Overcomplete tensor decomposition via convex optimization

Qiuwei Li, Ashley Prater, Lixin Shen, Gongguo Tang

Abstract: This work develops theories and computational methods for overcomplete, non-orthogonal tensor decomposition using convex optimization. Under an incoherence condition of the rank-one factors, we show that one can retrieve tensor decomposition by solving a convex, infinite-dimensional analog of ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> minimization on the space of measures. The optimal value of this optimization defines the tensor nuclear norm. Two computational schemes are proposed to solve the infinite-dimensional optimization: semidefinite programs based on sum-of-squares relaxations and nonlinear programs that are an exact reformulation of the tensor nuclear norm. The latter exhibits superior performance compared with the state-of-the-art tensor decomposition methods.