Category learning in deep neural networks: Information content and geometry of internal representations

In humans and other animals, category learning is associated with a better ability to discriminate between stimuli that are close to the category boundary, compared with stimuli well within a category. This perceptual within-category compression and between-category separation, called categorical perception, was also empirically observed in artificial neural networks trained on classification tasks. In previous modeling works based on empirical neuroscience data, we took an information-theoretic approach that shows that this expansion-compression is a necessary outcome of efficient learning. As a result, the impact of input or neuronal noise is reduced where it is most detrimental, namely, at the boundary between categories. Here we extend our theoretical framework to artificial feedforward networks. The Bayes cost that we consider is an average over the data distribution of the standard cross-entropy loss function. We show that minimizing this cost implies maximizing the mutual information between the set of categories and the neural activities prior to the decision layer. We then consider structured data, formalized by the assumption of an underlying feature space of small dimension. We show that, for wide networks, and more generally in situations of high signal-to-noise ratio, maximizing the mutual information implies (i) finding an appropriate projection space, and (ii) building a neural representation with the appropriate metric. The latter is based on a Fisher information matrix measuring the sensitivity of the neural activity to changes in the projection space. Optimal learning makes this neural Fisher information follow a category-specific Fisher information, measuring the sensitivity of the category membership to changes in the projection space. One consequence is that category learning induces the main neural correlate of categorical perception, an expansion of neural space near decision boundaries. To make this statement more precise, we characterize the properties of the categorical Fisher information. We show that its eigenvectors give the most discriminant directions at each point of the projection space. We find that, unexpectedly, its maxima are in general not exactly at, but near, the class boundaries. Considering toy models and the MNIST handwritten digits dataset, we numerically illustrate how, after learning, the two Fisher information matrices match and essentially align with the boundaries between categories. Finally, we provide a variety of supplemental analyses, in particular, we relate our approach to that of the information bottleneck, and we exhibit a bias-variance decomposition of the Bayes cost, which is of interest on its own.