通过近端交替线性化最小化实现耦合张量 - 矩阵恢复及其在劳动力技能和小企业健康估计中的应用
Coupled Tensor-Matrix Recovery via Proximal Alternating Linearized Minimization, with an Application to Workforce Skill and Small-Business Health Estimation
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中文总结 AI 辅助
研究从稀疏噪声观测中恢复共享一个模式的低秩张量和矩阵，通过近端交替线性化最小化方法，证明相关收敛性并给出样本复杂度结果，进行多实验验证，还应用于劳动力技能和小企业健康估计。
中文摘要 AI 辅助
我们研究从稀疏、有噪声的观测中恢复低秩张量$\mathcal{T}$和低秩矩阵$M$，二者共享一个模式。我们使用模式 - 1展开的核范数来放宽张量秩，该展开携带耦合且有精确近端算子。通过学习的线性算子$G$将$\mathcal{T}$和$M$耦合。证明了岭稳定惩罚目标存在极小值，近端交替线性化最小化（PALM）方案收敛到临界点。给出矩阵子问题的采样界，证明耦合问题的样本复杂度结果，提出猜想并实证检验。报告了多种子合成实验及结果，并将框架应用于劳动力技能和小企业健康估计。
英文摘要
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.
