[Paper Review] $C^*$-algebra Net: A New Approach Generalizing Neural Network Parameters to $C^*$-algebra

Yuka Hashimoto, Zhao Wang+1 more|arXiv (Cornell University)|Jun 20, 2022

Neural Networks and Applications4 citations

TL;DR

This paper introduces $C^*$-algebra Net, a novel framework that generalizes neural network parameters to $C^*$-algebra-valued weights, enabling continuous model aggregation and integration of multiple models. By leveraging Hilbert $C^*$-modules, the method supports gradient descent on function-valued parameters, demonstrating improved performance in density estimation and few-shot learning with limited data.

ABSTRACT

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.

Motivation & Objective

• To generalize neural network parameters from real numbers to $C^*$-algebra-valued functions, enabling continuous model aggregation.
• To develop a gradient descent method on Hilbert $C^*$-modules for optimizing $C^*$-algebra-valued parameters.
• To apply the framework to practical problems such as density estimation and few-shot learning with limited training data.
• To demonstrate the theoretical and practical advantages of using $C^*$-algebra structures in deep learning beyond classical real-valued models.

Proposed method

• Generalize real-valued neural network weights to elements of a $C^*$-algebra, specifically $C(\mathcal{Z})$, the space of continuous functions on a compact space $\mathcal{Z}$.
• Represent each network parameter as a function in $\mathcal{A} = C(\mathcal{Z})$, allowing continuous interpolation and integration across models.
• Formulate the learning objective using a loss functional over $\mathcal{A}$-valued parameters, enabling optimization via functional regression and integration.
• Adapt gradient descent to Hilbert $C^*$-modules by defining inner products and adjoint operations that generalize classical Hilbert space methods.
• Use kernel ridge regression with a parameter $\mu$ to regularize the functional parameterization, balancing model flexibility and generalization.
• Apply the framework to density estimation and few-shot learning by modeling the output as a function on $\mathcal{Z}$, with $\mathcal{Z}$ representing data points or latent variables.

Experimental results

Research questions

• RQ1 Can neural network parameters be generalized from $\mathbb{R}$ to $C^*$-algebra-valued functions to enable continuous model combination?
• RQ2 How can gradient descent be extended to optimize $C^*$-algebra-valued parameters using the structure of Hilbert $C^*$-modules?
• RQ3 Does the proposed framework improve performance in low-data regimes such as few-shot learning and density estimation?
• RQ4 To what extent can the framework generalize existing methods like ensemble learning, complex-valued networks, and distributional parameter learning?

Key findings

• The framework enables continuous aggregation of multiple models through $C^*$-algebra-valued parameters, outperforming standard ensemble learning by allowing interaction during training.
• In few-shot learning, the model achieves higher test accuracy with limited samples by learning functional parameterizations that generalize across tasks.
• For density estimation, the method effectively models complex data distributions using kernel ridge regression with $\mu$-regularized functional parameters.
• The optimal value of the regularization parameter $\mu$ balances generalization and accuracy, with performance degrading for both very small and very large $\mu$ values.
• The framework generalizes complex-valued networks and distributional parameter learning, subsuming existing approaches as special cases when $\mathcal{Z}$ is finite or distributions are restricted.
• Numerical results show that the $C^*$-algebra Net achieves state-of-the-art performance in few-shot learning tasks with $l=10$ few-shot examples, particularly when $\mu$ is tuned appropriately.