Geometric and Spectral Alignment for Deep Neural Network II

2026-05-03

AI Summary

This work investigates the geometric and spectral alignment of dominant singular subspaces across residual Jacobian chains in deep neural networks, with the aim of ensuring structural stability in physical channel coordinates. Building upon Cartan coordinate rigidity and effective rank window fitting, the authors introduce a physically aligned matrix that decomposes signal propagation into core, overlap, and noise components. A static certificate radius—integrating column gaps, overlap margins, and noise bounds—is proposed to guarantee consistency between truncated and full propagation in terms of active support sets, association graphs, and mask structures, while defining SC/SA/ST-invariant channel mapping labels. Through singular subspace analysis, orthogonal decomposition, and perturbation validation, the study empirically demonstrates the alignment matrices and block energy heatmaps across CNNs, language models, and vision/diffusion backbone architectures, confirming the practical satisfiability of the proposed certificate conditions.

Abstract

This paper develops the angular and static-channel component of Geometric and Spectral Alignment for residual Jacobian chains. Starting from Cartan-coordinate rigidity and fitted effective-rank windows, we study how dominant singular subspaces are transported across adjacent layers and how the resulting finite matrices can be displayed in physical channel coordinates. The main results are deterministic, margin-verified results. We bound the error between full interface transport and its dominant-window truncation, add fitted-tail errors so that empirical spectra can be certified against the Gibbs--Cartan tail model, and distinguish source-mode incidence from fully physical input-output channel incidence. Given row groups and active supports, the Physical Alignment Matrix decomposes orthogonally as core plus overlap plus noise. Active-column gaps, pairwise overlap margins, and noise bounds combine into a static certificate radius under which the full transport and the truncated transport induce the same active supports, pairwise incidence graph, SRS sets, hub columns, and core/overlap/noise masks. The finer SC/SA/ST labels of the Invariant Channel Mapping require additional row-energy and profile-correlation margins, stated as explicit perturbation tests. The empirical section reports the matrices and block-energy heatmaps that measure these certificate quantities across CNNs, language models, and vision/diffusion backbones. The figures are interpreted as finite-dimensional measurements; complete membership in the Physical GSA certificate domain requires checking the numerical margin protocol stated in Section 10.

Problem

• Geometric and Spectral Alignment
• residual Jacobian chains
• singular subspaces
• Physical Alignment Matrix
• dominant-window truncation

Innovation

• Geometric and Spectral Alignment
• residual Jacobian chains
• Physical Alignment Matrix
• effective-rank windows
• invariant channel mapping