Hilbert Space

Hilbert spaces, infinite-dimensional vector spaces with an inner product, are fundamental mathematical structures used to model diverse phenomena in science and engineering. Current research focuses on developing and analyzing algorithms for learning operators between Hilbert spaces, employing techniques like stochastic gradient descent and neural networks (including those with architectures optimized for specific operator types), often within the context of solving partial differential equations or reinforcement learning problems. These advancements are improving the efficiency and accuracy of solving complex problems in areas such as scientific computing, machine learning, and quantum computing, leading to more robust and powerful computational tools.

Papers

October 24, 2024

Learning dissipative Hamiltonian dynamics with reproducing kernel Hilbert spaces and random Fourier features
Torbjørn Smith, Olav Egeland
Kernel Hilbert Space Vector Field Kernel Extension Hilbert Space Random Fourier Feature Hamiltonian Dynamic Symplectic Group Master Equation

October 18, 2024

Spectral Representations for Accurate Causal Uncertainty Quantification with Gaussian Processes
Hugh Dance, Peter Orbanz, Arthur Gretton
Gaussian Process Uncertainty Quantification Spectral Representation Hilbert Space Unreliable Prediction Causal Bayesian Optimization

June 20, 2024

Optimal deep learning of holomorphic operators between Banach spaces
Ben Adcock, Nick Dexter, Sebastian Moraga
Deep Learning Generalization Bound Operator Learning Hilbert Space Banach Space

May 16, 2024

On Partially Unitary Learning
Mikhail Gennadievich Belov, Vladislav Gennadievich Malyshkin
Quantum Channel Hilbert Space Wave Function Unitary Gradient Neural Network

March 9, 2024

Statistical Efficiency of Distributional Temporal Difference Learning
Yang Peng, Liangyu Zhang, Zhihua Zhang
Policy Evaluation Distributional Reinforcement Learning Hilbert Space Quantile Temporal Difference

February 7, 2024

Learning Operators with Stochastic Gradient Descent in General Hilbert Spaces
Lei Shi, Jia-Qi Yang
Stochastic Gradient Descent Operator Learning Kernel Hilbert Space Hilbert Space

January 2, 2024

Learning solutions to some toy constrained optimization problems in infinite dimensional Hilbert spaces
Pinak Mandal
Practical Algorithm Optimization Problem Constrained Optimization Learning System Hilbert Space Lagrangian Method Constraint Function

November 25, 2023

A unified framework for learning with nonlinear model classes from arbitrary linear samples
Ben Adcock, Juan M. Cardenas, Nick Dexter
Generative Model Unified Framework Nonlinear Model Generalization Error Bound Hilbert Space Learning Guarantee

October 26, 2023

A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces
Jonathan W. Siegel, Stephan Wojtowytsch
Gradient Descent Gradient Flow Qualitative Difference Convex Function Hilbert Space Convex Objective

September 8, 2023

Online Infinite-Dimensional Regression: Learning Linear Operators
Vinod Raman, Unique Subedi, Ambuj Tewari
P$ Norm Hilbert Space Uniform Convergence Weak Convergence Linear Operator

August 31, 2023

Training Neural Networks Using Reproducing Kernel Space Interpolation and Model Reduction
Eric Arthur Werneburg
Neural Network Kernel Hilbert Space Model Reduction Hilbert Space Kernel Interpolation Kernel Space

August 21, 2023

Fat Shattering, Joint Measurability, and PAC Learnability of POVM Hypothesis Classes
Abram Magner, Arun Padakandla
Sample Complexity Bound Hilbert Space PAC Learnability Hypothesis Class Quantum Measurement Fat Shattering Dimension

July 19, 2023

A Dual Formulation for Probabilistic Principal Component Analysis
Henri De Plaen, Johan A. K. Suykens
Kernel Method Generative Framework Kernel Principal Component Analysis Hilbert Space Dual Formulation Probabilistic Principal Component Analysis

May 17, 2023

Deep quantum neural networks form Gaussian processes
Diego García-Martín, Martin Larocca, M. Cerezo
Neural Network Gaussian Process Quantum Neural Network Hilbert Space Deep Quantum Shallow Quantum Circuit

March 31, 2023

$\infty$-Diff: Infinite Resolution Diffusion with Subsampled Mollified States
Sam Bond-Taylor, Chris G. Willcocks
Diffusion Explainer Generative Diffusion Model Hidden State Hilbert Space Resolution Diffusion

March 3, 2023

On the complexity of PAC learning in Hilbert spaces
Sergei Chubanov
Complexity Matter Binary Classification Hilbert Space Polyhedral Complex

February 22, 2023

nSimplex Zen: A Novel Dimensionality Reduction for Euclidean and Hilbert Spaces
Richard Connor, Lucia Vadicamo
Dimensionality Reduction High Dimensional Space Euclidean Space Hilbert Space Multidimensional Scaling

February 12, 2023

Recursive Estimation of Conditional Kernel Mean Embeddings
Ambrus Tamás, Balázs Csanád Csáji
Kernel Hilbert Space Hilbert Space Kernel Mean Embeddings Recursive Bayesian

January 6, 2023

Reversibility of elliptical slice sampling revisited
Mareike Hasenpflug, Viacheslav Telezhnikov, Daniel Rudolf
Markov Chain Hilbert Space Reversible Dynamic Transition Kernel Slice Sampling

December 22, 2022

On Machine Learning Knowledge Representation In The Form Of Partially Unitary Operator. Knowledge Generalizing Operator
Vladislav Gennadievich Malyshkin
Form Like Document Quantum Channel Hilbert Space Knowledge Representation Learning Unitary Transformation Unitary Operator