# Noncommutative C^*-algebra Net: Learning Neural Networks with Powerful Product Structure in C^*-algebra

arXiv: https://arxiv.org/abs/2302.01191 · HTML: https://mlsanity.com/paper/2302.01191/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 53 references that are in the map.
- First posted on arXiv 26 Jan 2023.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [A Survey on Multi-Task Learning](https://mlsanity.com/paper/1707.08114/) (2017)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Ensemble deep learning: A review](https://mlsanity.com/paper/2104.02395/) (2021)
- [Learned Initializations for Optimizing Coordinate-Based Neural Representations](https://mlsanity.com/paper/2012.02189/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Decentralized Collaborative Learning of Personalized Models over Networks](https://mlsanity.com/paper/1610.05202/) (2016)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Personalized and Private Peer-to-Peer Machine Learning](https://mlsanity.com/paper/1705.08435/) (2017)
- [Deep Quaternion Networks](https://mlsanity.com/paper/1712.04604/) (2017)
- [Quaternion Convolutional Neural Networks](https://mlsanity.com/paper/1903.00658/) (2019)
- [Improved Generalization Bounds of Group Invariant / Equivariant Deep Networks via Quotient Feature Spaces](https://mlsanity.com/paper/1910.06552/) (2019)
- [Group Equivariant Capsule Networks](https://mlsanity.com/paper/1806.05086/) (2018)
- [Deep Learning with Functional Inputs](https://mlsanity.com/paper/2006.09590/) (2020)
- [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)
- [Learning in RKHM: a C^*-Algebraic Twist for Kernel Machines](https://mlsanity.com/paper/2210.11855/) (2022)

## Built on by (most cited first, 4 of 4)

- [Deep Learning with Kernels through RKHM and the Perron-Frobenius Operator](https://mlsanity.com/paper/2305.13588/) (2023)
- [C^*-Algebraic Machine Learning: Moving in a New Direction](https://mlsanity.com/paper/2402.02637/) (2024)
- [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)

## Abstract (arXiv)

We propose a new generalization of neural network parameter spaces with noncommutative C^*-algebra, which possesses a rich noncommutative structure of products. We show that this noncommutative structure induces powerful effects in learning neural networks. Our framework has a wide range of applications, such as learning multiple related neural networks simultaneously with interactions and learning equivariant features with respect to group actions. Numerical experiments illustrate the validity of our framework and its potential power.