaldson invariants of four-manifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric Yang…
We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal…
Defines intrinsically Hölder sections in metric spaces.
This paper introduces a new mathematical formulation and numerical approach for the computation of distances and geodesics between immersed planar curves. Our approach combines the general simplifying transform for first-order elastic metrics that was recently introduced by Kurtek and Needham, together with a relaxatio…