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In this paper, we develop a new classification method for manifold-valued data in the framework of probabilistic learning vector quantization. In many classification scenarios, the data can be naturally represented by symmetric positive…
We consider the problem of estimating the density of observations taking values in classical or nonclassical spaces such as manifolds and more general metric spaces. Our setting is quite general but also sufficiently rich in allowing the…
A common observation in data-driven applications is that high dimensional data has a low intrinsic dimension, at least locally. In this work, we consider the problem of estimating a $d$ dimensional sub-manifold of $\mathbb{R}^D$ from a…
3D point-cloud recognition with PointNet and its variants has received remarkable progress. A missing ingredient, however, is the ability to automatically evaluate point-wise importance w.r.t.\! classification performance, which is usually…
We solve a regularized weighted low-rank approximation problem by a stochastic gradient descent on a manifold. To guarantee the convergence of our stochastic gradient descent, we establish a convergence theorem on manifolds for…
We consider the problem of density estimation on Riemannian manifolds. Density estimation on manifolds has many applications in fluid-mechanics, optics and plasma physics and it appears often when dealing with angular variables (such as…
In statistical dimensionality reduction, it is common to rely on the assumption that high dimensional data tend to concentrate near a lower dimensional manifold. There is a rich literature on approximating the unknown manifold, and on…
In this paper, we present an algorithm for the computation of harmonic maps, and respectively, of the harmonic map heat flow between two closed Riemannian manifolds. Our approach is based on the totally geodesic embedding of the target…
While point-based neural architectures have demonstrated their efficacy, the time-consuming sampler currently prevents them from performing real-time reasoning on scene-level point clouds. Existing methods attempt to overcome this issue by…
We consider a class of nonsmooth optimization problems over the Stiefel manifold, in which the objective function is weakly convex in the ambient Euclidean space. Such problems are ubiquitous in engineering applications but still largely…
Neural networks for point clouds, which respect their natural invariance to permutation and rigid motion, have enjoyed recent success in modeling geometric phenomena, from molecular dynamics to recommender systems. Yet, to date, no model…
Deep neural networks have revolutionized 3D point cloud processing, yet efficiently handling large and irregular point clouds remains challenging. To tackle this problem, we introduce FastPoint, a novel software-based acceleration technique…
We obtain new curvature estimates and Bernstein type results for minimal $n-$submanifolds in $\ir{n+m},\, m\ge 2$ under the condition that the rank of its Gauss map is at most 2. In particular, this applies to minimal surfaces in Euclidean…
The study of the relation between the Weierstrass inducing formulae for constant mean curvature surfaces and the completely integrable euclidean nonlinear sigma-model suggests a connection among integrable sigma -models in a background and…
Measuring the similarity between data points often requires domain knowledge, which can in parts be compensated by relying on unsupervised methods such as latent-variable models, where similarity/distance is estimated in a more compact…
Manifold learning is a central task in modern statistics and data science. Many datasets (cells, documents, images, molecules) can be represented as point clouds embedded in a high dimensional ambient space, however the degrees of freedom…
Fitting an unknown number of hyperplanes to data is a fundamental yet challenging problem in machine learning, characterized by its non-convexity, non-differentiability, and unknown model order. Existing approaches often struggle with local…
A common method for estimating the Hessian operator from random samples on a low-dimensional manifold involves locally fitting a quadratic polynomial. Although widely used, it is unclear if this estimator introduces bias, especially in…
We introduce a method to successively locate equilibria (steady states) of dynamical systems on Riemannian manifolds. The manifolds need not be characterized by an a priori known atlas or by the zeros of a smooth map. Instead, they can be…
Point clouds obtained from 3D scans are typically sparse, irregular, and noisy, and required to be consolidated. In this paper, we present the first deep learning based edge-aware technique to facilitate the consolidation of point clouds.…