Related papers: Stratifying High Dimensional Data Based on Proximi…
A method for extracting multiscale geometric features from a data cloud is proposed and analyzed. The basic idea is to map each pair of data points into a real-valued feature function defined on $[0,1]$. The construction of these feature…
Distributional (or distribution-valued) data are a new type of data arising from several sources and are considered as realizations of distributional variables. A new set of fuzzy c-means algorithms for data described by distributional…
We study the following range searching problem: Preprocess a set $P$ of $n$ points in the plane with respect to a set $\mathcal{O}$ of $k$ orientations % , for a constant, in the plane so that given an $\mathcal{O}$-oriented convex polygon…
This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of…
In recent years new application areas have emerged in which one aims to capture the geometry of objects by means of three-dimensional point clouds. Often the obtained data consist of a dense sampling of the object's surface, containing many…
In this work, we present a new efficient method for convex shape representation, which is regardless of the dimension of the concerned objects, using level-set approaches. Convexity prior is very useful for object completion in computer…
Hyperdimensional computing (HDC) is a method to perform classification that uses binary vectors with high dimensions and the majority rule. This approach has the potential to be energy-efficient and hence deemed suitable for…
This paper presents a new O(nlog(n)) algorithm for computing the convex hull of a set of 3 dimensional points. The algorithm first sorts the point in (x,y,z) then incrementally adds sorted points to the convex hull using the constraint that…
Hypergraph spectral analysis has emerged as an effective tool processing complex data structures in data analysis. The surface of a three-dimensional (3D) point cloud and the multilateral relationship among their points can be naturally…
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…
Under any Multiclass Classification (MCC) setting defined by a collection of labeled point-cloud specified by a feature-set, we extract only stochastic partial orderings from all possible triplets of point-cloud without explicitly measuring…
Hierarchical and tree-like data sets arise in many applications, including language processing, graph data mining, phylogeny and genomics. It is known that tree-like data cannot be embedded into Euclidean spaces of finite dimension with…
Inspired by the classical fractional cascading technique, we introduce new techniques to speed up the following type of iterated search in 3D: The input is a graph $\mathbf{G}$ with bounded degree together with a set $H_v$ of 3D hyperplanes…
Finding valuable training data points for deep neural networks has been a core research challenge with many applications. In recent years, various techniques for calculating the "value" of individual training datapoints have been proposed…
We study the generalization of deep learning models in relation to the convex hull of their training sets. A trained image classifier basically partitions its domain via decision boundaries and assigns a class to each of those partitions.…
High dimensional data analysis is known to be as a challenging problem. In this article, we give a theoretical analysis of high dimensional classification of Gaussian data which relies on a geometrical analysis of the error measure. It…
Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…
In this paper, we investigate the geometric structure of activation spaces of fully connected layers in neural networks and then show applications of this study. We propose an efficient approximation algorithm to characterize the convex…
Convex clustering is a well-regarded clustering method, resembling the similar centroid-based approach of Lloyd's $k$-means, without requiring a predefined cluster count. It starts with each data point as its centroid and iteratively merges…
We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity.…