Multilinear Decomposition: Theory and Applications

Assume $\displaystyle \{a_{i_1,\dots,i_N}\}_{i_1,\dots,i_N=1}^{m_1, \dots, m_N} \in C^{m_1, \dots, m_N}$ is given ($N>2$), we are searching for

• $r \in N$,
• $\alpha_1, \dots, \alpha_r > 0$, \item
• $\displaystyle \forall p=1, \dots, N: B^{(p)} = \{ b_{i_p l} \}_{i_p=1, l=1}^{m_p,r}, ~~ \sum_{i_p}^{m_p} |b_{i_p l}|^2 = 1$,

that $$ \forall i_1,\dots,i_N: ~~ a_{i_1,\dots,i_N} \simeq \sum_{l=1}^r \alpha_l b_{i_1 l}^{(1)} \dots b_{i_N l}^{(N)}. $$ This approximation is called the multilinear decomposition and is the generalization of a singular value decomposition ($N=2$) for higher order arrays. The main goal of this talk is to give a short introduction to the theory of the multilinear decomposition, and to emphasize the difference between $N=2$ and $N>2$. Nowadays many practical applications explore multilinear approximation: we can find it in the compression of matrices, data approximation/compression in signal processing, fast algorithms for the matrix-by-matrix multiplication. During the talk several numerical algorithms developed for the multilinear decomposition (http://www.ilghiz.com/) will be demonstrated and we will see how those algorithms work in real time and present their parallel implementations. Those algorithms are stable and robust, it helps us to produce the last multilinear world record of four dimensional ($N=4$) data approximation with 350 skeletons ($r=350$) obtained on industrial data

Tugarinov V., Kay L., Ibraghimov I., Orekhov V., High-Resolution Four-Dimensional 1H-13C NOE Spectroscopy Using Methyl-TROSY, Sparse Data Acquisition, and Multilinear Decomposition. J. Am. Chem. Soc. 2005; 127: 2767--2775.