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

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.

• The paper introduces a framework for joint recovery of low-rank tensor and matrix data using PALM, leveraging a shared mode coupling to overcome sparse observations.
• It provides both theoretical guarantees for sequential coupling and empirical evidence, showing marked improvements over uncoupled methods.
• Empirical tests, including synthetic experiments, validate the method's effectiveness, with recovery errors reduced by up to an order of magnitude at higher sampling densities.

Coupled Tensor–Matrix Recovery via PALM: Theory, Algorithms, and Applications

Problem Formulation and Motivation

The paper introduces a framework for the joint recovery of a low-rank tensor $\mathcal{T}$ and a low-rank matrix $M$ from highly sparse and noisy observations, addressing the case where $\mathcal{T}$ and $M$ are connected by a shared mode and an explicit linear coupling operator. The motivation arises from scenarios such as workforce skill estimation and small-business health mapping, where multiple, imperfect sources encode complementary information over overlapping populations. Unlike classical matrix or tensor completion, this setting leverages mode-level coupling to propagate information into sparsely sampled regimes, exceeding the recovery capability of uncoupled approaches.

The nuclear norm of the mode-1 unfolding of the tensor acts as a convex surrogate for its non-unique rank, ensuring computational tractability and enabling an exact closed-form proximal operator via SVD. Coupling is implemented as $\mathcal{T}_{(1)} \approx M G^\top$ with a learned $G$, generalizing beyond factor-sharing schemes and enabling statistical transfer by direct linear mapping.

Convex Objective, Existence, and Optimization Algorithm

The core objective is a sum of squared data fidelity terms, a coupling penalty, nuclear-norm regularization for both $\mathcal{T}$ and $M$, and Tikhonov (ridge) regularization for $G$. Formally, the problem is

$$\min_{\mathcal{T},M,G} \frac{1}{2}\|P_{\Omega_\mathcal{T}}(\mathcal{T}) - y_\mathcal{T}\|_F^2 + \frac{1}{2}\|P_{\Omega_M}(M) - y_M\|_F^2 + \frac{\lambda_C}{2}\|\mathcal{T}_{(1)} - MG^\top\|_F^2 + \frac{\delta}{2}\|G\|_F^2 + \lambda_S \|\mathcal{T}\|_* + \lambda_R \|M\|_*,$$

where the nuclear norms and the linear coupling enforce the desired low-rank and information-sharing structure.

An existence theorem guarantees the well-posedness of the problem in finite dimensions, leveraging coercivity induced by the regularizers and the structure of the objective.

Optimization is performed with PALM (Proximal Alternating Linearized Minimization). Each variable block ($M$0, $M$1, $M$2) is updated via:

• Prox-gradient descent on the smooth part and an exact singular value thresholding proximal step for the nuclear norms,
• Ridge-regularized least squares for the $M$3-block.

A convergence theorem (built on [bolte2014]) establishes that the iterates converge to a critical point of the objective under standard block-wise stepsize rules when all approximations are exact.

Figure 1: Objective value vs.\ iteration, $M$4. The monotonic decrease confirms proper PALM behavior and convergence.

Recovery Guarantees and Sample Complexity

Theoretical sample-complexity results are dissected into three levels:

1. Matrix-Only Case (Uncoupled): Classical matrix completion theory applies, with recovery at $M$5 density $M$6 for bounded coherence.
2. Sequential Coupling (Provable Case, Theorem 2): If the coupling $M$7 is known a priori, $M$8 can be precisely recovered from only observations of $M$9, using a plug-in estimator. The error bound is proportional to the recovery error for $\mathcal{T}$0, scaled by the inverse minimal singular value of $\mathcal{T}$1, and holds even if $\mathcal{T}$2.
3. Fully Joint (Empirical/Open): When $\mathcal{T}$3 is not known and is estimated alternately with $\mathcal{T}$4, the coupling provides strict improvements at intermediate densities, as shown empirically. Formal guarantees for this challenging statistical regime remain conjectural.

This critical distinction emphasizes a substantial guarantee for sequential coupling and quantifies the empirical benefit and its limitations for the full joint recovery case.

Numerical Results: Empirical Validation

The paper presents synthetic experiments evaluating recovery as a function of sample density, model scale, and coupling strength. Strong numerical evidence is provided for the value of coupling, especially in settings with highly asymmetric sample densities across modalities.

• Coupled recovery gives lower errors for $\mathcal{T}$5 and $\mathcal{T}$6 as soon as moderate density is achieved on one side ($\mathcal{T}$7), exceeding uncoupled baselines by up to an order of magnitude at high densities.
• In extremely sparse regimes ($\mathcal{T}$8), coupling provides little benefit, as neither component contains enough information to propagate.
• For the provable sequential estimator with known $\mathcal{T}$9, the empirical error saturates at well below the (guaranteed) upper bound, and $M$0 is accurately reconstructed using no direct observations given sufficient observed $M$1.
• The scheme outperforms a CMTF (coupled matrix-tensor factorization) CP-factor-sharing baseline, demonstrating resilience to initialization and rank selection issues typical in CP/ALS frameworks.

  Figure 2: Relative recovery error vs.\ sampling density; the coupled method outperforms the uncoupled baseline for both $M$2 and $M$3, especially as density increases.

An anomaly is observed in the scale-up experiment: recovery error decreases with growing $M$4 at fixed per-entry density, which contradicts classical worst-case sample-complexity scaling in the matrix completion literature. This phenomenon is attributed to averaging across many low-dimensional rows, but remains an open topic for theoretical refinement.

Extensions, Schema Evolution, and Limitations

The formulation admits stronger regularizers (e.g., overlapped nuclear norm across all modes), but the convergence guarantee applies only to the mode-1-norm due to lack of closed-form proximal steps for the other cases.

A procedural mechanism for governance-coupled schema evolution is proposed, enabling the dynamic and auditable extension of the index sets as new categories or dimensions emerge—a critical component for real-world, evolving systems.

Limitations are transparent and significant: sample complexity for the full joint estimator is conjectural, not proved; only small- to moderate-scale synthetic experiments are presented; and application-specific results are labeled as design proposals, not validated findings. Validation on real deployment-scale data and analysis of the statistical properties of alternately estimated coupling remain open.

Implications and Outlook

This work formalizes and demonstrates a practical approach for coupled recovery in settings where two heterogeneous modalities share structure but differ in sample coverage. The theoretical advances—most notably, the rigorous guarantee for sequential coupling—clarify the boundaries of what can be definitively achieved with limited data. The differentiation between provable and conjectural phases is handled with technical care and empirical rigor.

Practically, the algorithmic choices (exact proximal steps, closed-form ridge solves) yield efficient and stable implementations. The explicit governance component anticipates real-world system needs, in which reproducibility and schema evolution are as central as optimization.

Potential future developments include:

• Extending convergence and recovery guarantees to overlapped nuclear norm settings and more general coupling functions,
• Deriving tight finite-sample rates for the alternately estimated coupling scenario,
• Application to real datasets in labor market or small business analytics, where resolution at sparse margins is a pivotal concern,
• Integration with side information or semi-supervised signals as in inductive matrix completion.

Conclusion

The paper delivers a precise characterization of coupled tensor–matrix recovery and its algorithmic realization via PALM, together with compelling evidence for joint statistical strength transfer in underobserved settings. The boundary between what is strictly guaranteed and what is empirically supported is clearly articulated. This work opens directions for both theoretical generalization and application to increasingly heterogeneous, evolving real-world systems.

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