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：给定尺寸分别为 $m \times 1 \times n$、$m \times p \times 1$ 和 $1\times p \times n$ 的张量 $\boldsymbol{\mathscr{A}}$、$\boldsymbol{\mathscr{B}}$、$\boldsymbol{\mathscr{C}}$，其Bhattacharya-Mesner（BM）乘积将生成一个维度为 $m \times p \times n$ 且BM秩为1的三阶张量（Mesner和Bhattacharya，1990）。因此，若三阶张量可表示为少量此类BM秩1项之和，则BM分解（BMD）可提供该张量的隐式压缩表示。本文首先提出一个生成模型，说明时空视频数据具有低BM秩的预期特性。随后，我们讨论BMD的非唯一性性质，并给出三阶张量BM秩的改进上界。我们提出并研究一种迭代算法的性质，该算法用于计算近似BMD，包括收敛行为以及能实现时空数据分解为平稳与非平稳分量的初始值选取策略。多组数值实验表明，我们的BMD算法在压缩视频数据的同时，能显著提取关键时序信息。特别地，我们将该方法与动态模态分解（DMD）进行对比：首先展示基于矩阵的DMD可重新表述为张量BMP形式，进而解释为何低BM秩分解能在产生更优压缩性能的同时，更有效地分离数据中的平稳与非平稳特征。最后，我们将低BM秩分解与其他两种张量分解方法（CP分解和t-SVDM分解）进行比较。