秩约束矩阵LASSO全局极小值的尖锐受限等距阈值

中文摘要 AI 辅助

我们确定了秩约束矩阵LASSO全局极小值处恢复的尖锐受限等距阈值。对于目标秩$r_{\star}$，若秩-$k$的RIP常数满足$\delta<\delta_{\mathrm{sharp}}(k/r_{\star})$，其中$\delta_{\mathrm{sharp}}(t)$在$0<t<4/3$时为$t/(4-t)$，在$t\ge4/3$时为$\sqrt{(t-1)/t}$，则对于所有满足$\lambda\gtrsim\\|\mathcal{A}^{*}(\xi)\\|_{\mathrm{op}}$的$\lambda$，以及每个搜索秩$r\ge r_{\star}$，所有全局极小值的Frobenius误差均满足$\lesssim\sqrt{r_{\star}}\lambda$。这些常数仅依赖于RIP常数和$t=k/r_{\star}$，且特别与搜索秩无关。当秩约束不起作用时，该结果可推广到普通凸矩阵LASSO。我们还得到了稀疏约束向量LASSO的类似结果。反之，由于存在全局极小值无法恢复真实值的反例，我们证明了阈值$\delta<\delta_{\mathrm{sharp}}(k/r_{\star})$无法改进。

英文摘要

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.

发表机构

• University of Illinois Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）

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