arXiv Computer Vision
Orthogonal Polynomial Approximation for Matrix Log Normalization in Global Covariance Pooling
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arXiv Machine Learning
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Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent
arXiv:2606. 02596v1 Announce Type: new Abstract: The curvature exponent $\alpha$ in $h_k \propto \sigma_k^\alpha$ -- governing how Hessian eigenvalues scale with gradient singular values -- varies systematically across layer types ($\alpha \approx 2$ for convolutions, $\approx 1$ for transformer attention, $< 1$ for MLP up-projections).
arXiv Machine Learning
Jun 5
Efficient Mean Curvature Computation on High-Dimensional Data Manifolds
arXiv:2606. 06329v1 Announce Type: new Abstract: Estimating local mean curvature at each point of a high-dimensional dataset is a key ingredient of geometry-aware machine learning algorithms, such as the Mean Curvature Boundary Points (MCBP) method.
arXiv Machine Learning
Jun 16
The Reverse Telescoping Coordinate System for Positive Definite Matrices: Geometry, Computation, and Generative Modeling
arXiv:2606. 15442v1 Announce Type: cross Abstract: We design a new unconstrained coordinate system where a $p\times p$ symmetric positive definite (SPD) matrix $\Theta$ is represented by a reverse telescoping map $\Theta(x)=\rm{RT}(x)$, with $x=(v,d,r)\in\mathbb{R}\times\mathbb{R}^{(p-1)}\times\mathbb{R}^{p(p-1)/2}$, representing respectively the log volume or log determinant; and the shape, as encoded by log relative diagonal scales and partial covariances among the nodes.
arXiv Machine Learning
Jul 28
Spectral-Aware Analytic Class-Incremental Learning for Long-Tailed Distributions
arXiv:2607. 22931v1 Announce Type: new Abstract: Analytic Continual Learning (ACL) offers a computationally efficient alternative to gradient-based approaches.
arXiv Machine Learning
Jul 16
Flatness and Gradient Alignment Are Both Necessary: Spectral-Aware Gradient-Aligned Exploration for Multi-Distribution Learning
arXiv:2605. 07914v2 Announce Type: replace Abstract: Sharpness-aware and gradient-alignment methods have been shown to improve generalization, however each family of methods targets a single geometric property of the loss landscape, while ignoring the other.
arXiv AI
Jul 29
Quotient Dynamics, Effective Curvature, and Implicit Bias in Positive Quadratic Networks
arXiv:2607. 25624v1 Announce Type: new Abstract: Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top.