On the Stability of the Jacobian Matrix in Deep Neural Networks

TL;DR

This paper provides a general stability theorem for the Jacobian matrix in deep neural networks, extending prior analyses to include sparsity and correlated weights, thereby broadening the theoretical understanding of spectral stability.

Contribution

It establishes a comprehensive stability theorem that applies to networks with structured, sparse, and correlated weights, surpassing previous limitations to i.i.d. weights.

Findings

Spectral stability guarantees for a broader class of network models.

Extension of initialization theory to structured and dependent weight matrices.

Rigorous bounds derived using recent random matrix theory advances.

Abstract

Deep neural networks are known to suffer from exploding or vanishing gradients as depth increases, a phenomenon closely tied to the spectral behavior of the input-output Jacobian. Prior work has identified critical initialization schemes that ensure Jacobian stability, but these analyses are typically restricted to fully connected networks with i.i.d. weights. In this work, we go significantly beyond these limitations: we establish a general stability theorem for deep neural networks that accommodates sparsity (such as that introduced by pruning) and non-i.i.d., weakly correlated weights (e.g. induced by training). Our results rely on recent advances in random matrix theory, and provide rigorous guarantees for spectral stability in a much broader class of network models. This extends the theoretical foundation for initialization schemes in modern neural networks with structured and dependent randomness.