De-biased Noisy Matrix Completion and De-shrunken Low-Rank Factor Uncertainty Quantification

A unified statistical inference and uncertainty quantification methodology (introduced by Yuxin Chen, Jianqing Fan, Cong Ma, and Yuling Yan in 2019) for noisy low-rank matrix completion $M_{ij} = M_{ij}^\star + E_{ij}$, $(i,j)\in\Omega$, $E_{ij}\stackrel{\text{i.i.d.}}{\sim}\mathcal{N}(0,\sigma^2)$ under uniform random sampling with probability $p$. Given either the convex nuclear-norm regularized estimator $\boldsymbol{Z}^{\text{cvx}} = \arg\min_{\boldsymbol{Z}} \frac{1}{2}\|\mathcal{P}_\Omega(\boldsymbol{Z} - \boldsymbol{M})\|_F^2 + \lambda\|\boldsymbol{Z}\|_*$ or the nonconvex factored gradient descent estimator $(\boldsymbol{X}^{\text{ncvx}},\boldsymbol{Y}^{\text{ncvx}})$ minimizing $\frac{1}{2}\|\mathcal{P}_\Omega(\boldsymbol{X}\boldsymbol{Y}^\top - \boldsymbol{M})\|_F^2 + \frac{\lambda}{2}(\|\boldsymbol{X}\|_F^2 + \|\boldsymbol{Y}\|_F^2)$, the method compensates for regularization shrinkage without sample splitting via two nearly equivalent estimators (Lemma 3): (1) the rank-$r$ projected *de-biased matrix estimator* $\boldsymbol{M}^{\text{d}} = \mathcal{P}_{\text{rank-}r}[\boldsymbol{Z} - \frac{1}{p}\mathcal{P}_\Omega(\boldsymbol{Z} - \boldsymbol{M})]$ (Equation 3.7), and (2) the *de-shrunken low-rank factor estimators* $\boldsymbol{X}^{\text{d}} = \boldsymbol{X}(\boldsymbol{I}_r + \frac{\lambda}{p}(\boldsymbol{X}^\top\boldsymbol{X})^{-1})^{1/2}$ and $\boldsymbol{Y}^{\text{d}} = \boldsymbol{Y}(\boldsymbol{I}_r + \frac{\lambda}{p}(\boldsymbol{Y}^\top\boldsymbol{Y})^{-1})^{1/2}$ satisfying $\boldsymbol{M}^{\text{d}} \approx \boldsymbol{X}^{\text{d}}\boldsymbol{Y}^{\text{d}\top}$ (Equations 3.8–3.9). Theorem 1 proves that up to global orthogonal rotation $\boldsymbol{H}^{\text{d}}$, the rows of $\boldsymbol{X}^{\text{d}}\boldsymbol{H}^{\text{d}} - \boldsymbol{X}^\star$ and $\boldsymbol{Y}^{\text{d}}\boldsymbol{H}^{\text{d}} - \boldsymbol{Y}^\star$ admit non-asymptotic i.i.d. $\mathcal{N}(\boldsymbol{0}, \frac{\sigma^2}{p}(\boldsymbol{\Sigma}^\star)^{-1})$ decompositions matching the oracle Cramér–Rao lower bound (Lemma 1). Theorem 2 and Corollary 1 establish that $M_{ij}^{\text{d}} - M_{ij}^\star = g_{ij} + \Delta_{ij}$ with $g_{ij}\sim\mathcal{N}(0, v_{ij}^\star)$, $v_{ij}^\star = \frac{\sigma^2}{p}(\|\boldsymbol{U}_{i,\cdot}^\star\|_2^2 + \|\boldsymbol{V}_{j,\cdot}^\star\|_2^2)$, yielding valid two-sided $(1-\alpha)$ confidence intervals $[M_{ij}^{\text{d}} \pm \Phi^{-1}(1-\alpha/2)\sqrt{v_{ij}}]$ and sharp Frobenius accuracy $\|\boldsymbol{M}^{\text{d}} - \boldsymbol{M}^\star\|_F^2 = (2+o(1))nr\sigma^2/p$ matching the oracle tangent-space lower bound including the pre-constant 2 (Theorems 3–4).