Geometric and Spectral Alignment for Deep Neural Network II

TL;DR

This paper introduces a geometric and spectral alignment method for analyzing residual Jacobian chains in deep neural networks, providing deterministic error bounds and empirical validation across various models.

Contribution

It develops a novel framework for physical channel alignment in neural networks, with explicit certificates and empirical analysis for CNNs, language models, and vision backbones.

Findings

Bounded error between full and truncated transport matrices.

Empirical spectra certified against Gibbs--Cartan tail model.

Matrices and heatmaps reveal alignment properties across models.

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.