We consider the Multiscale Single-Index Model (MSIM), first introduced in \cite{oymak2021learning}, as a stylized model for hierarchical learning with \emph{scale separation}. Each layer extracts a shared single-index feature at one physical scale and passes it to the next, thus defining a tractable setting in which to study how deep architectures learn multiscale representations. Under non-degeneracy and delocalization assumptions on the link function and planted features respectively, for fixed depth $K$ and local scale $d$, the first Wiener chaos of the target behaves as a perturbed spiked tensor, where the perturbation of order $d^{-1/2}$ comes from the non-linearity -- revealing the MSIM as a natural non-linear analogue of the Tensor PCA model \cite{montanari2014statistical}. While this perturbative picture is sufficient to enable efficient spectral recovery based on Tensor unfolding (as already observed in \cite{oymak2021learning}), it is not precise enough for the analysis of backpropagation gradient-based methods. In this work, we address this limitation by performing a fine-grained analysis of the Wiener chaos using Edgeworth expansions. In the first chaos, this gives a finite-rank hierarchy at scales $d^{-q/2}$. In higher chaoses, balanced flattenings exhibit staircase singular-value plateaus of size $d^{-ρ/2}$ and multiplicity $d^ρ$ under a natural higher-chaos non-cancellation condition. Using this higher-chaos structure, and under an additional slow Hermite-energy tail condition, we first establish shallow-network approximation lower bounds, quantifying the benefit of depth in this model. Next, and most importantly, we prove that online SGD on the correlation objective, where all layers evolve in the same timescale, achieves $1 - o_d(1)$ recovery with $n = \widetilde{O}( d^{K-1})$ samples, recovering the same sample complexity as in the linear counterpart.
Hierarchical Multiscale Structure-Function Coupling for Brain Connectome Integration
School of Medicine, University of Dundee, UK（邓迪大学医学院）；School of Science and Engineering, University of Dundee, UK（邓迪大学科学与工程学院）；College of Medicine and Biological Information Engineering, Northeastern University, China（东北大学医学院与生物信息工程学院）；College of Medical Informatics, Chongqing Medical University, China（重庆医科大学医学信息学院）；School of Airspace Science and Engineering, Shandong University, China（山东大学航空航天科学与工程学院）；Chinese Institute for Brain Research, China（中国脑科学研究院）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK（剑桥大学应用数学与理论物理系）；Department of Clinical Neurosciences, University of Cambridge, UK（剑桥大学临床神经科学系）
MCbiF: Measuring Topological Autocorrelation in Multiscale Clusterings via 2-Parameter Persistent Homology
Department of Mathematics, Imperial College London, UK（英国伦敦帝国理工学院数学系）；Department of Mathematical Modeling and Machine Learning, University of Zurich, Switzerland（瑞士苏黎世大学数学建模与机器学习系）
Multiscale Supervised Unbalanced Optimal Transport Flow Matching
Center for Quantitative Biology, Peking University（北京大学定量生物学中心）；National Engineering Laboratory for Big Data Analysis and Applications, Beijing, China（大数据分析与应用国家工程实验室）；AI for Science Institute, Beijing, China（人工智能科学研究院）
Prediction of Multiscale Features Using Deep Learning-based Preconditioner-Solver Architecture for Darcy Equation in High-Contrast Media
School of Mathematics and Physics, Xi'an Jiaotong-Liverpool University（数学与物理学院，西安交通大学利物浦大学）；Department of Mathematical Sciences, University of Liverpool（数学科学系，利物浦大学）；School of Mathematics and Information Science, Yantai University（数学与信息科学学院，烟台大学）
A Generalized Framework for Multiscale State-Space Modeling with Nested Nonlinear Dynamics: An Application to Bayesian Learning under Switching Regimes
School of Complex Adaptive Systems（复杂适应系统学院）；Arizona State University（亚利桑那州立大学）
