arXiv research

The paper classifies singularities of plane congruences and affine distance functions.

The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both theories. The concepts of the theory of affine spheres lead to a new definition of an …

The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on Rm. Atsushi Kameyama (Distances on Topological Self-Similar Sets, Proceedings of Symposia in Pure Mathematics, Volume 72.1, 2004) asked the following fundamental question: given a topological self-si…

Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefini…

Algorithm learns affine transformations robustly from corrupted samples.

New proof shows affine manifolds with parallel volume are Riemannian-flat.

The study examines lower and upper bounds of Wasserstein distances for affine transformations of random vectors.

Study equi-affine invariants for convex domains with asymptotes.

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is closely related to normal curvature, and from this we re-calculate the Minkowski Gau…

Estimates latent positions in 1D torus from noisy pairwise affinities.

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygo…

The paper establishes inequalities for convex curves and applies them to lattice point estimates.

Study on estimating distances between covariance operators and Gaussian processes.

The paper explores fully affine maximal curves and their properties.

New tiles in higher dimensions are shown to be homeomorphic to balls.

Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric convex body in $\mathbb{R}^2$ such that it has area $π$ and its $p$-affine perimeter is close en…

Training neural networks under a strict Lipschitz constraint is useful for provable adversarial robustness, generalization bounds, interpretable gradients, and Wasserstein distance estimation. By the composition property of Lipschitz functions, it suffices to ensure that each individual affine transformation or nonline…

In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information geometry), dually flat spaces play a key role. In a dually flat space, there exist dual a…

In this paper we tackle the issue of clustering trajectories of geolocalized observations. Using clustering technics based on the choice of a distance between the observations, we first provide a comprehensive review of the different distances used in the literature to compare trajectories. Then based on the limitation…

In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for classical knotoids in analogy to corresponding inv…

NucleusDiff models atomic nuclei interactions to prevent separation violations in drug design.

New algorithm estimates task affinities without repeated training, improving model performance and efficiency.

The volume distance from a point p to a convex hypersurface M of the (N+1)-dimensional space is defined as the minimum (N+1)-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane H(p) we ob…

Introduces new weighted floating functions and affine surface areas.

This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, …

We describe a probabilistic (generative) view of affinity matrices along with inference algorithms for a subclass of problems associated with data clustering. This probabilistic view is helpful in understanding different models and algorithms that are based on affinity functions OF the data. IN particular, we show how(…

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topolo…

Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.

Improved convergence rate for kNN graph Laplacians with adaptive bandwidth.

The abstract investigates how volume ratios relate to curvature in geometric surfaces.

Spectral clustering improves accuracy and efficiency for clustering discrete distributions.

The paper develops inequalities for log-concave functions and related surface areas.

We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We…

In statistical analysis, measuring a score of predictive performance is an important task. In many scientific fields, appropriate scores were tailored to tackle the problems at hand. A proper score is a popular tool to obtain statistically consistent forecasts. Furthermore, a mathematical characterization of the proper…

We study affine Jacobi structures on an affine bundle $π:A\to M$, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on $A$ and Lie algebroid structures on the vector bundle $A^+=\bigcup_{p\in M}Aff(A_p,\R)$ of affine functionals. Som…

Study infinite Euclidean distance discriminants of algebraic varieties.

This work presents a parametrized family of divergences, namely Alpha-Beta Log- Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infi…

New method preserves distances in time series data.

We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.

Paper explores properties of slice-matching operators for measure transfer.

Single particle reconstruction (SPR) from cryo-electron microscopy (EM) is a technique in which the 3D structure of a molecule needs to be determined from its contrast transfer function (CTF) affected, noisy 2D projection images taken at unknown viewing directions. One of the main challenges in cryo-EM is the typically…

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds $N^n$ contained in hypersurfaces $M^{n+1}$ of the $(n+2)$-space. We give condition under which this affine focal set is a regular hypersurface and, for curves in $3$-space, we describe its stable singulariti…

A Finsler function $F$ is affinely rigid if its canonical spray is uniquely metrizable, in the sense that if $\bar F$ is another Finsler function whose canonical spray is $S$, then $d(F/\bar F)=0$. In this short note we explore some sufficient conditions for a Finsler function to be affinely rigid, and discuss open pro…

Let us denote by $\mathcal K_n$ the hyperspace of all convex bodies of $\mathbb R^n$ equipped with the Hausdorff distance topology. An affine invariant point $p$ is a continuous and Aff(n)-equivariant map $p:\mathcal K_n\to \mathbb R^n$, where Aff(n) denotes the group of all nonsingular affine maps of $\mathbb R^n$. Fo…

New bounds on optimal transport regularization show faster convergence rates than previously known.

Study of Markov-modulated affine processes for richer models in finance.

After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail.

Python package for SPD matrix distances, reproducible and extensible.