相关论文: Geometry of the matching distance for 2D filtering…
One of the most animated themes of multidimensional persistence is the comparison between invariants. The matching distance between persistent Betti numbers functions (or rank invariants), is among the most studied metrics in this context,…
In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this…
Several important algorithms for machine learning and data analysis use pairwise distances as input. On Riemannian manifolds these distances may be prohibitively costly to compute, in particular for large datasets. To tackle this problem,…
The matching distance is a computationally tractable topological measure to compare multi-filtered simplicial complexes. We design efficient algorithms for approximating the matching distance of two bi-filtered complexes to any desired…
Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions…
Given a fixed closed manifold M, we exhibit an explicit formula for the distance function of the canonical L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the…
We introduce the convex matching distance, a novel metric for comparing functions with values in the real plane. This metric measures the maximal bottleneck distance between the persistence diagrams associated with the convex combinations…
We discuss some properties of the distance functions on Riemannian manifolds and we relate their behavior to the geometry of the manifolds. This leads to alternative proofs of some "classical" theorems connecting curvature and topology.
Suppose that there exists a discrete subset $X$ of a complete, connected, $n$-dimensional Riemannian manifold $M$ such that the Riemannian distances between points of $X$ correspond to the Euclidean distances of a net in $\mathbb{R}^{n}$.…
This paper defines a distance function that measures the dissimilarity between planar geometric figures formed with straight lines. This function can in turn be used in partial matching of different geometric figures. For a given pair of…
A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two…
To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback…
We first introduce a class of divergence measures between power spectral density matrices. These are derived by comparing the suitability of different models in the context of optimal prediction. Distances between "infinitesimally close"…
We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.
When matching parts of a surface to its whole, a fundamental question arises: Which points should be included in the matching process? The issue is intensified when using isometry to measure similarity, as it requires the validation of…
We present an extended version of Riemannian geometry suitable for the description of current formulations of double field theory (DFT). This framework is based on graded manifolds and it yields extended notions of symmetries, dynamical…
A "hidden symmetry" of a Riemannian manifold M is an isometry of a d-sheeted, 1<d<\infty, Riemannian cover of M which is not the lift of any isometry. In this paper we characterize the locally symmetric metric(s) on a closed, arithmetic…
The exact computation of the matching distance for multi-parameter persistence modules is an active area of research in computational topology. Achieving an easily obtainable exact computation of this distance would permit multi-parameter…
Over the past few years, symmetric positive definite (SPD) matrices have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the…
The Frechet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to handwriting recognition. More recently, the Frechet distance has been generalized to a…