Consider the setting where a $\rho$-sparse Rademacher vector is planted in a random $d$-dimensional subspace of $R^n$. A classical question is how to recover this planted vector given a random basis in this subspace. A recent result by [ZSWB21] showed that the Lattice basis reduction algorithm can recover the planted vector when $n\geq d+1$. Although the algorithm is not expected to tolerate inverse polynomial amount of noise, it is surprising because it was previously shown that recovery cannot be achieved by low degree polynomials when $n\ll \rho^2 d^{2}$ [MW21]. A natural question is whether we can derive an Statistical Query (SQ) lower bound matching the previous low degree lower bound in [MW21]. This will - imply that the SQ lower bound can be surpassed by lattice based algorithms; - predict the computational hardness when the planted vector is perturbed by inverse polynomial amount of noise. In this paper, we prove such an SQ lower bound. In particular, we show that super-polynomial number of VSTAT queries is needed to solve the easier statistical testing problem when $n\ll \rho^2 d^{2}$ and $\rho\gg \frac{1}{\sqrt{d}}$. The most notable technique we used to derive the SQ lower bound is the almost equivalence relationship between SQ lower bound and low degree lower bound [BBH+20, MW21].

翻译：考虑在随机$d$维子空间（嵌入$\mathbb{R}^n$）中植入一个$\rho$-稀疏Rademacher向量的场景。一个经典问题是如何在该子空间的随机基下恢复该植入向量。[ZSWB21]的最新结果表明，当$n\geq d+1$时，格基约简算法可恢复该植入向量。尽管该算法在抗逆多项式噪声方面表现欠佳，但这一发现仍令人惊讶——因为此前研究[MW21]已证明当$n\ll \rho^2 d^{2}$时，低阶多项式无法实现恢复。一个自然的问题是：能否推导出与[MW21]中低阶下界相匹配的统计查询（SQ）下界？这将——一方面揭示基于格的算法可以超越SQ下界；另一方面预测植入向量受逆多项式噪声扰动时的计算复杂度。本文证明了此类SQ下界。具体而言，我们表明当$n\ll \rho^2 d^{2}$且$\rho\gg \frac{1}{\sqrt{d}}$时，解决更简单的统计检验问题需要超多项式数量的VSTAT查询。我们推导SQ下界所采用的最显著技术是基于SQ下界与低阶下界[BBH+20, MW21]之间的近似等价关系。