# Fast and robust tensor decomposition with applications to dictionary learning

with . COLT 2017.

## abstract

We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy.

Our algorithms can decompose a 4-tensor with $$n$$-dimensional orthonormal components in the presence of error with constant spectral norm (when viewed as an $$n^2$$-by-$$n^2$$ matrix). The running time is $$n^5$$ which is close to linear in the input size $$n^4$$.

We also obtain algorithms with similar running time to learn sparsely-used orthogonal dictionaries even when feature representations have constant relative sparsity and non-independent coordinates.

The only previous polynomial-time algorithms to solve these problem are based on solving large semidefinite programs. In contrast, our algorithms are easy to implement directly and are based on spectral projections and tensor-mode rearrangements.

Or work is inspired by recent of Hopkins, Schramm, Shi, and Steurer (STOC’16) that shows how fast spectral algorithms can achieve the guarantees of SOS for average-case problems. In this work, we introduce general techniques to capture the guarantees of SOS for worst-case problems.

## keywords

sum-of-squares method, tensor computations, eigenvalues, machine learning, semidefinite programming.