Related papers: Intrinsic dimensional functional inequalities on m…
Given a matrix $D$ describing the pairwise dissimilarities of a data set, a common task is to embed the data points into Euclidean space. The classical multidimensional scaling (cMDS) algorithm is a widespread method to do this. However,…
The neural manifold hypothesis postulates that the activity of a neural population forms a low-dimensional manifold whose structure reflects that of the encoded task variables. In this work, we combine topological deep generative models and…
We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected.…
We study quantitative isoperimetric inequalities for two different perimeter-type functionals. We first consider classical capillarity functionals, which measure the perimeter of sets in a Euclidean half-space, assigning a constant weight…
The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of…
We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy…
The intrinsic unsharpness of a quantum observable is studied by introducing the notion of resolution width. This quantification of accuracy is shown to be closely connected with the possibility of making approximately repeatable…
Quantification of the number of variables needed to locally explain complex data is often the first step to better understanding it. Existing techniques from intrinsic dimension estimation leverage statistical models to glean this…
Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their…
The link between 3D spaces with (in general, non-constant) curvature and quantum deformations is presented. It is shown how the non-standard deformation of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that…
We consider the problem of reconstructing an embedding of a compact connected Riemannian manifold in a Euclidean space up to an almost isometry, given the information on intrinsic distances between points from its ``sufficiently large''…
The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space $\mathbb{R}^d$ ($d\geq 3$). These hemivariational inequalities have a variational structure and,…
In the framework of the Hartle-Hawking no-boundary proposal, we investigated quantum creation of the multidimensional universe with a cosmological constant ($\Lambda$) but without matter fields. We have found that the classical solutions of…
Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined…
The objective of this paper is to learn dense 3D shape correspondence for topology-varying generic objects in an unsupervised manner. Conventional implicit functions estimate the occupancy of a 3D point given a shape latent code. Instead,…
A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by…
We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh,…
Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the…
Infinite-dimensional manifolds modelled on arbitrary Hilbert spaces of functions are considered. It is shown that changes in model rather than changes of charts within the same model make coordinate formalisms on finite and…
The ongoing exponential rise in recording capacity calls for new approaches for analysing and interpreting neural data. Effective dimensionality has emerged as an important property of neural activity across populations of neurons, yet…