An unsupervised tour through the hidden pathways of deep neural networks
Diego Doimo
TL;DR
The thesis tackles understanding deep neural networks through unsupervised analysis of hidden representations and generalization. It introduces Gride, a scale-aware intrinsic-dimension estimator, and combines it with density-based methods to map how semantic structure emerges across layers, revealing hierarchical nucleation of density peaks that mirror category relations. It also uncovers a generalization mechanism in wide networks: redundant representations (clones) in the final layer interpolate training data and enable error decay as a $W^{-1/2}$ power law, contingent on regularization and interpolation. The work provides tools and empirical evidence linking intrinsic dimension, probability-density topology, and semantic organization across architectures (ResNet, ViT, iGPT), offering new insights for transfer learning, robustness, and interpretability. Collectively, the results advance unsupervised characterizations of representation geometry and illuminate practical implications for training, architecture design, and generalization limits in deep learning.
Abstract
The goal of this thesis is to improve our understanding of the internal mechanisms by which deep artificial neural networks create meaningful representations and are able to generalize. We focus on the challenge of characterizing the semantic content of the hidden representations with unsupervised learning tools, partially developed by us and described in this thesis, which allow harnessing the low-dimensional structure of the data. Chapter 2. introduces Gride, a method that allows estimating the intrinsic dimension of the data as an explicit function of the scale without performing any decimation of the data set. Our approach is based on rigorous distributional results that enable the quantification of uncertainty of the estimates. Moreover, our method is simple and computationally efficient since it relies only on the distances among nearest data points. In Chapter 3, we study the evolution of the probability density across the hidden layers in some state-of-the-art deep neural networks. We find that the initial layers generate a unimodal probability density getting rid of any structure irrelevant to classification. In subsequent layers, density peaks arise in a hierarchical fashion that mirrors the semantic hierarchy of the concepts. This process leaves a footprint in the probability density of the output layer, where the topography of the peaks allows reconstructing the semantic relationships of the categories. In Chapter 4, we study the problem of generalization in deep neural networks: adding parameters to a network that interpolates its training data will typically improve its generalization performance, at odds with the classical bias-variance trade-off. We show that wide neural networks learn redundant representations instead of overfitting to spurious correlation and that redundant neurons appear only if the network is regularized and the training error is zero.