# Learning in RKHM: a C^*-Algebraic Twist for Kernel Machines

arXiv: https://arxiv.org/abs/2210.11855 · HTML: https://mlsanity.com/paper/2210.11855/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 46 references that are in the map.
- First posted on arXiv 21 Oct 2022.

## Builds on (most cited first, 14 of 14)

- [A Survey of Convolutional Neural Networks: Analysis, Applications, and Prospects](https://mlsanity.com/paper/2004.02806/) (2020)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Convolutional Kernel Networks](https://mlsanity.com/paper/1406.3332/) (2014)
- [End-to-End Kernel Learning with Supervised Convolutional Kernel Networks](https://mlsanity.com/paper/1605.06265/) (2016)
- [Beyond Least-Squares: Fast Rates for Regularized Empirical Risk Minimization through Self-Concordance](https://mlsanity.com/paper/1902.03046/) (2019)
- [Globally Convergent Newton Methods for Ill-conditioned Generalized Self-concordant Losses](https://mlsanity.com/paper/1907.01771/) (2019)
- [Scalable Matrix-valued Kernel Learning for High-dimensional Nonlinear Multivariate Regression and Granger Causality](https://mlsanity.com/paper/1408.2066/) (2014)
- [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)
- [Universality of Group Convolutional Neural Networks Based on Ridgelet Analysis on Groups](https://mlsanity.com/paper/2205.14819/) (2022)
- [Reproducing kernel Hilbert C*-module and kernel mean embeddings](https://mlsanity.com/paper/2101.11410/) (2021)
- [C^*-algebra Net: A New Approach Generalizing Neural Network Parameters to C^*-algebra](https://mlsanity.com/paper/2206.09513/) (2022)
- [Entangled Kernels -- Beyond Separability](https://mlsanity.com/paper/2101.05514/) (2021)
- [Stability of Multi-Task Kernel Regression Algorithms](https://mlsanity.com/paper/1306.3905/) (2013)

## Built on by (most cited first, 7 of 7)

- [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)

Supervised learning in reproducing kernel Hilbert space (RKHS) and vector-valued RKHS (vvRKHS) has been investigated for more than 30 years. In this paper, we provide a new twist to this rich literature by generalizing supervised learning in RKHS and vvRKHS to reproducing kernel Hilbert C^*-module (RKHM), and show how to construct effective positive-definite kernels by considering the perspective of C^*-algebra. Unlike the cases of RKHS and vvRKHS, we can use C^*-algebras to enlarge representation spaces. This enables us to construct RKHMs whose representation power goes beyond RKHSs, vvRKHSs, and existing methods such as convolutional neural networks. Our framework is suitable, for example, for effectively analyzing image data by allowing the interaction of Fourier components.