## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, October 04, 2022, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Jingqiu Ding

## Fast algorithm for random overcomplete order-3 tensor decomposition

We develop the first fast spectral algorithm to decompose a random third-order tensor over Rd of rank up to O(d3/2/polylog(d)). Our algorithm only involves simple linear algebra operations and can recover all components in time O(d6.05) 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(d4/3/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. This is joint work with Tommaso D’Orsi, Chih-Hung Liu, Stefan Tiegel and David Steurer.

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