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# Fast algorithm for overcomplete order-3 tensor decomposition

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:3741-3799, 2022.

#### Abstract

We develop the first fast spectral algorithm to decompose a random third-order tensor over of rank up to $$O(d^{3/2}/polylog(d))$$. Our algorithm only involves simple linear algebra operations and can recover all components in time $$O(d^{6.05})$$ under the current matrix multiplication time. Prior to this work, comparable guarantees could only be achieved via sum-of-squares [Ma, Shi, Steurer 2016]. In contrast, fast algorithms [Hopkins, Schramm, Shi, Steurer 2016] could only decompose tensors of rank at most $$O(d^{4/3}/polylog(d))$$. Our algorithmic result rests on two key ingredients. A clean lifting of the third-order tensor to a sixth-order tensor, which can be expressed in the language of tensor networks. A careful decomposition of the tensor network into a sequence of rectangular matrix multiplications, which allows us to have a fast implementation of the algorithm.