Why Deep Jacobian Spectra Separate: Depth-Induced Scaling and Singular-Vector Alignment
arXiv:2602.12384v2 Announce Type: cross Abstract: Understanding why gradient-based training in deep networks exhibits strong implicit bias remains challenging, in part because tractable singular-value dynamics are typically available only for balanced deep linear models. We propose an alternative route based on two theoretically grounded and empirically testable signatures of deep Jacobians: depth-induced exponential scaling of ordered singular values and strong spectral separation. Adopting a fixed-gates view of piecewise-linear networks, where Jacobians reduce to products of masked linear maps within a single activation region, we prove the existence of Lyapunov exponents governing the top singular values at initialization, give closed-form expressions in a tractable masked model, and quantify finite-depth corrections. We further show that sufficiently strong separation forces singular-vector alignment in matrix products, yielding an approximately shared singular basis for intermedi
arXiv:2602.12384v2 Announce Type: cross Abstract: Understanding why gradient-based training in deep networks exhibits strong implicit bias remains challenging, in part because tractable singular-value dynamics are typically available only for balanced deep linear models. We propose an alternative route based on two theoretically grounded and empirically testable signatures of deep Jacobians: depth-induced exponential scaling of ordered singular values and strong spectral separation. Adopting a fixed-gates view of piecewise-linear networks, where Jacobians reduce to products of masked linear maps within a single activation region, we prove the existence of Lyapunov exponents governing the top singular values at initialization, give closed-form expressions in a tractable masked model, and quantify finite-depth corrections. We further show that sufficiently strong separation forces singular-vector alignment in matrix products, yielding an approximately shared singular basis for intermediate Jacobians. Together, these results motivate an approximation regime in which singular-value dynamics become effectively decoupled, mirroring classical balanced deep-linear analyses without requiring balancing. Experiments in fixed-gates settings validate the predicted scaling, alignment, and resulting dynamics, supporting a mechanistic account of emergent low-rank Jacobian structure as a driver of implicit bias.
Executive Summary
The article 'Why Deep Jacobian Spectra Separate: Depth-Induced Scaling and Singular-Vector Alignment' explores the implicit biases in gradient-based training of deep neural networks. The authors propose a novel approach to understand these biases by examining the singular-value dynamics of deep Jacobians, focusing on depth-induced exponential scaling and spectral separation. They adopt a fixed-gates view of piecewise-linear networks, proving the existence of Lyapunov exponents governing top singular values at initialization and providing closed-form expressions in a tractable masked model. The study also shows that strong spectral separation leads to singular-vector alignment, resulting in an approximately shared singular basis for intermediate Jacobians. Experiments validate these predictions, supporting a mechanistic account of emergent low-rank Jacobian structure as a driver of implicit bias.
Key Points
- ▸ The article introduces a new approach to understanding implicit biases in deep neural networks through the analysis of deep Jacobian spectra.
- ▸ Depth-induced exponential scaling and spectral separation are identified as key factors in the dynamics of singular values.
- ▸ The fixed-gates view of piecewise-linear networks is adopted to prove the existence of Lyapunov exponents and provide closed-form expressions for singular values.
- ▸ Strong spectral separation is shown to force singular-vector alignment, leading to a shared singular basis for intermediate Jacobians.
- ▸ Experiments validate the predicted scaling, alignment, and resulting dynamics, supporting the mechanistic account of emergent low-rank Jacobian structure.
Merits
Theoretical Rigor
The article provides a theoretically grounded analysis of deep Jacobian spectra, offering a novel perspective on the implicit biases in deep neural networks. The proofs and closed-form expressions contribute significantly to the theoretical understanding of the subject.
Empirical Validation
The study includes experiments that validate the predicted scaling, alignment, and resulting dynamics, strengthening the credibility of the theoretical findings.
Practical Implications
The findings have practical implications for the training and optimization of deep neural networks, potentially leading to more efficient and effective training methods.
Demerits
Complexity
The theoretical framework and mathematical proofs presented in the article are highly complex, which may limit the accessibility of the findings to a broader audience.
Scope Limitations
The study focuses on a specific view of piecewise-linear networks, which may not fully capture the complexities of all types of deep neural networks.
Empirical Generalization
While the experiments validate the predictions, the generalization of these findings to more diverse and complex neural network architectures remains to be thoroughly explored.
Expert Commentary
The article presents a significant advancement in the theoretical understanding of implicit biases in deep neural networks. By focusing on the singular-value dynamics of deep Jacobians, the authors provide a novel perspective that complements existing research. The adoption of the fixed-gates view of piecewise-linear networks is a clever approach that allows for tractable analysis and empirical validation. The findings have substantial practical implications, as they can inform the development of more efficient training algorithms and more robust neural network architectures. However, the complexity of the theoretical framework and the scope limitations of the study should be acknowledged. Future research should explore the generalization of these findings to more diverse and complex neural network architectures, ensuring that the insights gained are broadly applicable. Overall, this article is a valuable contribution to the field of machine learning and deep learning, offering both theoretical insights and practical implications.
Recommendations
- ✓ Future research should aim to extend the findings to more diverse and complex neural network architectures to ensure broader applicability.
- ✓ Efforts should be made to simplify the theoretical framework and make the findings more accessible to a wider audience, potentially through educational materials or simplified explanations.