This paper develops fast and efficient algorithms for computing Tucker decomposition with a given multilinear rank. By combining random projection and the power scheme, we propose two efficient randomized versions for the truncated high-order singular value decomposition (T-HOSVD) and the sequentially T-HOSVD (ST-HOSVD), which are two common algorithms for approximating Tucker decomposition. To reduce the complexities of these two algorithms, fast and efficient algorithms are designed by combining two algorithms and approximate matrix multiplication. The theoretical results are also achieved based on the bounds of singular values of standard Gaussian matrices and the theoretical results for approximate matrix multiplication. Finally, the efficiency of these algorithms are illustrated via some test tensors from synthetic and real datasets.

翻译：本文提出了在给定多线性秩条件下计算Tucker分解的快速高效算法。通过结合随机投影与幂迭代方案，我们为截断高阶奇异值分解（T-HOSVD）和顺序截断高阶奇异值分解（ST-HOSVD）两种常用的Tucker分解近似算法，设计了两种高效的随机化版本。为降低这两种算法的复杂度，我们通过融合两种算法与近似矩阵乘法，设计了快速高效的改进算法。理论分析基于标准高斯矩阵奇异值的界以及近似矩阵乘法的理论结果。最后，通过合成张量与真实数据集中的测试张量验证了这些算法的有效性。