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

TL;DR

This work develops a Bayesian-information theoretic account of category learning in deep networks, linking efficient learning to the mutual information between category labels and neural representations. It introduces two Fisher information matrices—the categorical F_cat, describing category sensitivity in feature space, and the neural F_code, capturing neural sensitivity to feature changes—and shows that maximizing I[Y,R] drives a matching of these matrices, leading to expansion near category boundaries (categorical perception). The analysis extends to wide networks, revealing that optimal coding aligns neural geometry with category structure, with principal discriminant directions defined by F_cat and CP maxima occurring near, but not always on, class boundaries. Numerical experiments on Gaussian continua and MNIST digits validate the theory, demonstrating alignment of F_code with F_cat after learning and a CP-like sharpening of discrimination near boundaries. The framework also connects to the Information Bottleneck, offering a bias-variance perspective on the Bayes cost and outlining implications for the geometry of internal representations in classification tasks.

Abstract

In humans and other animals, category learning enhances discrimination between stimuli close to the category boundary. This phenomenon, called categorical perception, was also empirically observed in artificial neural networks trained on classification tasks. In previous modeling works based on neuroscience data, we show that this expansion/compression is a necessary outcome of efficient learning. Here we extend our theoretical framework to artificial networks. We show that minimizing the Bayes cost (mean of the cross-entropy loss) implies maximizing the mutual information between the set of categories and the neural activities prior to the decision layer. Considering structured data with an underlying feature space of small dimension, we show that 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. Category learning thus induces an expansion of neural space near decision boundaries. We characterize the properties of the categorical Fisher information, showing 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 dataset, we numerically illustrate how after learning the two Fisher information matrices match, and essentially align with the category boundaries. Finally, we relate our approach to the Information Bottleneck one, and we exhibit a bias-variance decomposition of the Bayes cost, of interest on its own.

Figures (12)

• Figure 1: One-dimensional examples with two Gaussian categories: category boundary and categorical Fisher information. Top, (a) and (b): $a=1.5$, $\sigma=0.6$. Bottom, (c) and (d): $a=2$, $\sigma=1$. Left, (a) and (c): Density distribution of the two classes, their corresponding posterior probabilities, along with the location of the boundary $x_b$ and the location of the relevant maximum of the categorical Fisher information $F_{\text{cat}}(x)$. Right, (b) and (d): Density distribution of the two classes and categorical Fisher information $F_{\text{cat}}(x)$.
• Figure 2: One-dimensional example with two Gaussian categories: category boundary vs. argmax of categorical Fisher information. Location $x_b$ of the boundary and location $x_\text{cat}$ of the relevant maximum of the categorical Fisher information for various values of $a$ and $\sigma$.
• Figure 3: Two-dimensional example with two Gaussian categories: category boundary, maxima of the categorical Fisher information, and Principal discriminant curves. For this simple example, the covariance matrices are $\mathbf{\Sigma}_{-}= \sigma^2 \,\mathbb{I}, \;\mathbf{\Sigma}_{+}= a^2 \,\mathbf{\Sigma}_{-}$, with $a=1.2,\; \sigma=1.3$. In the $(x_1, x_2)$ plane, the blue and red squares localize the centers of the two categories. The circle in continuous line gives the category boundary. The $-$ category is the most probable inside the red circle. The circle in dashed line gives the location of the maxima of the categorical Fisher information. The dashed black lines are principal discrimination curves. The color map gives the density distribution of $\mathbf{x}$.
• Figure 4: Two-dimensional example with three Gaussian categories: Fisher information quantities. (a) Probability $P(\mathbf{x})$ (b) Visualization of the categorical Fisher information matrix $\mathbf{F}_{\text{cat}}(\mathbf{x})$ at each point on the $\mathbf{x} = (x_1, x_2)$ plane. The small line represents the direction at this point of the eigenvector of the Fisher information matrix associated with the largest eigenvalue $f_{\text{cat}}(\mathbf{x})$. The magnitude of this largest eigenvalue is represented by the color, the lighter the greater. (c) The quantity $P(\mathbf{x}) f_{\text{cat}}(\mathbf{x})$, quantifying the source of the potential classification errors in the $\mathbf{x}$-plane. (d) Visualization of the neural Fisher information matrix $\mathbf{F}_{\text{code}}(\mathbf{x})$, at each point on the $(x_1, x_2)$ plane, after learning. The graphic convention is the same as in (b).
• Figure 5: Two-dimensional example with three Gaussian categories: Fisher information along a one-diemnsional path. (a) Colored dots: training set, random samples from each of the categories. Background color: mix between the colors that correspond to each of three categories, proportionally to the posterior probabilities $P(y|\mathbf{x})$ as estimated by the neural network. Dark dots: a path interpolating between two samples from the blue and the red categories. (b) The dashed colored lines indicate the posterior probabilities, as found by the network, each color representing its respective category. The solid line is the scalar Fisher information along the one-dimensional path shown in (a).