A Unified Matrix-Spectral Framework for Stability and Interpretability in Deep Learning

TL;DR

This paper introduces a spectral analysis framework for deep neural networks that links stability and interpretability, providing diagnostics and regularization methods to enhance robustness and sensitivity control.

Contribution

It develops a unified matrix-spectral approach to analyze and improve stability and interpretability in deep learning models through spectral diagnostics and regularization.

Findings

Spectral regularization improves attribution stability.

Spectral quantities correlate with model robustness.

The framework offers practical stability diagnostics.

Abstract

We develop a unified matrix-spectral framework for analyzing stability and interpretability in deep neural networks. Representing networks as data-dependent products of linear operators reveals spectral quantities governing sensitivity to input perturbations, label noise, and training dynamics. We introduce a Global Matrix Stability Index that aggregates spectral information from Jacobians, parameter gradients, Neural Tangent Kernel operators, and loss Hessians into a single stability scale controlling forward sensitivity, attribution robustness, and optimization conditioning. We further show that spectral entropy refines classical operator-norm bounds by capturing typical, rather than purely worst-case, sensitivity. These quantities yield computable diagnostics and stability-oriented regularization principles. Synthetic experiments and controlled studies on MNIST, CIFAR-10, and CIFAR-100 confirm that modest spectral regularization substantially improves attribution stability even when global spectral summaries change little. The results establish a precise connection between spectral concentration and analytic stability, providing practical guidance for robustness-aware model design and training.