Many applications in data science and scientific computing involve
large-scale datasets that are expensive to store and compute with, but can be
efficiently compressed and stored in an appropriate tensor format. In recent
years, randomized matrix methods have been used to efficiently and accurately
compute low-rank matrix decompositions. Motivated by this success, we focus on
developing randomized algorithms for tensor decompositions in the Tucker
representation. Specifically, we present randomized versions of two well-known
compression algorithms, namely, HOSVD and STHOSVD. We present a detailed
probabilistic analysis of the error of the randomized tensor algorithms. We
also develop variants of these algorithms that tackle specific challenges posed
by large-scale datasets. The first variant adaptively finds a low-rank
representation satisfying a given tolerance and it is beneficial when the
target-rank is not known in advance. The second variant preserves the structure
of the original tensor, and is beneficial for large sparse tensors that are
difficult to load in memory. We consider several different datasets for our
numerical experiments: synthetic test tensors and realistic applications such
as the compression of facial image samples in the Olivetti database and word
counts in the Enron email dataset.

本文研究基于随机算法的张量分解方法在 Tucker 表达式下的应用，提出并分析了 HOSVD 和 STHOSVD 的随机算法，并针对大规模数据集提出了适用于不同需求的改进方案。