Related papers: The Slide Dimension of Point Processes
Many real objects are modeled as discrete sets of points, such as corners or other salient features. For our main applications in chemistry, points represent atomic centers in a molecule or a solid material. We study the problem of…
Statistical data depth plays an important role in the analysis of multivariate data sets. The main outcome is a center-outward ordering of the observations that can be used both to highlight features of the underlying distribution of the…
We consider stochastic differential equations in a Hilbert space, perturbed by the gradient of a convex potential. We investigate the problem of convergence of a sequence of such processes. We propose applications of this method to…
Following in the footsteps of P. Erd\H{o}s and A. R\'enyi we compute the Hausdorff dimension of sets of numbers whose digits with respect to their $Q$-Cantor series expansions satisfy various statistical properties. In particular, we…
Distances between probability distributions that take into account the geometry of their sample space,like the Wasserstein or the Maximum Mean Discrepancy (MMD) distances have received a lot of attention in machine learning as they can, for…
We show that the values of entropies of multidimensional shifts of finite type (SFTs) are characterized by a certain computation-theoretic property: a real number $h\geq 0$ is the entropy of such an SFT if and only if it is right…
We find numerically that the sample to sample fluctuation of the entropy, $\Delta S$, is a tool more sensitive in distinguishing how from high temperature behaviors, than the corresponding fluctuation in the free energy. In 1+1 dimensions…
We present a detailed analysis of random motions moving in higher spaces with a natural number of velocities. In the case of the so-called minimal random dynamics, under some wide assumptions, we show the joint distribution of the position…
We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$ for all…
The classical Hausdorff dimension of finite or countable metric spaces is zero. Recently, we defined a variant, called \emph{finite Hausdorff dimension}, which is not necessarily trivial on finite metric spaces. In this paper we apply this…
We define a new variant of exclusion processes in discrete time that has jump probabilities that depend on the last jump performed. In a particular limit for the jump probabilities and in suitable scaling limits for space and time, we…
This paper is devoted to the investigation of gradient flows in asymmetric metric spaces (for example, irreversible Finsler manifolds and Minkowski normed spaces) by means of discrete approximation. We study basic properties of curves and…
Shannon entropy is widely used to quantify the uncertainty of discrete random variables. But when normalized to the unit interval, as is often done in practice, it no longer conveys the alphabet sizes of the random variables being studied.…
Calculations and mechanistic explanations for the probabilistic movement of objects at the highly relevant cm length scales has been lacking and overlooked due to the complexity of current techniques. Predicting the final-configuration…
We study entropy-bounded computational geometry, that is, geometric algorithms whose running times depend on a given measure of the input entropy. Specifically, we introduce a measure that we call range-partition entropy, which unifies and…
The organization of high-dimensional probability spaces is a fundamental problem at the intersection of statistical physics and information theory. Here, we analyze the distributions populating level surfaces of the probability simplex…
The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite…
Simulating an arbitrary discrete distribution $D \in [0, 1]^n$ using fair coin tosses incurs trade-offs between entropy complexity and space and time complexity. Shannon's theory suggests that $H(D)$ tosses are necessary and sufficient, but…
Stein operators allow to characterise probability distributions via differential operators. Based on these characterisations, we develop a new method of point estimation for marginal parameters of strictly stationary and ergodic processes,…
We determine the Hausdorff dimension for the range of a class of pure jump Markov processes in $\mathbb{R}^d$, which turns out to be random and depends on the trajectories of these processes. The key argument is carried out through the SDE…