The paper investigates how machine learning models can regress scale‑free processes, such as earthquakes or avalanches, focusing on predicting rare, large events that require extrapolation. It studies two self‑similar systems: a 2‑dimensional fractional Gaussian field and the Abelian sandpile model. Experiments compare existing architectures (U‑net, Riesz network) with new proposals (wavelet‑based Graph Neural Network, Fourier embedding, Fourier‑Mellin Neural Operator) to identify spectral bias and coarse‑graining challenges and suggest inductive biases to address them.
By Anaclara Alvez-Canepa, Cyril Furtlehner, Fran\c{c}ois P. Landes
arXiv:2606. 04065v1 Announce Type: cross Abstract: We study simultaneous alternating power iteration for fixed-order asymmetric rank-one spiked tensor models.
By Yanjin Xiang, Zhihua Zhang
We investigate how each component of the Transformer feedforward block architecture design determines how much rank survives across depth at initialization. We reinterpret skip connections and normalization, long understood as controlling magnitude, as mechanisms for preserving gradient rank across depth, since the very matrix multiplications and nonlinear activations that make the network expressive also reduce the rank.
arXiv:2606. 05957v1 Announce Type: new Abstract: Singular learning theory and information geometry have studied the same parameter spaces in mostly separate vocabularies: the former computes Bayesian invariants in resolved coordinates, the latter works in original coordinates under a non-degeneracy assumption that overparameterised models routinely violate.
By Tejas Pradeep Shirodkar
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).
By Anherutowa Calvo
arXiv:2607. 14018v1 Announce Type: cross Abstract: We investigate how each component of the Transformer feedforward block architecture design determines how much rank survives across depth at initialization.
By Katie Everett
arXiv:2607. 28080v1 Announce Type: cross Abstract: We extend a recently introduced Entropy-Optimal Manifold Clustering (EOMC) to allow for a joint simultaneous identification of subsets and subspaces of relevant features in nonstationary and nonlinear regression problems.
By Illia Horenko
arXiv:2605. 18528v2 Announce Type: replace-cross Abstract: A growing lesson from neural network optimization is that optimizer design should respect how the model is parametrized.
By Jiayu Zhang, Tianyi Lin
arXiv:2607. 15702v1 Announce Type: cross Abstract: We prove a finite-sample formulation gap for physics-informed learning of nonlinear multiscale elliptic equations.
By Ronald Katende
arXiv:2608. 13335v1 Announce Type: new Abstract: Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly.
By Liu Ziyin, Yizhou Xu, Tomaso Poggio, Isaac Chuang
arXiv:2608. 28564v1 Announce Type: cross Abstract: We study kernel ridge regression under anisotropic Gaussian data, where the input covariance decays as a power law with exponent $\alpha\geq 0$ for polynomial inner-product kernels.
By Lorenzo Rizzi, Arie Wortsman Zurich, Bruno Loureiro
arXiv:2609.39855v1 Announce Type: new
Abstract: We study the width required for a randomly initialized hidden layer of a neural network to achieve rank lifting. Namely, given a dataset $X \in \mathbb...
By Luca Becchetti, Matteo Russo, Ruben Skorupinski