Kavli Affiliate: Eric Miller
| First 5 Authors: Fan Tian, Misha E. Kilmer, Eric Miller, Abani Patra,
| Summary:
Given tensors $boldsymbol{mathscr{A}}, boldsymbol{mathscr{B}},
boldsymbol{mathscr{C}}$ of size $m times 1 times n$, $m times p times 1$,
and $1times p times n$, 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.
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