How does training shape the Riemannian geometry of neural network representations?
This work addresses the problem of understanding geometric inductive biases in neural networks for researchers in machine learning, though it is incremental as it builds on prior studies of geometry in networks.
The study investigates how training alters the Riemannian geometry of neural network representations, showing that networks initially induce symmetric metrics but learn to magnify areas near decision boundaries during classification tasks, including in deep networks and self-supervised learning.
In machine learning, there is a long history of trying to build neural networks that can learn from fewer example data by baking in strong geometric priors. However, it is not always clear a priori what geometric constraints are appropriate for a given task. Here, we explore the possibility that one can uncover useful geometric inductive biases by studying how training molds the Riemannian geometry induced by unconstrained neural network feature maps. We first show that at infinite width, neural networks with random parameters induce highly symmetric metrics on input space. This symmetry is broken by feature learning: networks trained to perform classification tasks learn to magnify local areas along decision boundaries. This holds in deep networks trained on high-dimensional image classification tasks, and even in self-supervised representation learning. These results begin to elucidate how training shapes the geometry induced by unconstrained neural network feature maps, laying the groundwork for an understanding of this richly nonlinear form of feature learning.