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arXiv · 1402.5180

Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank-$1$ Updates

Animashree Anandkumar·Rong Ge·Majid Janzamin
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Abstract

In this paper, we provide local and global convergence guarantees for recovering CP (Candecomp/Parafac) tensor decomposition. The main step of the proposed algorithm is a simple alternating rank-$1$ update which is the alternating version of the tensor power iteration adapted for asymmetric tensors. Local convergence guarantees are established for third order tensors of rank $k$ in $d$ dimensions, when $k=o \bigl( d^{1.5} \bigr)$ and the tensor components are incoherent. Thus, we can recover overcomplete tensor decomposition. We also strengthen the results to global convergence guarantees under stricter rank condition $k \le \beta d$ (for arbitrary constant $\beta > 1$) through a simple initialization procedure where the algorithm is initialized by top singular vectors of random tensor slices. Furthermore, the approximate local convergence guarantees for $p$-th order tensors are also provided under rank condition $k=o \bigl( d^{p/2} \bigr)$. The guarantees also include tight perturbation analysis given noisy tensor.

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BibTeXRIS

Animashree Anandkumar, Rong Ge, Majid Janzamin. 2014-02-21. Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank-$1$ Updates. https://arxiv.org/abs/1402.5180

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