Deep learning estimation of the spectral density of functional time series on large domains

2026-01-01

AI Summary

This study addresses the computational bottleneck in spectral density estimation for functional time series observed on large grids—such as climate models or fMRI data—arising from the high-dimensional autocovariance kernel. To overcome this challenge, the authors propose a deep learning approach based on multilayer perceptrons that integrates spectral functional principal component analysis with neural networks for the first time. This method efficiently approximates the spectral density without explicitly computing the autocovariance kernel, leveraging the universal approximation capability of neural networks. Implemented with parallelization, the approach achieves substantial gains in computational efficiency—scaling to grid sizes on the order of \(10^5\)—while preserving estimation accuracy, as demonstrated in both simulated and real fMRI data experiments.

Technology Category

• Machine Learning: Kernel Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

We derive an estimator of the spectral density of a functional time series that is the output of a multilayer perceptron neural network. The estimator is motivated by difficulties with the computation of existing spectral density estimators for time series of functions defined on very large grids that arise, for example, in climate compute models and medical scans. Existing estimators use autocovariance kernels represented as large $G \times G$ matrices, where $G$ is the number of grid points on which the functions are evaluated. In many recent applications, functions are defined on 2D and 3D domains, and $G$ can be of the order $G \sim 10^5$, making the evaluation of the autocovariance kernels computationally intensive or even impossible. We use the theory of spectral functional principal components to derive our deep learning estimator and prove that it is a universal approximator to the spectral density under general assumptions. Our estimator can be trained without computing the autocovariance kernels and it can be parallelized to provide the estimates much faster than existing approaches. We validate its performance by simulations and an application to fMRI images.

Problem

• spectral density
• functional time series
• large domains
• autocovariance kernels
• computational complexity

Innovation

• deep learning
• spectral density estimation
• functional time series
• universal approximator
• high-dimensional grids