Related papers: Data depth and floating body
We show that a nontrivial graph isomorphism problem of two undirected graphs, and more generally, the permutation similarity of two given $n\times n$ matrices, is equivalent to equalities of volumes of the induced three convex bounded…
We introduce a novel projection depth for data lying in a general Hilbert space, called the regularized projection depth, with a focus on functional data. By regularizing projection directions, the proposed depth does not suffer from the…
We study points of density 1/2 of sets of finite perimeter in infinite-dimensional Gaussian spaces and prove that, as in the finite-dimensional theory, the surface measure is concentrated on this class of points. Here density 1/2 is…
We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the…
Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…
The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.
We introduce a new family of affine metrics on a locally strictly convex surface $M$ in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if $M$ is immersed in a…
In this paper, we investigate the similarity property of the sample projection depth contours. It turns out that some of these contours are of \emph{the same shape} with different sizes, following a similar fashion to the Mahalanobis depth…
We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.
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…
The lens depth of a point has been recently extended to general metric spaces, which is not the case for most depths. It is defined as the probability of being included in the intersection of two random balls centred at two random points X…
The domains of mesh functions are strict subsets of the underlying space of continuous independent variables. Spaces of partial maps between topological spaces admit topologies which do not depend on any metric. Such topologies…
We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…
The intrinsic volumes of a convex body are fundamental invariants that capture information about the average volume of the projection of the convex body onto a random subspace of fixed dimension. The intrinsic volumes also play a central…
Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a…
A median-radius framework for assessing centrality in multivariate data using median distances is proposed. Based on the proposed framework, a scale invariant measure of radial dispersion is defined and used to establish a depth function…
In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous;…
The simplicial depth, like other relevant multivariate statistical data depth functions, vanishes right outside the convex hull of the support of the distribution with respect to which the depth is computed. This is problematic when it is…
The merit of projecting data onto linear subspaces is well known from, e.g., dimension reduction. One key aspect of subspace projections, the maximum preservation of variance (principal component analysis), has been thoroughly researched…
Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article…