Inversion dynamics of class manifolds in deep learning reveals tradeoffs underlying generalisation

Supervised deep learning excels in the baffling task of disentangling the training data, so as to reach near-zero training error, while still achieving good accuracy on the classification of unseen data. How this feat is achieved, particularly in relation to the geometry and structure of the training data, is currently a topic of debate and partly still an open question [1-6]. Activations of hidden layers in response to input examples, i.e., the internal representations of the data, evolve during training to facilitate eventual linear separation in the last layer. This requires a gradual segregation of points belonging to different classes, in what can be pictured as a disentangling motion between their class manifolds. Segregation of class manifolds is a powerful conceptualisation that informs the design of distancebased losses in metric learning and contrastive learning [7-11] and underlies several approaches aimed at quantifying expressivity and generalisation, in artificial neural networks as well as in neuroscience [12-17]. Several recent efforts have leveraged this picture to characterise information processing along the layers of a deep network, particularly focusing on metrics such as intrinsic dimensionality and curvature [18-22]. In Ref. [19], for instance, two descriptors of manifold geometry, related to the intrinsic dimension and to the extension of the manifolds, are shown to undergo dramatic reduction as a result of training in deep convolutional neural networks. Such shrinking (together with intermanifold correlations, which we neglect in this manuscript) decisively supports the model's capacity in a memorisation task. Yet, this appears to be just one side of the coin.