We consider the task of certifying that a random $d$-dimensional subspace $X$ in $\mathbb{R}^n$ is well-spread - every vector $x \in X$ satisfies $c\sqrt{n} \|x\|_2 \leq \|x\|_1 \leq \sqrt{n}\|x\|_2$. In a seminal work, Barak et. al. showed a polynomial-time certification algorithm when $d \leq O(\sqrt{n})$. On the other hand, when $d \gg \sqrt{n}$, the certification task is information-theoretically possible but there is evidence that it is computationally hard [MW21,Cd22], a phenomenon known as the information-computation gap. In this paper, we give subexponential-time certification algorithms in the $d \gg \sqrt{n}$ regime. Our algorithm runs in time $\exp(\widetilde{O}(n^{\varepsilon}))$ when $d \leq \widetilde{O}(n^{(1+\varepsilon)/2})$, establishing a smooth trade-off between runtime and the dimension. Our techniques naturally extend to the related planted problem, where the task is to recover a sparse vector planted in a random subspace. Our algorithm achieves the same runtime and dimension trade-off for this task.

翻译：我们考虑验证随机$d$维子空间$X \subset \mathbb{R}^n$是否具有良好散布性的任务——即每个向量$x \in X$满足$c\sqrt{n} \|x\|_2 \leq \|x\|_1 \leq \sqrt{n}\|x\|_2$。在开创性工作中，Barak等人证明了当$d \leq O(\sqrt{n})$时存在多项式时间验证算法。另一方面，当$d \gg \sqrt{n}$时，验证任务在信息论上可行，但有证据表明其在计算上具有困难性[MW21,Cd22]，此现象被称为信息-计算鸿沟。本文给出了$d \gg \sqrt{n}$情形下的次指数时间验证算法。当$d \leq \widetilde{O}(n^{(1+\varepsilon)/2})$时，我们的算法运行时间为$\exp(\widetilde{O}(n^{\varepsilon}))$，建立了运行时间与维度之间的平滑权衡。该技术可自然延伸至相关植入问题——即从随机子空间中恢复植入稀疏向量的任务。针对此任务，我们的算法实现了相同的运行时间与维度权衡。