Best rank-one approximation is one of the most fundamental tasks in tensor computation. In order to fully exploit modern multi-core parallel computers, we build a bridge between the rank-one approximation of tensors and the eigenvector-dependent nonlinear eigenvalue problem (NEPv), and then develop an efficient decoupling algorithm, namely the higher-order self-consistent field (HOSCF) algorithm.

In order to present the scalability of HOSCF, we implement the HOSCF and HOPM(based ALS) algorithm in C++ language with OpenMP multi-threading and MPI multi-processes and the involved linear algebra operations such as eigensolver available from the Intel MKL library. The figure below illustrates the running time and speedup of the HOPM and HOSCF algorithms on 5,6,7,8-order(d) tensor as the number of processor cores increases from 1 to 256 (on 8 compute nodes, each node equipped with an Intel Xeon Platinum 8358P CPU of 2.60 GHz).

The HOSCF depends on blas and lapack. A convenient way to get this ready is oneapi.

1. install oneAPI Base and hpc Toolkit
2. source setvars.sh(perhaps at /opt/intel/oneapi/setvars.sh)

1. mkdir build
2. cmake ..
3. make -j 9
4. ./run -t 8 -n 6 -s 16 -r 10

./run -t 8 -n 6 -s 16 -r 10
[MPI] World size 1
[CONFIG] Number dim 6
[CONFIG] Shape 16
[Time] Avg time 1639.852300 ms
[Time] eigenpair J 46.879700 ms
Proportion 0.0285878
[Time] create J 1590.469900 ms
Proportion 0.969886

If you are interested our work, please find the detailed in paper

If you use HOSCF in your research, please cite our paper:

@misc{xiao2024hoscf,
      title={HOSCF: Efficient decoupling algorithms for finding the best rank-one approximation of higher-order tensors}, 
      author={Chuanfu Xiao and Zeyu Li and Chao Yang},
      year={2024},
      eprint={2403.01778},
      archivePrefix={arXiv},
      primaryClass={math.NA}
}