# C^*-algebra Net: A New Approach Generalizing Neural Network Parameters to C^*-algebra

arXiv: https://arxiv.org/abs/2206.09513 · HTML: https://mlsanity.com/paper/2206.09513/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 20 Jun 2022.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Meta-Learning with Latent Embedding Optimization](https://mlsanity.com/paper/1807.05960/) (2018)
- [Ensemble deep learning: A review](https://mlsanity.com/paper/2104.02395/) (2021)
- [DRAW: A Recurrent Neural Network For Image Generation](https://mlsanity.com/paper/1502.04623/) (2015)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [The Emergence of Spectral Universality in Deep Networks](https://mlsanity.com/paper/1802.09979/) (2018)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [A Survey of Complex-Valued Neural Networks](https://mlsanity.com/paper/2101.12249/) (2021)
- [TRADI: Tracking deep neural network weight distributions for uncertainty estimation](https://mlsanity.com/paper/1912.11316/) (2019)
- [Reproducing kernel Hilbert C*-module and kernel mean embeddings](https://mlsanity.com/paper/2101.11410/) (2021)

## Built on by (most cited first, 9 of 9)

- [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)
- [Noncompact uniform universal approximation](https://mlsanity.com/paper/2308.03812/) (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)
- [Localisation of Regularised and Multiview Support Vector Machine Learning](https://mlsanity.com/paper/2304.05655/) (2023)
- [Compact Matrix Quantum Group Equivariant Neural Networks](https://mlsanity.com/paper/2311.06358/) (2023)
- [Quantum Circuit C^*-algebra Net](https://mlsanity.com/paper/2404.06218/) (2024)
- [Algebraic Approach to Ridge-Regularized Mean Squared Error Minimization in Minimal ReLU Neural Network](https://mlsanity.com/paper/2508.17783/) (2025)

## Abstract (arXiv)

We propose a new framework that generalizes the parameters of neural network models to C^*-algebra-valued ones. C^*-algebra is a generalization of the space of complex numbers. A typical example is the space of continuous functions on a compact space. This generalization enables us to combine multiple models continuously and use tools for functions such as regression and integration. Consequently, we can learn features of data efficiently and adapt the models to problems continuously. We apply our framework to practical problems such as density estimation and few-shot learning and show that our framework enables us to learn features of data even with a limited number of samples. Our new framework highlights the potential possibility of applying the theory of C^*-algebra to general neural network models.