Robust polynomial time tensor decomposition

Tensor decomposition has recently become an invaluable algorithmic primitive. It has seen much use in new algorithms with provable guarantees for fundamental statistics and machine learning problems. In these settings, some low-rank k-tensor A ∑r i 1 a ⊗k i which wewould like to decompose into components a1, . . . , ar ∈ ’n is often not directly accessible. This could happen for many reasons; a common one is that A …X⊗k for some random variable X, and estimating A to high precision may require too many independent samples from X. In this lecture we will dig in to algorithms for robust tensor decomposition—that is, how to accomplish tensor decomposition efficiently in the presence of errors. Wewill focus on orthogonal tensor decompositionwhere components a1, . . . , ar ∈ ’n of the tensor A ∑r i 1 a ⊗k i to be decomposed are orthogonal unit vectors. Tensor decomposition is already both algorithmically nontrivial and quite useful in this setting—the orthogonal setting is good enough to give the best known algorithms for Gaussianmixtures, some kinds of dictionary learning, and the stochastic blockmodel. As we saw before, viawhitening if the covariance matrix ∑n i 1 ai a > i is known for non-orthogonl but linearly independent vectors a1, . . . , an then decomposing the tensor ∑n i 1 a ⊗3 i reduces to orthogonal decomposition.