Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent
arXiv:2606.02596v1 Announce Type: new Abstract: The curvature exponent $\alpha$ in $h_k \propto \sigma_k^\alpha$ -- governing how Hessian eigenvalues scale with gradient singular values -- varies systematically across layer types ($\alpha \approx 2$ for convolutions, $\approx 1$ for transformer attention, $< 1$ for MLP up-projections). Why? We prove the Spectral Alignment Decomposition: $\alpha = 2 + d\log\Phi_k / d\log\sigma_k$, where $\Phi_k$ measures alignment between Kronecker factor eigenbases and gradient singular directions. This reduces "why does $\alpha$ vary?" to a geometric question
This research provides a fundamental breakthrough in understanding the spectral properties of neural networks, coinciding with an increased demand for more efficient and robust AI models.
Understanding the curvature exponent's variation across neural network layers can lead to new architectural designs, optimization techniques, and theoretical foundations for AI, ultimately impacting model performance and resource efficiency.
The ability to decompose and explain the curvature exponent fundamentally changes how researchers and engineers will approach neural network architecture design and training algorithms, moving from empirical tweaking to theoretically guided development.
- · AI researchers
- · Deep learning framework developers
- · Companies building specialized AI hardware
- · AI model architects
- · Empirical-only AI development methodologies
This decomposition facilitates more principled design of neural network architectures tailored for specific tasks and hardware.
Improved theoretical understanding could accelerate breakthroughs in AI efficiency, reducing the computational and energy costs associated with large models.
More efficient and explainable AI models might enable deployment in resource-constrained environments or applications requiring high assurance.
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Read at arXiv cs.LG