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

arXiv: https://arxiv.org/abs/2602.01136 · HTML: https://mlsanity.com/paper/2602.01136/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 21 references that are in the map.
- First posted on arXiv 1 Feb 2026.

## Builds on (most cited first, 14 of 14)

- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Pruning Filters for Efficient ConvNets](https://mlsanity.com/paper/1608.08710/) (2016)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [On the Robustness of Interpretability Methods](https://mlsanity.com/paper/1806.08049/) (2018)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [Deep Information Propagation](https://mlsanity.com/paper/1611.01232/) (2016)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice](https://mlsanity.com/paper/1711.04735/) (2017)
- [Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolution Layers](https://mlsanity.com/paper/1802.00124/) (2018)
- [Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra](https://mlsanity.com/paper/2008.11865/) (2020)

## Abstract (arXiv)

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.