Deep Neural Networks with Random Gaussian Weights: A Universal Classification Strategy?

TL;DR

This paper demonstrates that deep neural networks with random Gaussian weights inherently perform a distance-preserving embedding, satisfying key classification properties and linking to compressed sensing, with theoretical proofs and validation on trained networks.

Contribution

It provides a formal proof that random Gaussian weight networks preserve data structure and relate to metric learning, connecting deep learning with compressed sensing theory.

Findings

Networks with random Gaussian weights perform distance-preserving embeddings.

Similar inputs tend to produce similar outputs in such networks.

Bounds on training set size for faithful representation of unseen data are derived.

Abstract

Three important properties of a classification machinery are: (i) the system preserves the core information of the input data; (ii) the training examples convey information about unseen data; and (iii) the system is able to treat differently points from different classes. In this work we show that these fundamental properties are satisfied by the architecture of deep neural networks. We formally prove that these networks with random Gaussian weights perform a distance-preserving embedding of the data, with a special treatment for in-class and out-of-class data. Similar points at the input of the network are likely to have a similar output. The theoretical analysis of deep networks here presented exploits tools used in the compressed sensing and dictionary learning literature, thereby making a formal connection between these important topics. The derived results allow drawing conclusions on the metric learning properties of the network and their relation to its structure, as well as providing bounds on the required size of the training set such that the training examples would represent faithfully the unseen data. The results are validated with state-of-the-art trained networks.