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Variational Seasonal-Trend Decomposition with Sparse Continuous-Domain Regularization

Julien Fageot

TL;DR

This work addresses recovering a continuous-domain signal $f_0=f_{0,T}+f_{0,S}$ from finite noisy measurements by formulating a variational problem with generalized total-variation regularization that promotes sparse, spline-like representations for both the trend $f_{0,T}$ and the 1-periodic seasonal part $f_{0,S}$. It develops a functional-analytic foundation, defines seasonal-trend native spaces, and proves a representer theorem guaranteeing composite spline solutions for both components, along with a $\Gamma$-convergence analysis showing grid-based discretizations converge uniformly to the continuous problem. The results establish, under broad conditions, that minimizers are sparse splines with explicitly characterized forms, and that practical grid-based algorithms reliably approximate the true continuous solutions. Overall, the paper provides a rigorous, scalable approach to seasonal-trend decomposition in noisy, limited-measurement regimes, with strong theoretical guarantees for both representation and discretization.

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 $Γ$-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.

Variational Seasonal-Trend Decomposition with Sparse Continuous-Domain Regularization

TL;DR

This work addresses recovering a continuous-domain signal from finite noisy measurements by formulating a variational problem with generalized total-variation regularization that promotes sparse, spline-like representations for both the trend and the 1-periodic seasonal part . It develops a functional-analytic foundation, defines seasonal-trend native spaces, and proves a representer theorem guaranteeing composite spline solutions for both components, along with a -convergence analysis showing grid-based discretizations converge uniformly to the continuous problem. The results establish, under broad conditions, that minimizers are sparse splines with explicitly characterized forms, and that practical grid-based algorithms reliably approximate the true continuous solutions. Overall, the paper provides a rigorous, scalable approach to seasonal-trend decomposition in noisy, limited-measurement regimes, with strong theoretical guarantees for both representation and discretization.

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 -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.
Paper Structure (33 sections, 15 theorems, 107 equations)

This paper contains 33 sections, 15 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Seasonal-Trend Sparse Inverse Problem
  3. Related Works
  4. Contributions and Outline
  5. Functional Analytic Preliminaries
  6. Tempered Generalized Functions
  7. Linear Shift-Invariant Operators
  8. Periodic and Non-Periodic Radon Measures
  9. Native Spaces for Seasonal-Trend Decomposition
  10. Trend Operators, Splines, and Native Spaces
  11. Trend Operators.
  12. Seasonal Operators, Splines, and Native Spaces
  13. Seasonal Operators.
  14. Periodic $\mathrm{L}_S$-Splines.
  15. Seasonal Native Spaces.
  16. ...and 18 more sections

Key Result

Proposition 1

Let ${\rm L}$ be a trend-admissible operator with trend-admissibility order $N_T \geq 0$. Then, the following facts hold:

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Definition 3
  • Proposition 2
  • proof
  • Definition 4
  • Proposition 3
  • proof
  • ...and 29 more