Related papers: A Hilbert Space of Stationary Ergodic Processes

We isolate a geometric mechanism that complements the dynamical suppression of macroscopic interference: In a high-dimensional Hilbert space, almost all state vectors are nearly orthogonal, accommodating an exponentially large reservoir of…

We develop a Hilbert space framework for a number of general multi-scale problems from dynamics. The aim is to identify a spectral theory for a class of systems based on iterations of a non-invertible endomorphism. We are motivated by the…

Using an information theoretic point of view, we investigate how a dynamics acting on a network can be coarse grained through the use of graph partitions. Specifically, we are interested in how aggregating the state space of a Markov…

A probabilistic approach of computing geometric rate of convergence of stochastic processes is introduced in this paper. The goal is to quantitatively compute both upper and lower bounds of the exponential rate of convergence to the…

Stochastic center manifolds theory are crucial in modelling the dynamical behavior of complex systems under stochastic influences. A multiplicative ergodic theorem on Hilbert space is proved to be satisfied to the exponential trichotomy…

Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However,…

Optimization techniques are at the core of many scientific and engineering disciplines. The steepest descent methods play a foundational role in this area. In this paper we studied a generalized steepest descent method on Riemannian…

We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel…

Recovering high-dimensional statistical structure from limited measurements is a fundamental challenge in hyperspectral imaging, where capturing full-resolution data is often infeasible due to sensor, bandwidth, or acquisition constraints.…

False and nuisance alarms in industrial fault detection systems are often triggered by uncertainty, causing normal process variable fluctuations to be erroneously identified as faults. This paper introduces a novel encoder-based residual…

These notes rigorously construct the stochastic integral of a Hilbert Space valued process driven by a Cylindrical Brownian Motion. We expand upon this stochastic calculus to present an introduction to stochastic differential equations in…

The Ornstein-Uhlenbeck process may be used to generate a noise signal with a finite correlation time. If a one-dimensional stochastic process is driven by such a noise source, it may be analysed by solving a Fokker-Planck equation in two…

Implicit sampling is a weighted sampling method that is used in data assimilation, where one sequentially updates estimates of the state of a stochastic model based on a stream of noisy or incomplete data. Here we describe how to use…

Modern multi-core systems have a large number of design parameters, most of which are discrete-valued, and this number is likely to keep increasing as chip complexity rises. Further, the accurate evaluation of a potential design choice is…

High-throughput data analyses are becoming common in biology, communications, economics and sociology. The vast amounts of data are usually represented in the form of matrices and can be considered as knowledge networks. Spectra-based…

Robotic applications require a comprehensive understanding of the scene. In recent years, neural fields-based approaches that parameterize the entire environment have become popular. These approaches are promising due to their continuous…

A key technique of machine learning and computer vision is to embed discrete weighted graphs into continuous spaces for further downstream processing. Embedding discrete hierarchical structures in hyperbolic geometry has proven very…

Any single system whose space of states is given by a separable Hilbert space is automatically equipped with infinitely many hidden tensor-like structures. This includes all quantum mechanical systems as well as classical field theories and…

Systems in nature are stochastic as well as nonlinear. In traditional applications, engineered filters aim to minimize the stochastic effects caused by process and measurement noise. Conversely, a previous study showed that the process…

Modern compression algorithms exploit complex structures that are present in signals to describe them very efficiently. On the other hand, the field of compressed sensing is built upon the observation that "structured" signals can be…