  Low-rank approximations of original samples are playing more and more an
important role in many recently proposed mathematical models from data science.
A natural and initial requirement is that these representations inherit
original structures or properties. With this aim, we propose a new
multi-symplectic method based on the Lanzcos bidiagonalization to compute the
partial singular triplets of JRS-symmetric matrices. These singular triplets
can be used to reconstruct optimal low-rank approximations while preserving the
intrinsic multi-symmetry. The augmented Ritz and harmonic Ritz vectors are used
to perform implicit restarting to obtain a satisfactory bidiagonal matrix for
calculating the $k$ largest or smallest singular triplets, respectively. We
also apply the new multi-symplectic Lanczos algorithms to color face
recognition and color video compressing and reconstruction. Numerical
experiments indicate their superiority over the state-of-the-art algorithms.
