arXiv research

Paper proposes KIIM to infer causal relationships from data.

A new method calculates intrinsic effective sample size for manifold-valued data.

Diffusion Maps framework is a kernel based method for manifold learning and data analysis that defines diffusion similarities by imposing a Markovian process on the given dataset. Analysis by this process uncovers the intrinsic geometric structures in the data. Recently, it was suggested to replace the standard kernel …

New method uses kernel deviance measures to discover causal relationships in heterogeneous data.

New method for long-term sampling of complex dynamics on curved spaces.

Identifies conditions for multiple invariant probabilities in Markov kernels.

Graph kernels for metric graphs using tropical algebra.

A new measure of dependence for various data types.

Study heat kernel on quaternionic contact manifolds, finding linear dependence of coefficients on curvature.

This paper studies rectifiability in Carnot groups and proves geometric area formulas.

Study provides bounds for estimating intrinsic dimension using Gaussian kernels.

New research optimizes HSIC estimation rate for translation-invariant kernels.

DID measures similarity invariant to diffeomorphisms.

New summary measures reveal geometric structure in weighted measures on manifolds.

Despite the success of the popular kernelized support vector machines, they have two major limitations: they are restricted to Positive Semi-Definite (PSD) kernels, and their training complexity scales at least quadratically with the size of the data. Many natural measures of similarity between pairs of samples are not…

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the $L^p$ norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds …

Paper advances sparse regularisation theory for measures with new kernel insights.

A new measure of causal influence quantifies intrinsic contributions in DAGs.

We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $L^p$-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvatu…

A new kernel for probability measures based on optimal transport.

Proposes IIKL for preserving geometric properties of non-Euclidean data.

We consider various notions of strains; quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of Patrizio Neff and co-workers , is to select a Riemannian metric on $\text{GL}_n$, and use its induced geodes…

The Weyl principle holds in some Finsler settings despite general failure.

We present in this work a new family of kernels to compare positive measures on arbitrary spaces $\Xcal$ endowed with a positive kernel $κ$, which translates naturally into kernels between histograms or clouds of points. We first cover the case where $\Xcal$ is Euclidian, and focus on kernels which take into account th…

Complex problems may require sophisticated, non-linear learning methods such as kernel machines or deep neural networks to achieve state of the art prediction accuracies. However, high prediction accuracies are not the only objective to consider when solving problems using machine learning. Instead, particular scientif…

We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is two-fold. First, we show, usi…

Random Gaussian fields on 4D Riemannian manifolds with conformal invariance.

Paper infers intrinsic dimension from quasi-convex measurements.

Physics-informed neural networks improve by measuring effective dimensionality of constraints.

Monetary risk measures are usually interpreted as the smallest amount of external capital that must be added to a financial position to make it acceptable. We propose a new concept: intrinsic risk measures and argue that this approach provides a direct path from unacceptable positions towards the acceptance set. Intrin…

Unified understanding of neural representation similarity measures.

Paper extends Bayesian Cramér-Rao bound with geometric considerations.

Data living on manifolds commonly appear in many applications. Often this results from an inherently latent low-dimensional system being observed through higher dimensional measurements. We show that under certain conditions, it is possible to construct an intrinsic and isometric data representation, which respects an …

Unified framework for singular statistical models using observable charts.

We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, $\{ν_t\}_{t>0},$ are also studied. We show that these he…

eDCF estimates intrinsic dimension using local connectivity.

New metric measure space theory for Lipschitz constants.

We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself.…

Diffusion maps are a commonly used kernel-based method for manifold learning, which can reveal intrinsic structures in data and embed them in low dimensions. However, as with most kernel methods, its implementation requires a heavy computational load, reaching up to cubic complexity in the number of data points. This l…

In-BO optimizes complex constrained domains using SIn-GP surrogate models.

It is classically known that generic smooth maps of R^2 into R^3 admit only cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap (u,uv,v^2) has non-trivial isometric deformations with infinite dimensional freedom. …

Graph Laplace operators uniquely identify metrics and densities on manifolds.

Researchers compute differential invariants for Carrollian spacetimes.

A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the spac…

CIRCE measures conditional independence for learning invariant features.

New risk measures for financial networks avoid external capital, reducing systemic risk.

A new method integrates autoencoders with geometry regularization for manifold learning.

Extends Mahalanobis distance to Banach spaces for anomaly detection.