Why Deep Jacobian Spectra Separate: Depth-Induced Scaling and Singular-Vector Alignment
#Jacobian Spectra #Gradient-based Training #Deep Neural Networks #Implicit Bias #Singular Values #Spectral Separation #arXiv
📌 Key Takeaways
- Research addresses the challenge of understanding implicit bias in gradient-based training of deep neural networks
- Current approaches are limited to balanced deep linear models, creating a knowledge gap
- The study introduces two signatures of deep Jacobians: depth-induced exponential scaling and spectral separation
- This theoretical framework offers new insights into the mathematical foundations of deep learning
📖 Full Retelling
Researchers published a groundbreaking study on arXiv on February 20, 2026, addressing the challenge of understanding why gradient-based training in deep neural networks exhibits strong implicit bias. The paper, titled 'Why Deep Jacobian Spectra Separate: Depth-Induced Scaling and Singular-Vector Alignment,' proposes an innovative theoretical framework that moves beyond the limitations of current approaches which are typically restricted to balanced deep linear models. The researchers introduce two empirically testable signatures of deep Jacobians—depth-induced exponential scaling of ordered singular values and strong spectral separation—as alternative pathways to understanding the complex dynamics of neural network training. This work represents a significant advancement in the mathematical understanding of deep learning systems, potentially leading to more efficient training methods and improved network architectures. The theoretical foundation provided by this research could help demystify some of the mysterious properties that have made deep learning so effective despite its apparent complexity.
🏷️ Themes
Deep Learning, Neural Networks, Mathematical Theory
📚 Related People & Topics
Singular value
Square roots of the eigenvalues of the self-adjoint operator
In mathematics, in particular functional analysis, the singular values of a compact operator T : X → Y {\displaystyle T:X\rightarrow Y} acting between Hilbert spaces X {\displaystyle ...
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Original Source
arXiv:2602.12384v1 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
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