# Geometric and Spectral Alignment for Deep Neural Network I

arXiv: https://arxiv.org/abs/2605.02108 · HTML: https://mlsanity.com/paper/2605.02108/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 15 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 8 of 8)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Fixup Initialization: Residual Learning Without Normalization](https://mlsanity.com/paper/1901.09321/) (2019)
- [Predicting trends in the quality of state-of-the-art neural networks without access to training or testing data](https://mlsanity.com/paper/2002.06716/) (2020)
- [Scaling ResNets in the Large-depth Regime](https://mlsanity.com/paper/2206.06929/) (2022)
- [An entropy formula for the Deep Linear Network](https://mlsanity.com/paper/2509.09088/) (2025)

## Built on by (most cited first, 1 of 1)

- [CHASE: Channel-Aligned Structure Exploitation for Geometry-Aware Model Engineering](https://mlsanity.com/paper/2610.09476/) (2026)

## Abstract (arXiv)

Deep residual architectures are modeled as products of near-identity Jacobians. This paper proves deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors, emphasizing a normalized top-radial Cartan coordinate and fitted power-law chart. Full-rank factors are mapped from GL(d) to the positive cone by Amapsto A^top A, then to ordered eigenvalue data. Under Frobenius normalization, exact power-law spectra form a trace-normalized Cartan orbit. This orbit is a Gibbs family on ranks, a Fisher information line, and a Bures--Wasserstein curve with line element d/4 times Fisher information. The main rigidity theorem is a slack-aware margin inequality: interface radial amplitude, non-backtracking slack, and signed residual variation control displacement of the fitted Cartan coordinate. In the exact-chart zero-slack case, a depth-L budget gives exponent drift of order (log M)/L; generally, slack and residual increments augment the bound. We separate scalar top-radial from full-Cartan spectral control, which also needs Bures/Hellinger residual variation. We prove approximate-power-law and metric-chart versions, converse lower bounds, Fisher--KL/Bures action estimates, and near-identity expansions for normalized residual chains. Near-identity results verify transport budgets; chart quality remains measurable. Effective rank is a spectral-energy quantile, giving finite-width power-law tail bounds and robust rank-window transition estimates. Empirical static-weight exponent profiles serve as diagnostics; full verification also requires interface budgets, slacks, and residuals for the same operator chain.