Uniqueness Guarantee of Solutions of Tensor Tubal-Rank Minimization Problem

This letter considers the recovery of a low-tubal-rank tensor from incomplete linear observations. It is shown that the unknown tensor <inline-formula><tex-math notation="LaTeX">$\boldsymbol{\mathcal {Z}}\in \mathbb {R}^{n_{1} \times n_{2} \times n_{3}}$</tex-math></inline-formula> of tubal-rank <inline-formula><tex-math notation="LaTeX">$r$</tex-math></inline-formula> can be reconstructed as a unique solution of a tractable method — tensor nuclear norm (TNN) minimization, provided that the number of Gaussian observations <inline-formula><tex-math notation="LaTeX">$m\geq 3r(n_{1} + n_{2} - r)n_{3}+1$</tex-math></inline-formula>. In this work, we examine the fundamental question of the minimal number of linear observations needed to reconstruct the tensor <inline-formula><tex-math notation="LaTeX">$\boldsymbol{\mathcal {Z}}$</tex-math></inline-formula> from these observations, regardless of the practicality of the reconstruction scheme. Consequently, we provide two benchmark results so that different reconstruction schemes including TNN minimization can be compared to each other. Specifically, we conclude that <inline-formula><tex-math notation="LaTeX">$m\geq 2r(n_{1} + n_{2} - 2r)n_{3}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$m\geq r(n_{1} + n_{2} - r)n_{3}+1$</tex-math></inline-formula> Gaussian observations are necessary and sufficient to guarantee uniform recovery and nonuniform recovery using tensor tubal-rank minimization method, respectively.

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