  High-Order, High-Dimension, and Sparse Tensor (HOHDST) data originates from
real industrial applications, i.e., social networks, recommender systems,
bio-information, and traffic information. Sparse Tensor Decomposition (STD) can
project the HOHDST data into low-rank space. In this work, a novel method for
STD of Kruskal approximating the core tensor and stochastic strategy for
approximating the whole gradient is proposed which comprises of the following
two parts: (1) the matrization unfolding order of the Kruskal product for the
core tensor follows the multiplication order of the factor matrix and then the
proposed theorem can reduce the exponential computational overhead into linear
one; (2) stochastic strategy adopts one-step random sampling set, the volume of
which is much smaller than original one, to approximate the whole gradient.
Meanwhile, this method can guarantee the convergence and save the memory
overhead. Due to the compactness of the same order matrix multiplication and
parallel access from stochastic strategy, the speed of cuFastTucker can be
further reinforced by GPU. Furthermore, %because large-scale HOHDST data cannot
be accommodated in a single GPU, a data division and communication strategy of
cuFastTucker is proposed for data accommodation on Multi-GPU. cuFastTucker can
achieve the fastest speed and keep the same accuracy and much lower memory
overhead than the SOTA algorithms, e.g., P$-$Tucker, Vest, and SGD$\_$Tucker.
The code and partial datasets are publically available on
"https://github.com/ZixuanLi-China/FastTucker".
