# Time Series Forecasting via Learning Convolutionally Low-Rank Models

arXiv: https://arxiv.org/abs/2104.11510 · HTML: https://mlsanity.com/paper/2104.11510/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 39 references that are in the map.
- First posted on arXiv 23 Apr 2021.

## Builds on (most cited first, 8 of 8)

- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Time Series Data Augmentation for Deep Learning: A Survey](https://mlsanity.com/paper/2002.12478/) (2020)
- [Guaranteed Matrix Completion via Non-convex Factorization](https://mlsanity.com/paper/1411.8003/) (2014)
- [Principles and Algorithms for Forecasting Groups of Time Series: Locality and Globality](https://mlsanity.com/paper/2008.00444/) (2020)
- [Complete Dictionary Recovery over the Sphere I: Overview and the Geometric Picture](https://mlsanity.com/paper/1511.03607/) (2015)
- [Incoherence-Optimal Matrix Completion](https://mlsanity.com/paper/1310.0154/) (2013)
- [Recovery of Future Data via Convolution Nuclear Norm Minimization](https://mlsanity.com/paper/1909.03889/) (2019)
- [Recovery of Coherent Data via Low-Rank Dictionary Pursuit](https://mlsanity.com/paper/1404.4032/) (2014)

## Built on by (most cited first, 10 of 10)

- [ImputeFormer: Low Rankness-Induced Transformers for Generalizable Spatiotemporal Imputation](https://mlsanity.com/paper/2312.01728/) (2023)
- [Laplacian Convolutional Representation for Traffic Time Series Imputation](https://mlsanity.com/paper/2212.01529/) (2022)
- [Forecasting Sparse Movement Speed of Urban Road Networks with Nonstationary Temporal Matrix Factorization](https://mlsanity.com/paper/2203.10651/) (2022)
- [A Novel Transformer Network with Shifted Window Cross-Attention for Spatiotemporal Weather Forecasting](https://mlsanity.com/paper/2208.01252/) (2022)
- [Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theory](https://mlsanity.com/paper/2501.15388/) (2025)
- [Time Series Forecasting via Semi-Asymmetric Convolutional Architecture with Global Atrous Sliding Window](https://mlsanity.com/paper/2301.13691/) (2023)
- [Adaptive Fuzzy Time Series Forecasting via Partially Asymmetric Convolution and Sub-Sliding Window Fusion](https://mlsanity.com/paper/2507.20641/) (2025)
- [Exact Recovery of Non-Random Missing Multidimensional Time Series via Temporal Isometric Delay-Embedding Transform](https://mlsanity.com/paper/2512.10191/) (2025)
- [Inductive Convolution Nuclear Norm Minimization for Tensor Completion with Arbitrary Sampling](https://mlsanity.com/paper/2604.17001/) (2026)
- [Robust Tensor Completion via Reflective Convolution Nuclear Norm Minimization](https://mlsanity.com/paper/2610.04979/) (2026)

## Abstract (arXiv)

Recently, Liu and Zhang studied the rather challenging problem of time series forecasting from the perspective of compressed sensing. They proposed a no-learning method, named Convolution Nuclear Norm Minimization (CNNM), and proved that CNNM can exactly recover the future part of a series from its observed part, provided that the series is convolutionally low-rank. While impressive, the convolutional low-rankness condition may not be satisfied whenever the series is far from being seasonal, and is in fact brittle to the presence of trends and dynamics. This paper tries to approach the issues by integrating a learnable, orthonormal transformation into CNNM, with the purpose for converting the series of involute structures into regular signals of convolutionally low-rank. We prove that the resultant model, termed Learning-Based CNNM (LbCNNM), strictly succeeds in identifying the future part of a series, as long as the transform of the series is convolutionally low-rank. To learn proper transformations that may meet the required success conditions, we devise an interpretable method based on Principal Component Pursuit (PCP). Equipped with this learning method and some elaborate data argumentation skills, LbCNNM not only can handle well the major components of time series (including trends, seasonality and dynamics), but also can make use of the forecasts provided by some other forecasting methods; this means LbCNNM can be used as a general tool for model combination. Extensive experiments on 100,452 real-world time series from Time Series Data Library (TSDL) and M4 Competition (M4) demonstrate the superior performance of LbCNNM.