Exploring the Common Principal Subspace of Deep Features in Neural Networks

TL;DR

This paper reveals that diverse deep neural networks trained on the same dataset share a common principal subspace in their learned features, regardless of architecture or training method, and introduces a new metric to measure this shared structure.

Contribution

The authors propose a novel $ ext{P}$-vector metric to quantify the principal subspace of deep features and demonstrate its effectiveness across various architectures and training paradigms.

Findings

Deep networks share a common principal subspace in learned features.

The $ ext{P}$-vector metric effectively measures angles between subspaces.

Angles decrease during training, indicating convergence of feature spaces.

Abstract

We find that different Deep Neural Networks (DNNs) trained with the same dataset share a common principal subspace in latent spaces, no matter in which architectures (e.g., Convolutional Neural Networks (CNNs), Multi-Layer Preceptors (MLPs) and Autoencoders (AEs)) the DNNs were built or even whether labels have been used in training (e.g., supervised, unsupervised, and self-supervised learning). Specifically, we design a new metric $\mathcal{P}$-vector to represent the principal subspace of deep features learned in a DNN, and propose to measure angles between the principal subspaces using $\mathcal{P}$-vectors. Small angles (with cosine close to $1.0$) have been found in the comparisons between any two DNNs trained with different algorithms/architectures. Furthermore, during the training procedure from random scratch, the angle decrease from a larger one ($70^\circ-80^\circ$ usually) to the small one, which coincides the progress of feature space learning from scratch to convergence. Then, we carry out case studies to measure the angle between the $\mathcal{P}$-vector and the principal subspace of training dataset, and connect such angle with generalization performance. Extensive experiments with practically-used Multi-Layer Perceptron (MLPs), AEs and CNNs for classification, image reconstruction, and self-supervised learning tasks on MNIST, CIFAR-10 and CIFAR-100 datasets have been done to support our claims with solid evidences. Interpretability of Deep Learning, Feature Learning, and Subspaces of Deep Features