Tensor Learning (张量学习)

Made by Xinyu Chen • 🌐 https://twitter.com/chenxy346

Python codes for tensor factorization, tensor completion, and tensor regression techniques with the following real-world applications:

• geotensor | Image inpainting
• transdim | Spatiotemporal traffic data imputation and prediction
• Recommender systems
• mats | Multivariate time series imputation and forecasting

In a hurry? Please check out our contents as follows.

Our Research

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We conduct extensive experiments on some real-world data sets:

• Middle-scale data sets:

  • PeMS (P) registers traffic speed time series from 228 sensors over 44 days with 288 time points per day (i.e., 5-min frequency). The tensor size is 228 x 288 x 44.
  • Guanghzou (G) contains traffic speed time series from 214 road segments in Guangzhou, China over 61 days with 144 time points per day (i.e., 10-min frequency). The tensor size is 214 x 144 x 61.
  • Electricity (E) records hourly electricity consumption transactions of 370 clients from 2011 to 2014. We use a subset of the last five weeks of 321 clients in our experiments. The tensor size is 321 x 24 x 35.

• Large-scale PeMS traffic speed data set registers traffic speed time series from 11160 sensors over 4/8/12 weeks (for PeMS-4W/PeMS-8W/PeMS-12W) with 288 time points per day (i.e., 5-min frequency) in California, USA. You can download this data set and place it at the folder of ../datasets.

  • Data size:
    • PeMS-4W: 11160 x 288 x 28 (contains about 90 million observations).
    • PeMS-8W: 11160 x 288 x 56 (contains about 180 million observations).
  • Data path example: ../datasets/California-data-set/pems-4w.csv.
  • Open data in Python with Pandas:

import pandas as pd

data = pd.read_csv('../datasets/California-data-set/pems-4w.csv', header = None)

mats

mats is a project in the tensor learning repository, and it aims to develop machine learning models for multivariate time series forecasting. In this project, we propose the following low-rank tensor learning models:

• Low-Rank Autoregressive Tensor Completion (LATC) (3-min introduction) for multivariate time series (middle-scale data sets like PeMS, Guangzhou, and Electricity) imputation and forecasting (Chen et al., 2020):

  • with nuclear norm (NN) minimization [Python code for imputation]
  • with truncated nuclear norm (TNN) minimization [Python code for imputation] [Python code for prediction]
  • with Schatten p-norm (SN) minimization [Python code for imputation]
  • with truncated Schatten p-norm (TSN) minimization [Python code for imputation]

• Low-Tubal-Rank Autoregressive Tensor Completion (LATC-Tubal) for large-scale spatiotemporal traffic data (large-scale data sets like PeMS-4W and PeMS-8W) imputation (Chen et al., 2020):

  • without autoregressive norm [Python code]
  • with autoregressive norm [Python code]

We write Python codes with Jupyter notebook and place the notebooks at the folder of ../mats. If you want to test our Python code, please run the notebook at the folder of ../mats. Note that each notebook is independent on others, you could run each individual notebook directly.

The baseline models include:

• on middle-scale data sets:

  • coming soon...

• on large-scale data sets:

  • Bayesian Probabilistic Matrix Factorization (BPMF, Salakhutdinov and Mnih, 2008) [Python code]

  • Bayesian Gaussian CP decomposition (BGCP, Chen et al., 2019) [Python code]

  • High-accuracy Low-Rank Tensor Completion (HaLRTC, Liu et al., 2013) [Python code]

  • Low-Rank Tensor Completion with Truncated Nuclear Norm minimization (LRTC-TNN, Chen et al., 2020) [Python code]

  • Tensor Nuclear Norm minimization with Discrete Cosine Transform (TNN-DCT, Lu et al., 2019) [Python code]

We write Python codes with Jupyter notebook and place the notebooks at the folder of ../baselines. If you want to test our Python code, please run the notebook at the folder of ../baselines. The notebook which reproduces algorithm on large-scale data sets is emphasized by Large-Scale-xx.

📖 Reproducing Literature in Python

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We reproduce some tensor learning experiments in the previous literature.

📖 Tutorial

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We summarize some preliminaries for better understanding tensor learning. They are given in the form of tutorial as follows.

• Foundations of Python Numpy Programming

  • Generating random numbers in Matlab and Numpy [Jupyter notebook] [blog post]

• Foundations of Tensor Computations

  • Kronecker product

• Singular Value Decomposition (SVD)

  • Randomized singular value decomposition [Jupyter notebook] [blog post]
  • Tensor singular value decomposition

If you find these codes useful, please star (★) this repository.

Helpful Material

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We believe that these material will be a valuable and useful source for the readers in the further study or advanced research.

• Vladimir Britanak, Patrick C. Yip, K.R. Rao (2006). Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations. Academic Press. [About the book]

• Ruye Wang (2010). Introduction to Orthogonal Transforms with Applications in Data Processing and Analysis. Cambridge University Press. [PDF]

• J. Nathan Kutz, Steven L. Brunton, Bingni Brunton, Joshua L. Proctor (2016). Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems. SIAM. [About the book]

• Yimin Wei, Weiyang Ding (2016). Theory and Computation of Tensors: Multi-Dimensional Arrays. Academic Press.

• Steven L. Brunton, J. Nathan Kutz (2019). Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press. [PDF] [data & code]

Quick Run

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• If you want to run the code, please
  • download (or clone) this repository,
  • open the .ipynb file using Jupyter notebook,
  • and run the code.

Citing

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This repository is from the following paper, please cite our paper if it helps your research.

• Xinyu Chen, Lijun Sun (2020). Low-rank autoregressive tensor completion for multivariate time series forecasting. arXiv: 2006.10436. [preprint] [data & Python code]

Acknowledgements

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This research is supported by the Institute for Data Valorization (IVADO).

License

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This work is released under the MIT license.