A Super-Resolution Framework for Tensor Decomposition

Abstract

This work considers a super-resolution framework for overcomplete tensor decomposition. Specifically, we view tensor decomposition as a super-resolution problem of recovering a sum of Dirac measures on the sphere and solve it by minimizing a continuous analog of the ${\ell }_1$ norm on the space of measures. The optimal value of this optimization defines the tensor nuclear norm. Similar to the separation condition in the super-resolution problem, by explicitly constructing a dual certificate, we develop incoherence conditions of the tensor factors so that they form the unique optimal solution of the continuous analog of ${\ell }_1$ norm minimization. Remarkably, the derived incoherence conditions are satisfied with high probability by random tensor factors uniformly distributed on the sphere, implying global identifiability of random tensor factors.

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Cited by in corpus (2)

• The Nonconvex Geometry of Linear Inverse Problems
• Exact nuclear norm, completion and decomposition for random overcomplete tensors via degree-4 SOS