Related papers: Computing Enclosing Depth
We study the problem of covering a given set of $n$ points in a high, $d$-dimensional space by the minimum enclosing polytope of a given arbitrary shape. We present algorithms that work for a large family of shapes, provided either only…
The paper is devoted to studying the performance of a computational pipeline, the number of simultaneously executing stages of which at each time is bounded from above by a fixed number. A look at the restriction as a structural hazard…
We design an efficient data structure for computing a suitably defined approximate depth of any query point in the arrangement $\mathcal{A}(S)$ of a collection $S$ of $n$ halfplanes or triangles in the plane or of halfspaces or simplices in…
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…
Data depths are score functions that quantify in an unsupervised fashion how central is a point inside a distribution, with numerous applications such as anomaly detection, multivariate or functional data analysis, arising across various…
Clustering high-dimensional datasets is hard because interpoint distances become less informative in high-dimensional spaces. We present a clustering algorithm that performs nonlinear dimensionality reduction and clustering jointly. The…
During the past two decades there has been a lot of interest in developing statistical depth notions that generalize the univariate concept of ranking to multivariate data. The notion of depth has also been extended to regression models and…
Endoscopic depth estimation is a critical technology for improving the safety and precision of minimally invasive surgery. It has attracted considerable attention from researchers in medical imaging, computer vision, and robotics. Over the…
A fundamental problem in shape matching and geometric similarity is computing the maximum area overlap between two polygons under translation. For general simple polygons, the best-known algorithm runs in $O((nm)^2 \log(nm))$ time [Mount,…
In many high-impact applications, it is important to ensure the quality of output of a machine learning algorithm as well as its reliability in comparison with the complexity of the algorithm used. In this paper, we have initiated a…
This study presents a novel algorithm for identifying the set of extreme points that constitute the exact convex hull of a point set in high-dimensional Euclidean space. The proposed method iteratively solves a sequence of dynamically…
Tukey depth function is one of the most famous multivariate tools serving robust purposes. It is also very well known for its computability problems in dimensions $p \ge 3$. In this paper, we address this computing issue by presenting two…
The depth rule is a level truncation of tensor product coefficients expected to be sufficient for the evaluation of fusion coefficients. We reformulate the depth rule in a precise way, and show how, in principle, it can be used to calculate…
We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade…
In this paper, we study the problem of computing the diameter of a set of $n$ points in $d$-dimensional Euclidean space for a fixed dimension $d$, and propose a new $(1+\varepsilon)$-approximation algorithm with $O(n+ 1/\varepsilon^{d-1})$…
We introduce derivation depth-a computable metric of the reasoning effort needed to answer a query based on a given set of premises. We model information as a two-layered structure linking abstract knowledge with physical carriers, and…
High-dimensional datasets typically cluster around lower-dimensional manifolds but are also often marred by severe noise, obscuring the intrinsic geometry essential for downstream learning tasks. We present a quantum algorithm for…
A new depth-based clustering procedure for directional data is proposed. Such method is fully non-parametric and has the advantages to be flexible and applicable even in high dimensions when a suitable notion of depth is adopted. The…
Counting is among the most fundamental operations in computing. For example, counting the pth frequency moment has been a very active area of research, in theoretical computer science, databases, and data mining. When p=1, the task (i.e.,…
We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…