Related papers: Principal Autoparallel Analysis: Data Analysis in …
The (parallel linear) transports in tensor spaces generated by derivations of the tensor algebra along paths are axiomatically described. Certain their properties are investigated. Transports along paths defined by derivations of the tensor…
We provide a general theory for parallel transport on non-collapsed ${\sf RCD}$ spaces obtaining both existence and uniqueness results. Our theory covers the case of geodesics and, more generally, of curves obtained via the flow of…
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…
Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of…
Spatial statistics is an area of study devoted to the statistical analysis of data that have a spatial label associated with them. Geographers often refer to the "location information" associated with the "attribute information," whose…
Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient…
This handbook chapter provides an essential introduction to the field of spatial econometrics, offering a comprehensive overview of techniques and methodologies for analysing spatial data in the social sciences. Spatial econometrics…
In many robot motion planning problems, task objectives and physical constraints induce non-Euclidean geometry on the configuration space, yet many planners operate using Euclidean distances that ignore this structure. We address the…
We perform statistical analysis of the single-vehicle data measured on the Dutch freeway A9 and discussed in Ref. [2]. Using tools originating from the Random Matrix Theory we show that the significant changes in the statistics of the…
We express parallel transport for several common matrix Lie groups with a family of pseudo-Riemannian metrics in terms of matrix exponential and exponential actions. The metrics are constructed from a deformation of a bi-invariant metric…
Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order…
Let $M$ be a complete connected Riemannian manifold. For $n \geq 0$, we endow the Wasserstein space $P^{(n)}_2(M) = P_2(\ldots P_2(M)\ldots)$, equipped with the Wasserstein distance $W_2$, with a variational structure that generalizes the…
Many phenomena are naturally characterized by measuring continuous transformations such as shape changes in medicine or articulated systems in robotics. Modeling the variability in such datasets requires performing statistics on Lie groups,…
It has been shown beneficial for many types of data which present an underlying hierarchical structure to be embedded in hyperbolic spaces. Consequently, many tools of machine learning were extended to such spaces, but only few…
Traffic flow is a very prominent example of a driven non-equilibrium system. A characteristic phenomenon of traffic dynamics is the spontaneous and abrupt drop of the average velocity on a stretch of road leading to congestion. Such a…
Learning useful representations is a key ingredient to the success of modern machine learning. Currently, representation learning mostly relies on embedding data into Euclidean space. However, recent work has shown that data in some domains…
Latent space geometry provides a rigorous and empirically valuable framework for interacting with the latent variables of deep generative models. This approach reinterprets Euclidean latent spaces as Riemannian through a pull-back metric,…
Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family. We propose to study this…
The auto-parallel equation over spaces with affine connections and metrics is considered as a result of the application of the method of Lagrangians with covariant derivatives (MLCD) on a given Lagrangian density.
Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction -- how to assemble discrete samples into global…