On the Stability of the Jacobian Matrix in Deep Neural Networks

• URL: http://arxiv.org/abs/2506.08764v1
• Date: Tue, 10 Jun 2025 13:04:42 GMT
• Title: On the Stability of the Jacobian Matrix in Deep Neural Networks
• Authors: Benjamin Dadoun, Soufiane Hayou, Hanan Salam, Mohamed El Amine Seddik, Pierre Youssef,
• Abstract summary: We establish a general stability theorem for deep neural networks that accommodates sparsity and weakly correlated weights.<n>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.
• Score: 9.617753464544606
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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