Related papers: Curvature of high-dimensional data

Evidence that the Universe may be close to the critical density, required for its expansion eventually to be halted, comes principally from dynamical studies of large-scale structure. These studies either use the observed peculiar velocity…

Point cloud filtering and normal estimation are two fundamental research problems in the 3D field. Existing methods usually perform normal estimation and filtering separately and often show sensitivity to noise and/or inability to preserve…

Bayesian model averaging is a procedure to obtain parameter constraints that account for the uncertainty about the correct cosmological model. We use recent cosmological observations and Bayesian model averaging to derive tight limits on…

How can we tell complex point clouds with different small scale characteristics apart, while disregarding global features? Can we find a suitable transformation of such data in a way that allows to discriminate between differences in this…

Recent progress of semantic point clouds analysis is largely driven by synthetic data (e.g., the ModelNet and the ShapeNet), which are typically complete, well-aligned and noisy free. Therefore, representations of those ideal synthetic…

Extracting high-level structural information from 3D point clouds is challenging but essential for tasks like urban planning or autonomous driving requiring an advanced understanding of the scene at hand. Existing approaches are still not…

The success of deep learning in high-dimensional settings is often attributed to the presence of low-dimensional structure in real-world data. While standard theoretical models typically assume that this structure lies in the target…

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted…

For manifold learning, it is assumed that high-dimensional sample/data points are embedded on a low-dimensional manifold. Usually, distances among samples are computed to capture an underlying data structure. Here we propose a metric…

Many approaches in the field of machine learning and data analysis rely on the assumption that the observed data lies on lower-dimensional manifolds. This assumption has been verified empirically for many real data sets. To make use of this…

Most existing approaches for point cloud normal estimation aim to locally fit a geometric surface and calculate the normal from the fitted surface. Recently, learning-based methods have adopted a routine of predicting point-wise weights to…

Despite the extensive usage of point clouds in 3D vision, relatively limited data are available for training deep neural networks. Although data augmentation is a standard approach to compensate for the scarcity of data, it has been less…

Three dimensional (3D) object recognition is becoming a key desired capability for many computer vision systems such as autonomous vehicles, service robots and surveillance drones to operate more effectively in unstructured environments.…

We study mean convex mean curvature flow $M_s$ of local spacelike graphs in the flat slicing of de Sitter space. We show that if the initial slice is of non-negative time and is graphical over a large enough ball, and if $M_s$ is of bounded…

We consider density estimators based on the nearest neighbors method applied to discrete point distibutions in spaces of arbitrary dimensionality. If the density is constant, the volume of a hypersphere centered at a random location is…

We analyse the clustering of matter on large scales in an extension of the concordance model that allows for spatial curvature. We develop a consistent approach to curvature and wide-angle effects on the galaxy 2-point correlation function…

It is now practically the norm for data to be very high dimensional in areas such as genetics, machine vision, image analysis and many others. When analyzing such data, parametric models are often too inflexible while nonparametric…

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties…

This work presents an accurate and robust method for estimating normals from point clouds. In contrast to predecessor approaches that minimize the deviations between the annotated and the predicted normals directly, leading to direction…

Normalizing Flows (NFs) are universal density estimators based on Neural Networks. However, this universality is limited: the density's support needs to be diffeomorphic to a Euclidean space. In this paper, we propose a novel method to…