Related papers: Learning functions, operators and dynamical system…

We provide a framework for learning of dynamical systems rooted in the concept of representations and Koopman operators. The interplay between the two leads to the full description of systems that can be represented linearly in a finite…

The study of mathematical connections between operator-theoretic formulations of classical dynamics and quantum mechanics began at least as early as the 1930s in work of Koopman and von Neumann and was developed in later decades by many…

Machine learning encompasses a set of tools and algorithms which are now becoming popular in almost all scientific and technological fields. This is true for molecular dynamics as well, where machine learning offers promises of extracting…

This paper presents an active learning strategy for robotic systems that takes into account task information, enables fast learning, and allows control to be readily synthesized by taking advantage of the Koopman operator representation. We…

We present a general framework to learn functions in tensor product reproducing kernel Hilbert spaces (TP-RKHSs). The methodology is based on a novel representer theorem suitable for existing as well as new spectral penalties for tensors.…

We propose an alternating optimization algorithm to the nonconvex Koopman operator learning problem for nonlinear dynamic systems. We show that the proposed algorithm will converge to a critical point with rate $O(1/T)$ and $O(\frac{1}{\log…

Lecture note topics: 1. Some tools from real and complex analysis, 2. Hilbert spaces, 3. Banach spaces, 4. Compact operators and their spectra, 5. Intermezzo: reproducing kernel Hilbert spaces, 6. Banach algebras ,7. Spectral theory of…

This work proposes kernel transform learning. The idea of dictionary learning is well known; it is a synthesis formulation where a basis is learnt along with the coefficients so as to generate or synthesize the data. Transform learning is…

This paper presents a new method for learning dissipative Hamiltonian dynamics from a limited and noisy dataset. The method uses the Helmholtz decomposition to learn a vector field as the sum of a symplectic and a dissipative vector field.…

We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations…

Metric and kernel learning are important in several machine learning applications. However, most existing metric learning algorithms are limited to learning metrics over low-dimensional data, while existing kernel learning algorithms are…

Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of…

The Koopman operator is a mathematical tool that allows for a linear description of non-linear systems, but working in infinite dimensional spaces. Dynamic Mode Decomposition and Extended Dynamic Mode Decomposition are amongst the most…

Koopman operators provide a linear framework for data-driven analyses of nonlinear dynamical systems, but their infinite-dimensional nature presents major computational challenges. In this article, we offer an introductory guide to Koopman…

Finding an embedding space for a linear approximation of a nonlinear dynamical system enables efficient system identification and control synthesis. The Koopman operator theory lays the foundation for identifying the nonlinear-to-linear…

Molecular dynamics simulations are an important tool for describing the evolution of a chemical system with time. However, these simulations are inherently held back either by the prohibitive cost of accurate electronic structure theory…

We introduce an algebraic analogue of dynamical systems, based on term rewriting. We show that a recursive function applied to the output of an iterated rewriting system defines a formal class of models into which all the main architectures…

Learning kernels in operators from data lies at the intersection of inverse problems and statistical learning, providing a powerful framework for capturing non-local dependencies in function spaces and high-dimensional settings. In contrast…

Kernel methods map data into high-dimensional spaces, enabling linear algorithms to learn nonlinear functions without explicitly storing the feature vectors. Quantum kernel methods promise efficient learning by encoding feature maps into…

These lecture notes endeavour to collect in one place the mathematical background required to understand the properties of kernels in general and the Random Fourier Features approximation of Rahimi and Recht (NIPS 2007) in particular. We…