Efficient Tensor Decomposition

TL;DR

This paper explores efficient algorithms for tensor decomposition, focusing on overcoming NP-hardness through provable guarantees and advanced analysis frameworks like smoothed analysis.

Contribution

It introduces novel algorithms for tensor decomposition that are efficient and come with theoretical guarantees under mild assumptions.

Findings

Algorithms achieve polynomial-time performance under certain conditions

Provable guarantees improve reliability of tensor decomposition methods

Smoothed analysis offers insights into practical efficiency

Abstract

This chapter studies the problem of decomposing a tensor into a sum of constituent rank one tensors. While tensor decompositions are very useful in designing learning algorithms and data analysis, they are NP-hard in the worst-case. We will see how to design efficient algorithms with provable guarantees under mild assumptions, and using beyond worst-case frameworks like smoothed analysis.