A theory of collective variables for learning systems

Within the last few years, neural networks have become more and more prominent in many areas of life and research. However, their widespread adoption and deployment have raised concerns about their lack of interpretability, hindering our ability to understand developed models and trace their behavior back to the used data.

To address this critical challenge, researchers have turned their attention towards developing techniques to extract interpretable information from the states of neural networks, allowing to gain insights into the underlying mechanisms and processes. While most research provides insights into the local behavior of the network, they do not consider global mechanisms and patterns that drive the network’s decision-making. Just as in systems investigated in statistical physics, relying solely on a single entity can be insufficient to extract system-wide observables. In order to adress this, the concept of collective variables is applied to neural networks.

Collective variables (CVs) refer to macroscopic quantities that emerge from the interplay of microscopic degrees of freedom. In the context of neural networks, these observables are quantities that arise from the interactions between neurons and data, allowing one to capture the global behavior of the network. This approach is based on the neural tangent kernel and this work focuses on finite-sized models.

By utilizing the CVs, the initial state and the evolution of neural networks during training can be analyzed and interpreted. This offers possible new insights into learning behavior and generalization properties. Starting with simple toy systems, the studies are systematically extended to state-of-the-art models. Since the principle of CVs only relies on a gradientbased learning mechanism, the research is expanded to encompass other learnable systems, providing a basis for comparison with neural networks.