What is the meaning of "best-rank"?

Answer 1

it is the #1 on the list. it is ahead of send tier.

Answer 2

The best rank (R1, R2, R3,.....Rn) approximation problem is a multilinear generalization of the best rank-R approximation problem for matrices: how can a given tensor (``multidimensional matrix'') be approximated, in an optimal least-squares sense, by a tensor with pre-specified column rank, row rank, etc.

For matrices, the best rank-R approximation is obtained by setting the smallest singular values equal to zero, while keeping the R largest ones. Truncation of the Higher-Order Singular Value Decomposition (HOSVD) may yield a good tensor approximation, but this approximation is generically suboptimal. In order to improve the fit, we have investigated gradient-based optimization over the Stiefel manifold of column-wise orthogonal matrices. We have also devised an Alternating Least-Squares (ALS) algorithm, in which the modes are updated two by two; this algorithm is an order of magnitude faster than the techniques that are currently available. For a further improvement of the efficiency, we have proposed a preprocessing step, based on a higher-order generalization of the Hessenberg Decomposition