Variational Seasonal-Trend Decomposition with Sparse Continuous-Domain Regularization

TL;DR

This paper introduces a variational method for recovering continuous seasonal and trend components from noisy measurements, with theoretical guarantees on spline representation and convergence of discretizations.

Contribution

It develops a novel variational framework with convex regularizations promoting sparsity, and proves convergence of discrete approximations to the continuous solution.

Findings

Minimizers are splines in both components.

Discrete approximations converge to the continuous solution.

Framework effectively handles noise and limited measurements.

Abstract

We consider the inverse problem of recovering a continuous-domain function from a finite number of noisy linear measurements. The unknown signal is modeled as the sum of a slowly varying trend and a periodic or quasi-periodic seasonal component. We formulate a variational framework for their joint recovery by introducing convex regularizations based on generalized total variation, which promote sparsity in spline-like representations. Our analysis is conducted in an infinite-dimensional setting and leads to a representer theorem showing that minimizers are splines in both components. To make the approach numerically feasible, we introduce a family of discrete approximations and prove their convergence to the original problem in the sense of $\Gamma$-convergence. This further ensures the uniform convergence of discrete solutions to their continuous counterparts. The proposed framework offers a principled approach to seasonal-trend decomposition in the presence of noise and limited measurements, with theoretical guarantees on both representation and discretization.