Fast algorithm for overcomplete order-3 tensor decomposition
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We develop the first fast spectral algorithm to decompose a random third-order tensor over R^d 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.
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