Related papers: Riemannian geometry for efficient analysis of prot…
Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a…
This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…
Finding constrained saddle points on Riemannian manifolds is significant for analyzing energy landscapes arising in physics and chemistry. Existing works have been limited to special manifolds that admit global regular level-set…
Computing geodesics for Riemannian manifolds is a difficult task that often relies on numerical approximations. However, these approximations tend to be either numerically unstable, have slow convergence, or scale poorly with manifold…
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions,…
Covariance matrices have attracted attention for machine learning applications due to their capacity to capture interesting structure in the data. The main challenge is that one needs to take into account the particular geometry of 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,…
We provide a full characterization of geodesic completeness for spaces of configurations of landmarks with smooth Riemannian metrics that satisfy a rotational and translation invariance and which are induced from metrics on subgroups of the…
This paper proposes a general framework of Riemannian adaptive optimization methods. The framework encapsulates several stochastic optimization algorithms on Riemannian manifolds and incorporates the mini-batch strategy that is often used…
Human motion generation is often learned in Euclidean spaces, although valid motions follow structured non-Euclidean geometry. We present Riemannian Motion Generation (RMG), a unified framework that represents motion on a product manifold…
Modeling distributions on Riemannian manifolds is a crucial component in understanding non-Euclidean data that arises, e.g., in physics and geology. The budding approaches in this space are limited by representational and computational…
The fundamental laws of physics are intrinsically geometric, dictating the evolution of systems through principles of symmetry and conservation. While modern machine learning offers powerful tools for modeling complex dynamics from data,…
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of…
In recent decades, advancements in motion learning have enabled robots to acquire new skills and adapt to unseen conditions in both structured and unstructured environments. In practice, motion learning methods capture relevant patterns and…
Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations…
The Riemannian geometry is one of the main theoretical pieces in Modern Mathematics and Physics. The study of Riemann Geometry in the relevant literature is performed by using a well defined analytical path. Usually it starts from the…
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…
This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is…
We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold reconstruction where a smooth $n$-dimensional submanifold…
Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a smooth manifold $\mathcal{M}$. Algorithms designed in this framework usually require some geometrical…