# C^*-Algebraic Machine Learning: Moving in a New Direction

arXiv: https://arxiv.org/abs/2402.02637 · HTML: https://mlsanity.com/paper/2402.02637/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 46 references that are in the map.
- First posted on arXiv 4 Feb 2024.

## Builds on (most cited first, 18 of 18)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Meta-Learning with Latent Embedding Optimization](https://mlsanity.com/paper/1807.05960/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Ensemble deep learning: A review](https://mlsanity.com/paper/2104.02395/) (2021)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Stochastic Particle Gradient Descent for Infinite Ensembles](https://mlsanity.com/paper/1712.05438/) (2017)
- [A representer theorem for deep kernel learning](https://mlsanity.com/paper/1709.10441/) (2017)
- [Autoencoding any Data through Kernel Autoencoders](https://mlsanity.com/paper/1805.11028/) (2018)
- [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)
- [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)
- [Efficient displacement convex optimization with particle gradient descent](https://mlsanity.com/paper/2302.04753/) (2023)
- [Noncommutative C^*-algebra Net: Learning Neural Networks with Powerful Product Structure in C^*-algebra](https://mlsanity.com/paper/2302.01191/) (2023)
- [Kernel Mean Embeddings of Von Neumann-Algebra-Valued Measures](https://mlsanity.com/paper/2007.14698/) (2020)

## Built on by (most cited first, 1 of 1)

- [Quantum Circuit C^*-algebra Net](https://mlsanity.com/paper/2404.06218/) (2024)

## Abstract (arXiv)

Machine learning has a long collaborative tradition with several fields of mathematics, such as statistics, probability and linear algebra. We propose a new direction for machine learning research: C^*-algebraic ML - a cross-fertilization between C^*-algebra and machine learning. The mathematical concept of C^*-algebra is a natural generalization of the space of complex numbers. It enables us to unify existing learning strategies, and construct a new framework for more diverse and information-rich data models. We explain why and how to use C^*-algebras in machine learning, and provide technical considerations that go into the design of C^*-algebraic learning models in the contexts of kernel methods and neural networks. Furthermore, we discuss open questions and challenges in C^*-algebraic ML and give our thoughts for future development and applications.