Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the $\sqrtn$ Dimension Threshold
<|reference_start|>Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the $\sqrtn$ Dimension Threshold: 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.<|reference_end|>
@article{guruswami2024certifying,
    title={Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond
  the $\sqrt{n}$ Dimension Threshold},
    author={Venkatesan Guruswami, Jun-Ting Hsieh, Prasad Raghavendra},
    journal={arXiv preprint arXiv:2405.05373},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.05373},
    primaryClass={cs.DS cs.CC math.MG}
}