Given tensors $\boldsymbol{\mathscr{A}}, \boldsymbol{\mathscr{B}}, \boldsymbol{\mathscr{C}}$ of size $m \times 1 \times n$, $m \times p \times 1$, and $1\times p \times n$, respectively, their Bhattacharya-Mesner (BM) product will result in a third order tensor of dimension $m \times p \times n$ and BM-rank of 1 (Mesner and Bhattacharya, 1990). Thus, if a third-order tensor can be written as a sum of a small number of such BM-rank 1 terms, this BM-decomposition (BMD) offers an implicitly compressed representation of the tensor. Therefore, in this paper, we give a generative model which illustrates that spatio-temporal video data can be expected to have low BM-rank. Then, we discuss non-uniqueness properties of the BMD and give an improved bound on the BM-rank of a third-order tensor. We present and study properties of an iterative algorithm for computing an approximate BMD, including convergence behavior and appropriate choices for starting guesses that allow for the decomposition of our spatial-temporal data into stationary and non-stationary components. Several numerical experiments show the impressive ability of our BMD algorithm to extract important temporal information from video data while simultaneously compressing the data. In particular, we compare our approach with dynamic mode decomposition (DMD): first, we show how the matrix-based DMD can be reinterpreted in tensor BMP form, then we explain why the low BM-rank decomposition can produce results with superior compression properties while simultaneously providing better separation of stationary and non-stationary features in the data. We conclude with a comparison of our low BM-rank decomposition to two other tensor decompositions, CP and the t-SVDM.

翻译：给定大小为$m \times 1 \times n$、$m \times p \times 1$和$1\times p \times n$的张量$\boldsymbol{\mathscr{A}}, \boldsymbol{\mathscr{B}}, \boldsymbol{\mathscr{C}}$，其Bhattacharya-Mesner (BM)乘积将产生维度为$m \times p \times n$、BM秩为1的三阶张量（Mesner和Bhattacharya, 1990）。因此，若一个三阶张量可表示为少量此类BM秩1项之和，则这种BM分解（BMD）提供了张量的隐式压缩表示。为此，本文给出一个生成模型，说明时空视频数据可预期具有低BM秩。随后，我们讨论BMD的非唯一性性质，并给出三阶张量BM秩的改进边界。我们提出并研究了一种用于求解近似BMD的迭代算法的性质，包括收敛行为以及允许将时空数据分解为平稳与非平稳分量的初始猜测的合理选择。多项数值实验表明，我们的BMD算法在压缩数据的同时，能出色地提取视频中的重要时间信息。特别地，我们将该方法与动态模式分解（DMD）进行对比：首先展示基于矩阵的DMD如何以张量BMP形式重新解释，然后解释为何低BM秩分解能在生成具有更优压缩性能的结果的同时，更好地分离数据中的平稳与非平稳特征。最后，我们将低BM秩分解与另外两种张量分解（CP和t-SVDM）进行比较。