Symmetry in Machine Learning: A Fundamental Concept and Its Applications

In this paper, we explore the idea of symmetry in machine learning and how it can be used to improve the performance of neural networks. Symmetry is a fundamental concept that helps us understand when an object is unchanged after some transformation has been applied to it. In the context of machine learning, using symmetry as an inductive bias can lead to powerful insights and practical breakthroughs.
One challenge with symmetry in machine learning is that it can be difficult to maintain when dealing with real-world data, which often have symmetries that are not explicitly defined. To address this issue, we propose a new notion of "relaxed equivariance," which allows us to break the symmetry of inputs and map them to arbitrary orbit types. This allows us to handle symmetric inputs beyond the constraints imposed by equivariance, making it possible to train neural networks that are more accurate and robust.
To illustrate how relaxed equivariance works, let’s consider an example of a neural network that is trained to classify objects based on their shape. If the objects have symmetries, such as being mirrored or rotated, the neural network should be able to recognize these symmetries and adjust its classification accordingly. However, in some cases, it may not be desirable to enforce equivariance too strictly, especially when dealing with real-world data that often have complex symmetries. By allowing for relaxed equivariance, we can train neural networks that are more flexible and accurate in these situations.
In summary, this paper introduces a new notion of relaxed equivariance to handle symmetric inputs in machine learning, allowing us to break the symmetry of inputs and map them to arbitrary orbit types. This makes it possible to train neural networks that are more accurate and robust when dealing with real-world data that often have complex symmetries. By demystifying complex concepts using everyday language and engaging metaphors or analogies, we hope to provide a clear understanding of this important research in the field of machine learning.