Asymptotic Log-Det Sum-of-Ranks Minimization via Tensor (Alternating) Iteratively Reweighted Least Squares

Abstract: Affine sum-of-ranks minimization (ASRM) generalizes the affine rank minimization (ARM) problem from matrices to tensors. Here, the interest lies in the ranks of a family $\mathcal{K}$ of different matricizations. Transferring our priorly discussed results on asymptotic log-det rank minimization, we show that iteratively reweighted least squares with weight strength $p = 0$ remains a, theoretically and practically, particularly viable method denoted as $\mathrm{IRLS}$-$0\mathcal{K}$. As in the matrix case, we prove global convergence of asymptotic minimizers of the log-det sum-of-ranks function to desired solutions. Further, we show local convergence of $\mathrm{IRLS}$-$0\mathcal{K}$ in dependence of the rate of decline of the therein appearing regularization parameter $\gamma \searrow 0$. For hierarchical families $\mathcal{K}$, we show how an alternating version ($\mathrm{AIRLS}$-$0\mathcal{K}$, related to prior work under the name $\mathrm{SALSA}$) can be evaluated solely through tensor tree network based operations. The method can thereby be applied to high dimensions through the avoidance of exponential computational complexity. Further, the otherwise crucial rank adaption process becomes essentially superfluous even for completion problems. In numerical experiments, we show that the therefor required subspace restrictions and relaxation of the affine constraint cause only a marginal loss of approximation quality. On the other hand, we demonstrate that $\mathrm{IRLS}$-$0\mathcal{K}$ allows to observe the theoretical phase transition also for generic tensor recoverability in practice. Concludingly, we apply $\mathrm{AIRLS}$-$0\mathcal{K}$ to larger scale problems.