[Paper Review] Deep learning estimation of the spectral density of functional time series on large domains

Neda Mahvash Mohammadi, Soham Sarkar+1 more|arXiv (Cornell University)|Jan 1, 2026

Functional Brain Connectivity Studies0 citations

TL;DR

Presents a neural-network based estimator for the spectral density of functional time series on large spatial grids, bypassing autocovariance kernel computations and enabling fast, parallelizable estimation with universal approximation guarantees.

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 $G imes G$ 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.

Motivation & Objective

• Motivate spectral density estimation for high-resolution functional time series defined on very large grids.
• Develop a neural network framework that avoids explicit autocovariance kernel computation.
• Prove universal approximation properties for the neural estimator under general conditions.
• Propose a practical estimation algorithm that is parallelizable and computationally efficient.

Proposed method

• Formulate spectral density in the functional time series setting using the spectral density operator in L2(Q).
• Construct neural network architectures (shallow, deep, and shared) to approximate the spectral density kernel via Fourier transforms of network outputs.
• Define the output layer to approximate the cospectrum and quadspectrum components and ensure nonnegative, Hermitian kernels.
• Introduce a Fourier-transform based representation (via g_m,h) and an output layer that yields a spectral density kernel consistent with the theory.
• Develop a loss function that compares empirical spectral density estimators built from observed data with those from the neural network surrogate, avoiding direct autocovariance calculations.
• Provide an algorithm (Spectral-NN estimator) that trains networks and computes estimated spectral density without forming large autocovariance matrices.

Figure 1: The magnitude of the fitted spectral-NN estimator for the 3D fMRI data. The spectral-NN model was fitted with $M=L=10$ , depth $=4$ , width $=20$ and $q=20$ .

Experimental results

Research questions

• RQ1 Can a neural network-based framework approximate the spectral density operator of a functional time series on large domains without computing autocovariances?
• RQ2 Under what conditions do the proposed networks provide universal approximation of the spectral density?
• RQ3 How can the Fourier-transform representation of network outputs be leveraged to construct valid spectral density kernels?
• RQ4 Is the resulting estimator computationally scalable and parallelizable for high-resolution functional data (e.g., fMRI or climate models)?

Key findings

• The neural network estimator can approximate the spectral density operator in integrated Hilbert-Schmidt norm.
• The approach circumvents computation of large G×G autocovariance kernels, enabling scalable estimation on grids with tens to hundreds of thousands of points.
• The estimator is compatible with a frequency-domain principal components framework and can be trained directly on grid values without estimating autocovariances.
• A practical algorithm (Spectral-NN) is proposed to train networks, compute cospectrum and quadspectrum, and produce spectral density estimates.
• Numerical validation includes simulations and an application to fMRI images, demonstrating computational efficiency and feasibility on large domains.