Joint recovery conjecture for coupled tensor-matrix estimation

Let $\mathcal{T}$ be a tensor whose mode-1 unfolding $\mathcal{T}_{(1)}$ is observed on $\Omega_{\mathcal{T}}$, let $M$ be a matrix observed on $\Omega_M$, and let $G$ be the unknown coupling matrix in the fully coupled recovery problem. Assume that $G$ is well-conditioned, and let $C$ denote the constant from the matrix-completion result for recovering $\mathcal{T}_{(1)}$. Joint recovery conjecture. There exists $C' < C$ such that, whenever $\Omega_{\mathcal{T}}$ satisfies the density condition of that matrix-completion result for $\mathcal{T}_{(1)}$, the matrix $M$ is recoverable from observations satisfying

$$|\Omega_M| \geq C' \mu r n_1 \log^2(n_1).$$

This bound is strictly smaller than the uncoupled requirement, even though $G$ is estimated jointly rather than treated as known. The conjecture concerns the fully coupled problem with nuclear-norm regularization, linear coupling, an ordinal constraint, and an unknown coupling matrix; the coupling term is expected to transfer information from the well-sampled tensor to $M$.