  Transform Invariant Low-Rank Textures, referred to as TILT, can accurately
and robustly extract textural or geometric information in a 3D from
user-specified windows in 2D in spite of significant corruptions and warping.
It was discovered that the task can be characterized, both theoretically and
numerically, by solving a sequence of matrix nuclear-norm and $\ell_1$-norm
involved convex minimization problems. For solving this problem, the direct
extension of Alternating Direction Method of Multipliers (ADMM) in an usual
Gauss-Seidel manner often performs numerically well in practice but there is no
theoretical guarantee on its convergence. In this paper, we resolve this
dilemma by using the novel symmetric Gauss-Seidel (sGS) based ADMM developed by
Li, Sun \& Toh (Math. Prog. 2016). The sGS-ADMM is guaranteed to converge and
we shall demonstrate in this paper that it is also practically efficient than
the directly extended ADMM. When the sGS technique is applied to this
particular problem, we show that only one variable needs to be re-updated, and
this updating hardly imposes any excessive computational cost. The sGS
decomposition theorem of Li, Sun \& Toh (arXiv: 1703.06629) establishes the
equivalent between sGS-ADMM and the classical ADMM with an additional
semi-proximal term, so the convergence result is followed directly. Extensive
experiments illustrate that the sGS-ADMM and its generalized variant have
superior numerical efficiency over the directly extended ADMM.
