This repository provides an example of tensor-train (TT) decomposition using the Xerus library, demonstrating its effectiveness in compressing higher-order tensors. To reproduce the results, follow the instructions in Usage.md.
Higher-order tensors present significant computational challenges due to the curse of dimensionality, leading to high storage requirements and expensive computations. Developing efficient compression techniques is crucial for handling such large-scale data.
In their work, Ballard, Kolda, and Lindstrom use the Miranda Turbulent Flow Dataset to illustrate the successful application of Tucker decomposition for tensor compression. This repository aims to showcase how tensor-train decomposition can achieve even better performance.
Additionally, we highlight the simplicity and usability of the Xerus library, which offers efficient tensor computation tools with strong support for the tensor-train format and general tensor networks.
The Miranda Turbulent Flow Dataset represents a Rayleigh-Taylor instability simulation, where two fluids of different densities accelerate against each other, leading to turbulent mixing. The dataset captures a late-stage time step of this instability, computed using the Miranda software from Lawrence Livermore National Laboratory (LLNL) and is published by the Scientific Data Reduction Benchmark (SDRBench). We utilize the pre-processed subtensor from Ballard et al. with dimensions 2048 x 256 x 256.
density.mat- Input MATLAB file, uploaded with Git LFS.Miranda.cpp- C++ implementation for computing the tensor-train compressions. Requires Xerus, Matio (available on Ubuntu viasudo apt install libmatio-dev) and at least 7+ GB of RAM.xerusTensors/- Contains the uncompressed Xerus tensor and tensor-train tensor for faster execution ofMiranda.cpp. Both are uploaded with Git LFS.compressionResults/- Stores the MAT files of the compressed tensors (not in TT format) generated for different thresholds1e-x. Also uses Git LFS.visualization/- Includes the Python script for 3D visualization and the resulting PNG image.
We first computed the full-rank Tensor-Train decomposition and then used the Xerus round() function to compress the tensor to the desired threshold. While this function will find a sufficient approximation, it does not necessarily yield the best estimate. In accordance with Ballard et al., we used the squared error in Frobenius norm (
Our results show that even for a relative error of 1e-8, the tensor-train approximation takes less than 4% of the original storage size. The Tucker decomposition was only evaluated up to a threshold of 1e-4, and for these values, the comparison reveals 10 to 100 times better compression ratios for the TT decomposition.
| Threshold | Rel. Error | TT-Ranks | Entries | Compression Ratio | % of Original Size |
|---|---|---|---|---|---|
| 1e-1 | 1.8e-2 | [1, 1] | 2,560 | 52,416.00 | 0.0019 |
| 1e-2 | 8.9e-3 | [3, 3] | 9,216 | 14,559.11 | 0.0069 |
| 1e-3 | 9.5e-4 | [72, 19] | 344,576 | 389.57 | 0.2568 |
| 1e-4 | 9.7e-5 | [245, 43] | 1,152,960 | 116.36 | 0.8590 |
| 1e-5 | 9.8e-6 | [458, 72] | 2,089,984 | 64.24 | 1.5569 |
| 1e-6 | 9.7e-7 | [682, 103] | 3,083,264 | 43.54 | 2.2970 |
| 1e-7 | 9.9e-8 | [904, 133] | 4,090,880 | 32.80 | 3.0499 |
| 1e-8 | 1.0e-8 | [1120, 161] | 5,127,936 | 26.18 | 3.8200 |
In our visualization, we show the results of the first five compressions together with the original tensor.
- G. Ballard, T. G. Kolda, and P. Lindstrom, Miranda Turbulent Flow Dataset, https://gitlab.com/tensors/tensor_data_miranda_sim, 2022.
- K. Zhao, S. Di, X. Liang, S. Li, D. Tao, J. Bessac, Z. Chen, and F. Cappello, SDRBench: Scientific Data Reduction Benchmark for Lossy Compressors, 2020 IEEE International Conference on Big Data (Bigdata20), 2020, https://doi.org/10.1109/bigdata50022.2020.9378449
- W. H. Cabot, and A. W. Cook, Reynolds number effects on Rayleigh-Taylor instability with possible implications for type Ia supernovae, Nature Physics, Vol. 2, No. 8, pp. 562-568, July 2006, https://doi.org/10.1038/nphys361
- B. Huber, and S. Wolf, Xerus - A General Purpose Tensor Library, 2014-2017, https://libxerus.org/