Related papers: Riemannian Statistics for Any Type of Data
We introduce novel estimators for computing the curvature, tangent spaces, and dimension of data from manifolds, using tools from diffusion geometry. Although classical Riemannian geometry is a rich source of inspiration for geometric data…
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,…
Regression on manifolds, and, more broadly, statistics on manifolds, has garnered significant importance in recent years due to the vast number of applications for non Euclidean data. Circular data is a classic example, but so is data in…
While classical data analysis has addressed observations that are real numbers or elements of a real vector space, at present many statistical problems of high interest in the sciences address the analysis of data that consist of more…
Non-Euclidean constraints are inherent in many kinds of data in computer vision and machine learning, typically as a result of specific invariance requirements that need to be respected during high-level inference. Often, these geometric…
A generalisation of Riemannian geometry is considered, based exclusively on the minimal assumptions that the line element $ds$ is a regular function of position and direction and that the distance of every point from itself is equal to…
Riemannian manifolds have been widely employed for video representations in visual classification tasks including video-based face recognition. The success mainly derives from learning a discriminant Riemannian metric which encodes the…
Several important algorithms for machine learning and data analysis use pairwise distances as input. On Riemannian manifolds these distances may be prohibitively costly to compute, in particular for large datasets. To tackle this problem,…
Classical archetypal analysis is appealing for its interpretability, but its linear geometry can limit performance on data with strongly non-linear structure; at the same time, existing neural extensions improve flexibility while often…
This paper studies how to generalize Tukey's depth to problems defined in a restricted space that may be curved or have boundaries, and to problems with a nondifferentiable objective. First, using a manifold approach, we propose a broad…
We revisit the task of learning a Euclidean metric from data. We approach this problem from first principles and formulate it as a surprisingly simple optimization problem. Indeed, our formulation even admits a closed form solution. This…
Manifold-valued parameters routinely arise in modern statistical applications such as in medical imaging, robotics, and computer vision, to name a few. While traditional Bayesian approaches are applicable to such settings by considering an…
Random geometric graphs are random graph models defined on metric measure spaces. A random geometric graph is generated by first sampling points from a metric space and then connecting each pair of sampled points independently with a…
Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance…
Manifold learning is a fundamental task at the core of data analysis and visualisation. It aims to capture the simple underlying structure of complex high-dimensional data by preserving pairwise dissimilarities in low-dimensional…
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…
Statistical inference for spatial processes from partially realized or scattered data has seen voluminous developments in diverse areas ranging from environmental sciences to business and economics. Inference on the associated rates of…
These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian…
We introduce a novel geometric approach to the image labeling problem. Abstracting from specific labeling applications, a general objective function is defined on a manifold of stochastic matrices, whose elements assign prior data that are…
The space of embedded submanifolds plays an important role in applications such as computational anatomy and shape analysis. We can define two different classes on Riemannian metrics on this space: so-called outer metrics are metrics that…