Related papers: Exact computation of the halfspace depth
Let $\RR$ be a real closed field (e.g. the field of real numbers) and $\mathscr{S} \subset \RR^n$ be a semi-algebraic set defined as the set of points in $\RR^n$ satisfying a system of $s$ equalities and inequalities of multivariate…
Processing point clouds using deep neural networks is still a challenging task. Most existing models focus on object detection and registration with deep neural networks using point clouds. In this paper, we propose a deep model that learns…
The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion a ranking of observations which is absent in more than one dimension. Motivated by the rapid…
We introduce a natural notion of depth that applies to individual cutting planes as well as entire families. This depth has nice properties that make it easy to work with theoretically, and we argue that it is a good proxy for the practical…
Metric magnitude is a measure of the "size" of point clouds with many desirable geometric properties. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization…
Private data analysis suffers a costly curse of dimensionality. However, the data often has an underlying low-dimensional structure. For example, when optimizing via gradient descent, the gradients often lie in or near a low-dimensional…
3D point cloud completion, the task of inferring the complete geometric shape from a partial point cloud, has been attracting attention in the community. For acquiring high-fidelity dense point clouds and avoiding uneven distribution,…
What is the dimension of a network? Here, we view it as the smallest dimension of Euclidean space into which nodes can be embedded so that pairwise distances accurately reflect the connectivity structure. We show that a recently proposed…
The halfspace depth is a prominent tool of nonparametric multivariate analysis. The upper level sets of the depth, termed the trimmed regions of a measure, serve as a natural generalization of the quantiles and inter-quantile regions to…
The development of practical applications, such as autonomous driving and robotics, has brought increasing attention to 3D point cloud understanding. While deep learning has achieved remarkable success on image-based tasks, there are many…
This paper introduces {\em fusion subspace clustering}, a novel method to learn low-dimensional structures that approximate large scale yet highly incomplete data. The main idea is to assign each datum to a subspace of its own, and minimize…
We propose a novel measure of statistical depth, the metric spatial depth, for data residing in an arbitrary metric space. The measure assigns high (low) values for points located near (far away from) the bulk of the data distribution,…
When 3D-point clouds from overhead sensors are used as input to remote sensing data exploitation pipelines, a large amount of effort is devoted to data preparation. Among the multiple stages of the preprocessing chain, estimating the…
Depth is a concept that measures the `centrality' of a point in a given data cloud or in a given probability distribution. Every depth defines a family of so-called trimmed regions. For statistical applications it is desirable that with…
Point cloud stands as the most widely adopted format for representing 3D shapes and scenes due to its simplicity and geometric fidelity. However, its inherent unordered and irregular nature, exacerbated by sensor noise and occlusions,…
Geometry and topology have generated impacts far beyond their pure mathematical primitive, providing a solid foundation for many applicable tools. Typically, real-world data are represented as vectors, forming a linear subspace for a given…
The digital factory provides undoubtedly a great potential for future production systems in terms of efficiency and effectivity. A key aspect on the way to realize the digital copy of a real factory is the understanding of complex indoor…
Tukey's depth offers a powerful tool for nonparametric inference and estimation, but also encounters serious computational and methodological difficulties in modern statistical data analysis. This paper studies how to generalize and compute…
Point cloud compression has garnered significant interest in computer vision. However, existing algorithms primarily cater to human vision, while most point cloud data is utilized for machine vision tasks. To address this, we propose a…
Point cloud analysis has a wide range of applications in many areas such as computer vision, robotic manipulation, and autonomous driving. While deep learning has achieved remarkable success on image-based tasks, there are many unique…