OpenNovelty.org

Characteristic Root Analysis and Regularization for Linear Time Series Forecasting

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
long term time series forecastinglinear modelcharacteristic rootsmodesnoise robustnessrank reductionroot purge
Abstract:

Time series forecasting remains a critical challenge across numerous domains, yet the effectiveness of complex models often varies unpredictably across datasets. Recent studies highlight the surprising competitiveness of simple linear models, suggesting that their robustness and interpretability warrant deeper theoretical investigation. This paper presents a systematic study of linear models for time series forecasting, with a focus on the role of characteristic roots in temporal dynamics. We begin by analyzing the noise-free setting, where we show that characteristic roots govern long-term behavior and explain how design choices such as instance normalization and channel independence affect model capabilities. We then extend our analysis to the noisy regime, revealing that models tend to produce spurious roots. This leads to the identification of a key data-scaling property: mitigating the influence of noise requires disproportionately large training data, highlighting the need for structural regularization. To address these challenges, we propose two complementary strategies for robust root restructuring. The first uses rank reduction techniques, including Reduced-Rank Regression and Direct Weight Rank Reduction, to recover the low-dimensional latent dynamics. The second, a novel adaptive method called Root Purge, encourages the model to learn a noise-suppressing null space during training. Extensive experiments on standard benchmarks demonstrate the effectiveness of both approaches, validating our theoretical insights and achieving state-of-the-art results in several settings. Our findings underscore the potential of integrating classical theories for linear systems with modern learning techniques to build robust, interpretable, and data-efficient forecasting models.

Disclaimer
This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper contributes a systematic theoretical framework analyzing characteristic roots in linear time series models, focusing on how roots govern long-term dynamics and how design choices like instance normalization affect model capabilities. It resides in the Autoregressive Root Analysis leaf, which contains eight papers within the Characteristic Root and Eigenstructure Methods branch. This represents a moderately populated research direction within a taxonomy of fifty papers across thirty-six topics, suggesting focused but not overcrowded attention to root-based stability analysis in linear forecasting.

The taxonomy reveals neighboring leaves including Singular Spectrum Analysis with Root Extraction (two papers) and Neural Eigenstructure Methods (two papers), indicating that eigenstructure-based forecasting extends beyond classical AR analysis. The broader Decomposition-Based Forecasting branch encompasses Dynamic Mode Decomposition, Principal Component Decomposition, and Hankel Matrix methods, representing alternative eigenvalue-driven approaches that do not explicitly analyze characteristic polynomial roots. The paper's emphasis on root restructuring and noise-induced spurious roots distinguishes it from these decomposition methods, which typically focus on signal separation rather than root-level stability diagnostics.

Among thirty candidates examined through semantic search, none clearly refuted the three core contributions. The theoretical analysis of characteristic roots examined ten candidates with zero refutable overlaps; rank reduction techniques examined ten with zero refutations; and the Root Purge adaptive training method examined ten with zero refutations. This limited search scope suggests that within the top-thirty semantically similar papers, no prior work explicitly combines root-based theoretical analysis with rank reduction and adaptive training strategies for mitigating noise-induced spurious roots, though the search does not claim exhaustive coverage of all relevant literature.

Based on the limited search scope of thirty candidates, the work appears to occupy a distinct position within autoregressive root analysis by integrating theoretical root dynamics with practical regularization strategies. The absence of refutable candidates among examined papers suggests novelty in the specific combination of contributions, though the moderately populated taxonomy leaf indicates active prior research on characteristic polynomial methods. The analysis does not cover broader literature beyond top-thirty semantic matches or citation-expanded candidates.

Taxonomy

Core-task Taxonomy Papers
38
3
Claimed Contributions
30
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: linear time series forecasting with characteristic root analysis. The field encompasses a diverse set of approaches organized into six main branches. Characteristic Root and Eigenstructure Methods focus on analyzing the stability and dynamics of autoregressive processes through their eigenvalues and roots, providing theoretical insights into model behavior. Decomposition-Based Forecasting leverages techniques that break down complex signals into interpretable components for prediction. Neural Network Forecasting explores modern deep learning architectures tailored to temporal patterns. Uncertainty Quantification and Optimization addresses probabilistic modeling and parameter tuning challenges. Theoretical Foundations and Extensions develop the mathematical underpinnings of time series models, including stationarity conditions and asymptotic properties. Domain-Specific Applications translate these methods to fields such as hydrology, finance, and environmental monitoring, where specialized constraints and data characteristics arise. Within Characteristic Root and Eigenstructure Methods, a small cluster of works examines autoregressive root analysis to understand model stability and forecast reliability. Characteristic Root Analysis and[0] situates itself in this tradition, emphasizing how characteristic polynomial roots inform linear forecasting performance. Nearby studies such as Application of characteristic polynomial[20] and Order determination of multivariate[25] similarly exploit polynomial structure for model identification and diagnostics, though they may prioritize different aspects like order selection or multivariate extensions. In contrast, works like Non-linear time series model[31] and Inference for time series[32] broaden the scope beyond linear frameworks or focus on inferential rather than purely predictive goals. The interplay between classical eigenstructure analysis and modern computational methods remains an active area, with ongoing questions about how root-based diagnostics scale to high-dimensional or nonstationary settings and how they complement data-driven neural approaches.

Claimed Contributions

Theoretical analysis of characteristic roots in linear time series forecasting

The authors develop a theoretical framework examining how characteristic roots govern temporal dynamics in linear forecasting models. They analyze design choices like instance normalization and channel independence in noise-free settings, then extend to noisy regimes where they identify a data-scaling property showing that mitigating noise requires disproportionately large training data.

10 retrieved papers
Rank reduction techniques for robust root identification

The authors propose using Reduced-Rank Regression (RRR) and Direct Weight Rank Reduction (DWRR) to recover low-dimensional latent dynamics by constraining the weight matrix rank. These methods project input and output data onto learned low-dimensional subspaces to suppress noise while preserving signal components.

10 retrieved papers
Root Purge adaptive training method

The authors introduce Root Purge, an adaptive method that modifies the training loss to encourage learning a noise-suppressing null space during optimization. This approach dynamically adjusts model rank through the rank-nullity theorem, balancing signal fitting with noise suppression without requiring fixed rank assumptions.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Theoretical analysis of characteristic roots in linear time series forecasting

The authors develop a theoretical framework examining how characteristic roots govern temporal dynamics in linear forecasting models. They analyze design choices like instance normalization and channel independence in noise-free settings, then extend to noisy regimes where they identify a data-scaling property showing that mitigating noise requires disproportionately large training data.

Contribution

Rank reduction techniques for robust root identification

The authors propose using Reduced-Rank Regression (RRR) and Direct Weight Rank Reduction (DWRR) to recover low-dimensional latent dynamics by constraining the weight matrix rank. These methods project input and output data onto learned low-dimensional subspaces to suppress noise while preserving signal components.

Contribution

Root Purge adaptive training method

The authors introduce Root Purge, an adaptive method that modifies the training loss to encourage learning a noise-suppressing null space during optimization. This approach dynamically adjusts model rank through the rank-nullity theorem, balancing signal fitting with noise suppression without requiring fixed rank assumptions.

Table of Contents