Related papers: The statistical Minkowski distances: Closed-form f…
This work proposes a distance that combines Minkowski and Chebyshev distances and can be seen as an intermediary distance. This combination not only achieves efficient run times in neighbourhood iteration tasks in Z^2, but also obtains good…
We consider a discrete latent variable model for two-way data arrays, which allows one to simultaneously produce clusters along one of the data dimensions (e.g. exchangeable observational units or features) and contiguous groups, or…
In this paper, we introduce the concept of N-dimensional generalized Minkowski space, i.e. a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical coodinates. This is the first…
The clustering of bounded data presents unique challenges in statistical analysis due to the constraints imposed on the data values. This paper introduces a novel method for model-based clustering specifically designed for bounded data.…
We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities…
In this paper we give explicit gauge invariant Lagrangian formulation for massive theories based on mixed symmetry tensors \Phi_{[\mu\nu],\alpha}, T_{[\mu\nu\alpha],\beta} and R_{[\mu\nu],[\alpha\beta]} both in Minkowski as well as in…
Exploiting the geometric nature of statistical divergences, we devise a way to define associated induced uncertainty measures for discrete and finite probability distributions. We also report new uncertainty measures and discuss their…
Comparing metric measure spaces (i.e. a metric space endowed with aprobability distribution) is at the heart of many machine learning problems. The most popular distance between such metric measure spaces is theGromov-Wasserstein (GW)…
Dirichlet Process mixture models (DPMM) in combination with Gaussian kernels have been an important modeling tool for numerous data domains arising from biological, physical, and social sciences. However, this versatility in applications…
Given two points $p,q$ in the real plane, the signed area of the rectangle with the diagonal $[pq]$ equals the square of the Minkowski distance between the points $p,q$. We prove that $N>1$ points in the Minkowski plane $\R^{1,1}$ generate…
We propose a new approach for deriving probabilistic inequalities based on bounding likelihood ratios. We demonstrate that this approach is more general and powerful than the classical method frequently used for deriving concentration…
In this paper, we explore the connection between convergence in distribution and Mallows distance in the context of positively associated random variables. Our results extend some known invariance principles for sequences with FKG property.…
We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections…
We propose a novel method for the description of spatial patterns formed by a coverage of point sets representing galaxy samples. This method is based on a complete family of morphological measures known as Minkowski functionals, which…
The aim of this paper is to introduce new statistical criterions for estimation, suitable for inference in models with common continuous support. This proposal is in the direct line of a renewed interest for divergence based inference tools…
Computational difficulty of quadratic matching and the Gromov-Wasserstein distance has led to various approximation and relaxation schemes. One of such methods, relying on the notion of distance profiles, has been widely used in practice,…
In this note we apply the general Reilly formula established in \cite{QX} to the solution of a Neumann boundary value problem to prove an optimal Minkowski type inequality in space forms.
We prove bounds on statistical distances between high-dimensional exchangeable mixture distributions (which we call \emph{permutation mixtures}) and their i.i.d. counterparts. Our results are based on a novel method for controlling $\chi^2$…
The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland [1976] show that…
Discrepancy measures between probability distributions, often termed statistical distances, are ubiquitous in probability theory, statistics and machine learning. To combat the curse of dimensionality when estimating these distances from…