Related papers: Riemannian Statistics for Any Type of Data
For robots to work alongside humans and perform in unstructured environments, they must learn new motion skills and adapt them to unseen situations on the fly. This demands learning models that capture relevant motion patterns, while…
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''…
We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach…
Circular and non-flat data distributions are prevalent across diverse domains of data science, yet their specific geometric structures often remain underutilized in machine learning frameworks. A principled approach to accounting for the…
Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and…
The paper proposes a Riemannian Manifold Hamiltonian Monte Carlo sampler to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The…
Medical data range from genomic sequences and retinal photographs to structured laboratory results and unstructured clinical narratives. Although these modalities appear disparate, many encode convergent information about a single…
We propose a geometrical approach to the investigation of Hamiltonian systems on (Pseudo) Riemannian manifolds. A new geometrical criterion of instability and chaos is proposed. This approach is more generic than well known reduction to the…
The problem of determining the configuration of points from partial distance information, known as the Euclidean Distance Geometry (EDG) problem, is fundamental to many tasks in the applied sciences. In this paper, we propose two algorithms…
Score-based generative models exhibit state of the art performance on density estimation and generative modeling tasks. These models typically assume that the data geometry is flat, yet recent extensions have been developed to synthesize…
The notions of (metric) hypersurface data were introduced in [Mars,2013] as a tool to analyze, from an abstract viewpoint, hypersurfaces of arbitrary signature in pseudo-riemannian manifolds. In this paper, general geometric properties of…
Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean…
The increasing application of deep-learning is accompanied by a shift towards highly non-linear statistical models. In terms of their geometry it is natural to identify these models with Riemannian manifolds. The further analysis of the…
This paper studies large-scale optimization problems on Riemannian manifolds whose objective function is a finite sum of negative log-probability losses. Such problems arise in various machine learning and signal processing applications. By…
In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of geodesics). By `universal' we mean without curvature…
In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…
Diffusion models have demonstrated remarkable empirical success in the recent years and are considered one of the state-of-the-art generative models in modern AI. These models consist of a forward process, which gradually diffuses the data…
The space of all non degenerate bilinear structures on a manifold $M$ carries a one parameter family of pseudo Riemannian metrics. We determine the geodesic equation, covariant derivative, curvature, and we solve the geodesic equation…
In statistical applications it has become increasingly common to encounter data structures that live on non-linear spaces such as manifolds. Classical linear regression, one of the most fundamental methodologies of statistical learning,…
An information-geometric approach to sensor management is introduced that is based on following geodesic curves in a manifold of possible sensor configurations. This perspective arises by observing that, given a parameter estimation problem…