Frobenius
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Related Articles from SNS
Adjacency Spectral Radius Under Laplacian Sparsification: Deterministic and Probabilistic Bounds
arXiv:2606.07459v1 Announce Type: cross Abstract: Spielman-Srivastava spectral sparsification preserves Laplacian quadratic forms to within (1 +/- epsilon), but does not directly control the adjacency spectral radius lambda_1, which governs the NIMFA epidemic threshold and arises in spectral clustering. We prove |lambda_1(A_H) - lambda_1(A_G)| <= epsilon(2 Delta - lambda_1) deterministically, with a sharp epsilon*lambda_1 bound for reweighting sparsifiers via Perron-Frobenius monotonicity....
Low-Rank Decay for Grokking in Scale-Invariant Transformers: A Spectral-Geometric View
Announce Type: new Abstract: Modern Transformer architectures frequently employ normalization mechanisms such as RMSNorm and Query-Key Normalization, making parts of the model approximately scale-invariant with respect to weight magnitudes. In this regime, standard Frobenius-norm weight decay acts purely along the radial direction of the weight space and cannot directly simplify the function represented by the normalized layer. We study grokking in small algorithmic tasks through this lens...
Relativity from the Perspectives of Observers
Announce Type: new Abstract: This paper reviews the role of observers in the development of relativity theory, from special relativity to general relativity, emphasizing that observer-dependent descriptions are as fundamental as the covariance of physical laws. This paper reviews the role of observers in the development of relativity theory, from special relativity to general relativity, emphasizing that observer-dependent descriptions are as fundamental as the covariance of physical laws....
ResMerge: Residual-based Spectral Merging of Large Language Models
arXiv:2606.02252v1 Announce Type: new Abstract: Model merging offers a training-free way to combine multiple post-trained expert models, but merging experts obtained through reinforcement learning (RL) remains challenging. Existing spectral merging methods often assume that leading singular directions contain the main task signal, while lower-energy residual components can be compressed, selected, or attenuated to reduce interference. We find that this assumption does not hold for RL task...
The Spectral Dynamics and Noise Geometry of Muon
arXiv:2606.08388v1 Announce Type: new Abstract: Muon replaces a matrix gradient $G=U\Sigma V^\top$ by its polar factor $UV^\top$. This keeps the singular directions selected by the gradient, but makes the update spectrum flat. We study the optimization bias created by this operation. Under explicit alignment assumptions, we prove that the polar update is the one-step entropy-maximizing choice among bounded updates that use the gradient singular directions and do not adapt to the current...
Riemannian Optimization for Hadamard Products of Low-Rank Matrices
arXiv:2606.01216v1 Announce Type: new Abstract: The elementwise Hadamard product of two low-rank matrices provides a parameter-efficient model for data with multiplicative structure, but its modeling is challenging due to the presence of additional symmetries under coupled row/column scalings between the two factors. In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold. We propose a novel block-diagonal...
Non-Uniform Codebook Design for Optical IRS-Assisted VLC Systems
Announce Type: new Abstract: Optical intelligent reflecting surfaces (OIRS) can improve the coverage of indoor visible light communication (VLC) systems, however, practical deployment requires a finite offline codebook to avoid repeated real-time optimisation of mirror orientations. A uniform codebook with fixed angular steps does not provide uniform coverage on the user plane, because the mapping from steering angles to reflection locations on the user plane is nonlinear. To address this...
Relational Epipolar Graphs for Robust Relative Camera Pose Estimation
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Rethinking Bregman Divergences in Kronecker-Factored Optimizers
arXiv:2606.00542v2 Announce Type: replace Abstract: Shampoo-style optimizers approximate gradient covariance matrices using Kronecker-factored structures. Recent work~\cite{lin2026understanding} showed that such approximations can be viewed as projections under Bregman matrix divergences, leading to different Kronecker-factored preconditioners. However, it remains unclear what role the choice of divergence plays when the covariance is not exactly Kronecker-factored.
Spectral Methods in Microeconomics
arXiv:2502.12309v2 Announce Type: replace-cross Abstract: Matrices often appear in formal models of social and economic behavior, especially models involving networks. Such models are used to study subjects ranging from opinion dynamics to pollution-mitigation negotiations to the regulation of large marketplace platforms. Matrices are used to capture the focal economic structure in each case.