Related papers: A Random Bockstein Operator
Topological Data Analysis has grown in popularity in recent years as a way to apply tools from algebraic topology to large data sets. One of the main tools in topological data analysis is persistent homology. This paper uses undergraduate…
The restricted Boltzmann machine is a network of stochastic units with undirected interactions between pairs of visible and hidden units. This model was popularized as a building block of deep learning architectures and has continued to…
The theory of random sets is demonstrated to prove useful for the theory of random operators. A random operator is here defined by requiring the graph to be a random set. It is proved that the spectrum and the set of eigenvalues of random…
Machine learning provides algorithms that can learn from data and make inferences or predictions on data. Stochastic acceptors or probabilistic automata are stochastic automata without output that can model components in machine learning…
Studies the cohomology of p-central, powerful, p-groups with a certain extension property. These groups are naturally associated to Lie algebras. The paper develops a machinery that calculates the first few terms of the Bockstein spectral…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…
Ensembles of random stochastic and bistochastic matrices are investigated. While all columns of a random stochastic matrix can be chosen independently, the rows and columns of a bistochastic matrix have to be correlated. We evaluate the…
Recent developments have found unexpected connections between non-commutative probability theory and algebraic topology. In particular, Boolean cumulants functionals seem to be important for describing morphisms of homotopy operadic…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
This article proposes a co-variance operator for Banach valued random elements using the concept of $U$-statistic. We then study the asymptotic distribution of the proposed co-variance operator along with related large sample properties.…
We develop sampling methodology aimed at determining stochastic operators that satisfy a support size restriction on the autocorrelation of the operators stochastic spreading function. The data that we use to reconstruct the operator (or,…
Topological measurements are increasingly being accepted as an important tool for quantifying complex structures. In many applications, these structures can be expressed as nodal domains of real-valued functions and are obtained only…
The analytic properties of the Markov operator associated to a random walk are common tools in the study of the behaviour and some probabilistic features related to the walk. In this paper we consider a class of Markov operators which…
Topological data analysis refers to approaches for systematically and reliably computing abstract ``shapes'' of complex data sets. There are various applications of topological data analysis in life and data sciences, with growing interest…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
We study topological boundedness of order-to-topology bounded and order-to-topology continuous operators from ordered vector spaces to topological vector spaces. The uniform boundedness principle for such operators is investigated.
We deal with the general structure of (noncommutative) stochastic processes by using the standard techniques of Operator Algebras. Any stochastic process is associated to a state on a universal object, i.e. the free product $C^*$-algebra in…
The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data…
Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful tool to complement traditional scientific computing,…
Topological data analysis is a relatively new branch of machine learning that excels in studying high dimensional data, and is theoretically known to be robust against noise. Meanwhile, data objects with mixed numeric and categorical…