Coupled Tensor Matrix Recovery via Proximal Alternating Linearized Minimization with an Application to Workforce Skill and Small Business Health Estimation

Jul 16, 2026

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Recovery of a low-rank tensor T and a low-rank matrix M is studied from sparse, noisy observations when the two objects share one mode. Tensor rank is relaxed using the nuclear norm of the mode-1 unfolding, whose proximal operator is given exactly by singular value thresholding and preserves the coupling. A learned linear operator G links T and M, yielding existence of a minimizer for a ridge-stabilized objective and PALM convergence to a critical point. Sampling and sample-complexity results are provided for key sub-cases, followed by synthetic tests and an application to workforce-skill and small-business-health estimation.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages15

Reading Time38 min

Introduction

Contributions

Problem Formulation

Rank and its convex relaxation

Experiments and Application

Synthetic experiments

Application: workforce skill and small-business health estimation

arXiv :2607 . 10163v1 [math .OC] 11 Jul 2026 Coupled Tensor–Matrix Recovery via Proximal Alternating Linearized Minimization, with an Application to Workforce Skill and Small-Business Health Estimation Analee Miranda Department of Mathematics, Pace University New York, NY 10038 July 14, 2026 Email: [amiranda2@pace.edu](amiranda2@pace.edu) Abstract We study recovery of a low-rank tensor T and a low-rank matrix M from sparse, noisy observations. 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 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 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. Subject classification: math.OC (Optimization and Control); cross-list math.NA (Numerical Analysis), cs.CY (Computers and Society) 1 Introduction Recovering a partially observed multi-way array from sparse, noisy data is well studied in matrix and tensor completion (Cand`es and Recht, 2012; Cand`es and Tao, 2010; Liu et al., 2012; Kolda and Bader, 2009) . We study a variant with two arrays: a tensor T and a matrix M. They are observed independently. They share structure through one mode. We ask how to recover both jointly, so that the shared mode helps recovery of whichever object is more sparsely sampled. The closest prior formulation is coupled matrix–tensor factorization (CMTF) (Acar et al., 2011), which couples a tensor and matrices through shared CP factor matrices. Our formulation differs in three ways. We relax rank using the nuclear norm of the mode-1 unfolding, not a fixed CP rank; this extends to the overlapped nuclear norm across all modes, at the cost of an open convergence guarantee. We couple the two objects through a learned linear operator, not a shared factor. We impose an ordinal constraint on one mode of the matrix. We motivate and instantiate the framework with one running application: joint estimation of a workforce-skill landscape and a small-business-health landscape. Local workforce boards need to see both. AI is reshaping tasks within job functions, at different rates across industries. The same industries contain small businesses whose health is unevenly reported: solo operators and micro businesses file less often than mid-market firms. A board with only skill data cannot seethe business impact. A board with only business filings cannot see which skill shifts are driving it. This paper couples the two. Section 2 gives the specific tensor and matrix structure. Every application-specific claim here is a proposed design, not a validated finding; we have not run the framework on deployed data. 1.1 Contributions 1. A convex formulation using the nuclear norm of

FAQs

What problem does the paper address?

It addresses recovering a low-rank tensor T and a low-rank matrix M from sparse, noisy observations when they share one mode and can be observed independently.

How is tensor rank handled in the proposed approach?

Tensor mode-1 rank is relaxed using the nuclear norm of the mode-1 unfolding, enabling an exact proximal operator via singular value thresholding.

What algorithm is proposed and what convergence is shown?

A proximal alternating linearized minimization (PALM) scheme is analyzed, and convergence to a critical point is established by verifying conditions from a known nonconvex block-coordinate convergence theorem.

Jul 16, 2026

Upload by Oliver