# Block Toeplitz Sparse Precision Matrix Estimation for Large-Scale Interval-Valued Time Series Forecasting

arXiv: https://arxiv.org/abs/2504.03322 · HTML: https://mlsanity.com/paper/2504.03322/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 47 references that are in the map.
- First posted on arXiv 4 Apr 2025.

## Builds on (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Deep learning for time series classification: a review](https://mlsanity.com/paper/1809.04356/) (2018)
- [Time Series Forecasting With Deep Learning: A Survey](https://mlsanity.com/paper/2004.13408/) (2020)
- [A review on outlier/anomaly detection in time series data](https://mlsanity.com/paper/2002.04236/) (2020)
- [AdaRNN: Adaptive Learning and Forecasting of Time Series](https://mlsanity.com/paper/2108.04443/) (2021)
- [Shape and Time Distortion Loss for Training Deep Time Series Forecasting Models](https://mlsanity.com/paper/1909.09020/) (2019)
- [See Better Before Looking Closer: Weakly Supervised Data Augmentation Network for Fine-Grained Visual Classification](https://mlsanity.com/paper/1901.09891/) (2019)
- [Toeplitz Inverse Covariance-Based Clustering of Multivariate Time Series Data](https://mlsanity.com/paper/1706.03161/) (2017)
- [Highly comparative feature-based time-series classification](https://mlsanity.com/paper/1401.3531/) (2014)
- [An Overview on the Estimation of Large Covariance and Precision Matrices](https://mlsanity.com/paper/1504.02995/) (2015)
- [Graphical Elastic Net and Target Matrices: Fast Algorithms and Software for Sparse Precision Matrix Estimation](https://mlsanity.com/paper/2101.02148/) (2021)

## Abstract (arXiv)

Modeling and forecasting interval-valued time series (ITS) have attracted considerable attention due to their growing presence in various contexts. To the best of our knowledge, there have been no efforts to model large-scale ITS. In this paper, we propose a feature extraction procedure for large-scale ITS, which involves key steps such as auto-segmentation and clustering, and feature transfer learning. This procedure can be seamlessly integrated with any suitable prediction models for forecasting purposes. Specifically, we transform the automatic segmentation and clustering of ITS into the estimation of Toeplitz sparse precision matrices and assignment set. The majorization-minimization algorithm is employed to convert this highly non-convex optimization problem into two subproblems. We derive efficient dynamic programming and alternating direction method to solve these two subproblems alternately and establish their convergence properties. By employing the Joint Recurrence Plot (JRP) to image subsequence and assigning a class label to each cluster, an image dataset is constructed. Then, an appropriate neural network is chosen to train on this image dataset and used to extract features for the next step of forecasting. Real data applications demonstrate that the proposed method can effectively obtain invariant representations of the raw data and enhance forecasting performance.