Related papers: On multidimensional record patterns
This article provides an overview on the statistical modeling of complex data as increasingly encountered in modern data analysis. It is argued that such data can often be described as elements of a metric space that satisfies certain…
For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…
Stochastic processes offer a flexible mathematical formalism to model and reason about systems. Most analysis tools, however, start from the premises that models are fully specified, so that any parameters controlling the system's dynamics…
In this paper, we consider the direct and inverse problems of the description of lattice positive random fields by various systems of finite-dimensional (as well as one-point) probability distributions parameterized by boundary conditions.…
Using statistical physics methods, we study generative diffusion models in the regime where the dimension of space and the number of data are large, and the score function has been trained optimally. Our analysis reveals three distinct…
The statistics of records for a time series generated by a continuous time random walk is studied, and found to be independent of the details of the jump length distribution, as long as the latter is continuous and symmetric. However, the…
We simulate static memory materials on a two-dimensional lattice. The bulk properties of such materials depend on boundary conditions. Considerable information can be stored in various local patterns. We observe local probabilities…
The diffraction of various random subsets of the integer lattice $\mathbb{Z}^{d}$, such as the coin tossing and related systems, are well understood. Here, we go one important step beyond and consider random point sets in $\mathbb{R}^{d}$.…
It is a standard assumption that datasets in high dimension have an internal structure which means that they in fact lie on, or near, subsets of a lower dimension. In many instances it is important to understand the real dimension of the…
We consider the persistent exclusion process in which a set of persistent random walkers interact via hard-core exclusion on a hypercubic lattice in $d$ dimensions. We work within the ballistic regime whereby particles continue to hop in…
Using an inverse of the standard linear congruential random number generator, large randomly occupied lattices can be visited by a random walker without having to determine the occupation status of every lattice site in advance. In seven…
Random feature maps are ubiquitous in modern statistical machine learning, where they generalize random projections by means of powerful, yet often difficult to analyze nonlinear operators. In this paper, we leverage the "concentration"…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
Consider that the coordinates of $N$ points are randomly generated along the edges of a $d$-dimensional hypercube (random point problem). The probability that an arbitrary point is the $m$th nearest neighbor to its own $n$th nearest…
Graphical models have become a very popular tool for representing dependencies within a large set of variables and are key for representing causal structures. We provide results for uniform inference on high-dimensional graphical models…
This paper considers distributed statistical inference for general symmetric statistics %that encompasses the U-statistics and the M-estimators in the context of massive data where the data can be stored at multiple platforms in different…
The exact statistics of an arbitrary quantum observable is analytically obtained. Due to the probabilistic nature of a sequence of intermediate measurements and stochastic fluctuations induced by the interaction with the environment, the…
This paper considers the maximum likelihood estimation of factor models of high dimension, where the number of variables (N) is comparable with or even greater than the number of observations (T). An inferential theory is developed. We…
We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and show that it can be repeated recursively any number $n$ of generations. In two dimensions, we determine the percolation…
A new methodology is proposed for generating realizations of a random vector with values in a finite-dimensional Euclidean space that are statistically consistent with a data set of observations of this vector. The probability distribution…