给出矩阵LASSO全局最小解的精确恢复条件，关键阈值可计算。
Sharp Restricted Isometry Thresholds for Global Minima of Rank-Restricted Matrix LASSO
- 基于秩受限矩阵LASSO，推导出全局最小解的精确恢复阈值。
- 当RIP常数小于临界值时，所有全局最小解误差不超过√r⋆λ。
- 理论严谨，适用于低秩矩阵恢复，也推广到稀疏向量情形。
我们确定了秩受限矩阵LASSO在全局最小值处恢复的精确受限等距阈值。对于目标秩 $r_{ar{}}$，若秩-$k$ 的RIP常数满足 $δ<δ_{\mathrm{sharp}}(k/r_{\star})$，其中 $δ_{\mathrm{sharp}}(t)=t/(4-t)$（当 $0<t<4/3$）和 $δ_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$（当 $t\ge4/3$），则所有全局最小解在任意搜索秩 $r\ge r_{\star}$ 下，其Frobenius误差 $\ extless\sim\sqrt{r_{\star}}λ$，且对所有 $λ\gtrsim\|\mathcal{A}^{*}(ξ)\|_{\mathrm{op}}$ 成立。常数仅依赖于RIP常数和 $t=k/r_{\star}$，与搜索秩无关。当秩限制失效时，结果退化为普通凸矩阵LASSO。我们还得到了稀疏性受限向量LASSO的类似结论。反之，由于存在反例表明全局最小解无法恢复真值，该阈值不可改进。
原文摘要 · Abstract (English)
We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $δ<δ_{\mathrm{sharp}}(k/r_{\star})$, where $δ_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $δ_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}λ$ for all $λ\gtrsim\|\mathcal{A}^{*}(ξ)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $δ<δ_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.
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