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新论文 · v1arXiv:2611.06990

ToroFit: Fitting Wedge Sums of Tori to High-Dimensional Data

Tanishq Bhatia, Jose A. Perea, Tarek M. Zikry

原文摘要

Data arising in scientific applications often exhibits complex nonlinear structure. Identifying and parameterizing said nonlinearities can enable better modeling, enhance predictive analysis, data interpolation, denoising, and anomaly detection. In this paper, we develop theoretical and computational methods to detect and parametrize Euclidean point cloud data lying near a wedge sum of tori. Specifically, we use persistent cohomology and cup products to identify the distinct toroidal components, and on each component, we optimize over low-energy multivariate trigonometric polynomials for best data fit. The ensuing optimization problem reduces to generalized ridge regression, which can be solved efficiently via standard techniques. We demonstrate the utility of the proposed framework in both synthetic and real datasets, including interpolation of low frame-rate recurrent videos, attractor recovery in quasiperiodic dynamical systems, and the parametrization of gene-space cycle data for parasites and humans.

stat.ML · cs.LG · math.AT

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2605.06395

Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves

Kartik Tandon, Julian Gould, Tanishq Bhatia, Francesca Dominici, Alejandro Ribeiro, Claudio Battiloro

原文摘要

Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as \textit{HilbNets}. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin \& Niyogi \cite{BELKIN20081289} convergence result for the graph Laplacian to the manifold Laplacian, a theoretical cornerstone of geometric learning methods. Second, we discretize the signals and prove that the discretized (implementable) HilbNets converge to the underlying continuous architectures and are transferable across different samplings of the same bundle, providing consistency for learning. Finally, we validate our framework on synthetic and real-world tasks. Overall, our results broaden the scope of geometric learning as a whole by lifting classical Laplacian-based frameworks to settings where the signal at each point lives in its own Hilbert space.

cs.LG · cs.AI · eess.SP

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arXiv ↗PDF ↗