The paper introduces a unified matrix‑spectral framework that treats deep neural networks as data‑dependent products of linear operators, enabling the analysis of stability and interpretability through spectral quantities. It defines 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, and shows that spectral entropy provides a more realistic measure of sensitivity than worst‑case bounds. Experiments on MNIST, CIFAR‑10, and CIFAR‑100 demonstrate that modest spectral regularization can significantly improve attribution stability without large changes to global spectral summaries.
arXiv:2606. 04767v1 Announce Type: new Abstract: The robustness of deep neural networks is crucial for safety-critical deployments, yet existing evaluation methods are often attack-dependent and lack interpretability.
arXiv:2506. 08764v3 Announce Type: replace Abstract: Deep neural networks are known to suffer from exploding or vanishing gradients as depth increases, a phenomenon closely tied to the spectral behavior of the input-output Jacobian.
arXiv:2607. 16329v1 Announce Type: cross Abstract: Lipschitz continuity is a fundamental property of neural networks that characterizes their sensitivity to input perturbations.
arXiv:2607. 13631v1 Announce Type: new Abstract: The Hessian matrix is an important quantity of interest when it comes to studying the loss landscape and optimization dynamics in deep learning, as well as designing measures of generalization, second-order learning algorithms, etc.
arXiv:2606. 16883v1 Announce Type: cross Abstract: Generalization is a critical property of data-driven models, particularly deep learning models deployed in safety-critical applications.
The paper introduces a method for constructing L-Lipschitz deep residual networks (ResNets) using a Linear Matrix Inequality (LMI) framework. By reformulating the ResNet architecture as a pseudo-tridiagonal LMI and applying the Gershgorin circle theorem, the authors derive closed‑form constraints on network parameters that guarantee Lipschitz continuity. The work also presents a compositional framework for handling recursive systems in hierarchical architectures, while noting that the Gershgorin-based approximations can over‑constrain the system, reducing expressive capacity.
arXiv:2602. 04078v2 Announce Type: replace-cross Abstract: Deep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence.
The Hessian matrix is an important quantity of interest when it comes to studying the loss landscape and optimization dynamics in deep learning, as well as designing measures of generalization, second-order learning algorithms, etc. Prior works have focused on empirical results or pursued a theoretical treatment under overly simplified settings.
arXiv:2604. 14669v2 Announce Type: replace Abstract: Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored.
arXiv:2607. 25494v1 Announce Type: cross Abstract: Finite-precision arithmetic unavoidably introduces numerical approximation errors.
arXiv:2506. 13139v3 Announce Type: replace-cross Abstract: Modern Machine Learning (ML) and Deep Neural Networks (DNNs) often operate on high-dimensional data and rely on overparameterized models, where classical low-dimensional intuitions break down.
arXiv:2506.17874v3 Announce Type: replace-cross Abstract: In many real-world applications, ensuring the robustness and stability of deep neural networks (DNNs) is crucial, particularly for image clas...