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In this work, we propose to study the global geometrical properties of generative models. We introduce a new Riemannian metric to assess the similarity between any two data points. Importantly, our metric is agnostic to the parametrization…
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…
Murphy and the second author showed that a generic closed Riemannian manifold has no totally geodesic submanifolds, provided it is at least four dimensional. Lytchak and Petrunin established the same thing in dimension 3. For the higher…
Difficulties in replication and reproducibility of empirical evidences in machine learning research have become a prominent topic in recent years. Ensuring that machine learning research results are sound and reliable requires…
In this report, the explicit probability density functions of the random Euclidean distances associated with equilateral triangles are given, when the two endpoints of a link are randomly distributed in 1) the same triangle, 2) two adjacent…
This paper is concerned with the study of a circular random distribution called geodesic Normal distribution recently proposed for general manifolds. This distribution, parameterized by two real numbers associated to some specific location…
We derive the isoperimetric profile of Gaussian type for an absolutely continuous probability measure on Euclidean spaces with respect to the Lebesgue measure, whose density is a radial function.The key is a generalization of the Poincar\'e…
In the past decades, the central limit theorem (CLT) has been generalized to non-Euclidean data spaces. Some years ago, it was found that for some random variables on the circle, the sample Fr\'echet mean fluctuates around the population…
We formulate a geometric measurement theory of dynamical classical systems possessing both continuous and discrete degrees of freedom. The approach is covariant with respect to choices of clocks and canonically incorporates laboratories.…
Deep neural networks have shown great potential in image reconstruction problems in Euclidean space. However, many reconstruction problems involve imaging physics that are dependent on the underlying non-Euclidean geometry. In this paper,…
For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…
We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional…
The main purpose of this paper is to propose an ergodic theoretic approach to the study of entire holomorphic curves. Brody curves are one-Lipschitz holomorphic maps from the complex plane to the complex projective space. They naturally…
In this paper we describe the geometry of distributions by their symmetries, and present a simplified proof of the Frobenius theorem and some related corollaries. Then, we study the geometry of solutions of $F-$Gordon equation; A PDE which…
Implicit Neural Representations (INRs) have emerged as a powerful tool for geometric representation, yet their suitability for physics-based simulation remains underexplored. While metrics like Hausdorff distance quantify surface…
Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the…
We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this…
A formalism is presented for analytically obtaining the probability density function, (P_{n}(s)), for the random distance (s) between two random points in an (n)-dimensional spherical object of radius (R). Our formalism allows (P_{n}(s)) to…
We present new exact solutions for two-dimensional geometries generated by continuous distributions of topological defects within a conformal metric framework. By reformulating Einstein's equations in two dimensions as a Poisson equation…
It is known that any target function is realized in a sufficiently small neighborhood of any randomly connected deep network, provided the width (the number of neurons in a layer) is sufficiently large. There are sophisticated theories and…