Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent
The paper introduces a Spectral Alignment Decomposition explaining why the curvature exponent α varies across layer types (≈2 for convolutions, ≈1 for attention, <1 for MLP up-projections). It reduces this variation to geometric alignment between Kronecker factor eigenbases and gradient singular directions. A spectral transfer identity s=αγ is derived, predicting Hessian decay exponent s with ~2% median error across 93 layers without free parameters. An architecture-adaptive preconditioner T(σ;α) yields Spectral Newton, outperforming AdamW on vision benchmarks.
[2606.02596] Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent
[Submitted on 22 May 2026]
Title:Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent
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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, $
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