The Multi-Dimensional Decomposition with Constraints

Abstract

We search for the best fit in Frobenius norm of $A\in {\mathbb{C}}^{m\times n}$ by a matrix product $B{C}^{\ast }$, where $B\in {\mathbb{C}}^{m\times r}$ and $C\in {\mathbb{C}}^{n\times r}$, $r\le m$ so $B=\{b_{ij}\}$, ($i=1,\dots ,m$,~ $j=1,\dots ,r$) definite by some unknown parameters ${\sigma }_1,\dots ,{\sigma }_k$, $k<<mr$ and all partial derivatives of $\frac{\delta b_{ij}}{\delta {\sigma }_l}$ are definite, bounded and can be computed analytically. We show that this problem transforms to a new minimization problem with only $k$ unknowns, with analytical computation of gradient of minimized function by all $\sigma$. The complexity of computation of gradient is only 4 times bigger than the complexity of computation of the function, and this new algorithm needs only $3mr$ additional memory. We apply this approach for solution of the three-way decomposition problem and obtain good results of convergence of Broyden algorithm.

References in corpus (1)

• The ELEGANT NMR Spectrometer