Coupled Tensor-Matrix Recovery via Proximal Alternating Linearized Minimization, with an Application to Workforce Skill and Small-Business Health Estimation
2026-07-11 • Computers and Society
Computers and Society
AI summaryⓘ
The authors study how to recover a low-rank tensor and a low-rank matrix that share one dimension, from limited and noisy data. They use a mathematical tool called the nuclear norm on the tensor's mode-1 unfolding to connect it with the matrix, and introduce a learned operator to link them. They prove that their algorithm converges and show when recovery is possible, including cases where only one of the objects is observed. They test their methods on synthetic data and propose application ideas, but have not validated the approach on real-world deployed data.
low-rank tensorlow-rank matrixnuclear normmode-1 unfoldingproximal operatorPALM algorithmsample complexitymatrix completionnonconvex optimizationcoupling operator
Authors
Analee Miranda
Abstract
We study recovery of a low-rank tensor $\mathcal{T}$ and a low-rank matrix $M$ from sparse, noisy observations. $\mathcal{T}$ and $M$ share one mode. We relax tensor rank using the nuclear norm of the mode-1 unfolding. This unfolding carries the coupling. It also has an exact proximal operator. We couple $\mathcal{T}$ and $M$ through a learned linear operator $G$. We prove a minimizer exists for the ridge-stabilized penalized objective. We prove that a proximal alternating linearized minimization (PALM) scheme converges to a critical point, for the algorithm as implemented, by verifying the hypotheses of a known nonconvex block-coordinate convergence theorem against our objective and identifying which conditions come from this problem's structure. For the matrix-only sub-problem, we state a proven sampling bound from matrix completion theory. For the coupled problem, we prove a sample-complexity result for a sequential sub-case: a separately-known coupling operator recovers $M$ from $\mathcal{T}$'s recovery accuracy alone, with no observations of $M$ needed. For the fully joint, alternately-estimated case, we state a conjecture and test it empirically, including a low-density regime where coupling does not help. We report multi-seed synthetic experiments with mean and standard deviation across sampling densities, an asymmetric-density experiment, and convergence curves, and we explain why recovery error stays high at low density. We apply the framework to workforce-skill and small-business-health estimation. Every application-specific choice is a proposed design, not a validated result; we have not run the framework on deployed data.