# Reproducing kernel Hilbert C*-module and kernel mean embeddings

arXiv: https://arxiv.org/abs/2101.11410 · HTML: https://mlsanity.com/paper/2101.11410/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 63 references that are in the map.
- First posted on arXiv 27 Jan 2021.

## Builds on (most cited first, 11 of 11)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [A Unifying Framework in Vector-valued Reproducing Kernel Hilbert Spaces for Manifold Regularization and Co-Regularized Multi-view Learning](https://mlsanity.com/paper/1401.8066/) (2014)
- [Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables](https://mlsanity.com/paper/1808.10551/) (2018)
- [Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators](https://mlsanity.com/paper/1805.12324/) (2018)
- [Krylov Subspace Method for Nonlinear Dynamical Systems with Random Noise](https://mlsanity.com/paper/1909.03634/) (2019)
- [Kernel Mean Matching for Content Addressability of GANs](https://mlsanity.com/paper/1905.05882/) (2019)
- [The Matrix Hilbert Space and Its Application to Matrix Learning](https://mlsanity.com/paper/1706.08110/) (2017)

## Built on by (most cited first, 10 of 10)

- [C^*-algebra Net: A New Approach Generalizing Neural Network Parameters to C^*-algebra](https://mlsanity.com/paper/2206.09513/) (2022)
- [Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2403.02524/) (2024)
- [Learning in RKHM: a C^*-Algebraic Twist for Kernel Machines](https://mlsanity.com/paper/2210.11855/) (2022)
- [Deep Learning with Kernels through RKHM and the Perron-Frobenius Operator](https://mlsanity.com/paper/2305.13588/) (2023)
- [Noncommutative C^*-algebra Net: Learning Neural Networks with Powerful Product Structure in C^*-algebra](https://mlsanity.com/paper/2302.01191/) (2023)
- [C^*-Algebraic Machine Learning: Moving in a New Direction](https://mlsanity.com/paper/2402.02637/) (2024)
- [Spectral Truncation Kernels: Noncommutativity in C^*-algebraic Kernel Machines](https://mlsanity.com/paper/2405.17823/) (2024)
- [Position: Quantum Kernel Machines Should Move Beyond Scalar-Valued Kernels to Realize Their Potential](https://mlsanity.com/paper/2506.03779/) (2025)
- [Localisation of Regularised and Multiview Support Vector Machine Learning](https://mlsanity.com/paper/2304.05655/) (2023)
- [Quantum Circuit C^*-algebra Net](https://mlsanity.com/paper/2404.06218/) (2024)

## Abstract (arXiv)

Kernel methods have been among the most popular techniques in machine learning, where learning tasks are solved using the property of reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis framework with reproducing kernel Hilbert C^*-module (RKHM) and kernel mean embedding (KME) in RKHM. Since RKHM contains richer information than RKHS or vector-valued RKHS (vvRKHS), analysis with RKHM enables us to capture and extract structural properties in such as functional data. We show a branch of theories for RKHM to apply to data analysis, including the representer theorem, and the injectivity and universality of the proposed KME. We also show RKHM generalizes RKHS and vvRKHS. Then, we provide concrete procedures for employing RKHM and the proposed KME to data analysis.