The planted clique problem is well-studied in the context of observing, explaining, and predicting interesting computational phenomena associated with statistical problems. When equating computational efficiency with the existence of polynomial time algorithms, the computational hardness of (some variant of) the planted clique problem can be used to infer the computational hardness of a host of other statistical problems. Is this ability to transfer computational hardness from (some variant of) the planted clique problem to other statistical problems robust to changing our notion of computational efficiency to space efficiency? We answer this question affirmatively for three different statistical problems, namely Sparse PCA, submatrix detection, and testing almost k-wise independence. The key challenge is that space efficient randomized reductions need to repeatedly access the randomness they use. Known reductions to these problems are all randomized and need polynomially many random bits to implement. Since we can not store polynomially many random bits in memory, it is unclear how to implement these existing reductions space efficiently. There are two ideas involved in circumventing this issue and implementing known reductions to these problems space efficiently. 1. When solving statistical problems, we can use parts of the input itself as randomness. 2. Secret leakage variants of the planted clique problem with appropriate secret leakage can be more useful than the standard planted clique problem when we want to use parts of the input as randomness. (abstract shortened due to arxiv constraints)

翻译：植入团问题在观察、解释和预测与统计问题相关的有趣计算现象方面已被广泛研究。当将计算效率等同于多项式时间算法的存在性时，植入团问题（或其变体）的计算难度可用于推断许多其他统计问题的计算难度。这种从植入团问题（或其变体）向其他统计问题转移计算难度的能力，在将计算效率的概念改变为空间效率时是否依然稳健？我们针对三个不同的统计问题（即稀疏主成分分析、子矩阵检测和几乎k-wise独立性检验）对此给出肯定答案。关键挑战在于空间高效的随机化归约需要重复访问其使用的随机性。已知针对这些问题的归约均为随机化方法，且需要多项式数量的随机比特来实现。由于我们无法在内存中存储多项式数量的随机比特，因此如何以空间高效的方式实现这些现有归约尚不明确。解决这一问题并实现这些已知问题的空间高效归约涉及两个思路：1. 在解决统计问题时，我们可以将输入本身的部分内容用作随机性；2. 当需要将输入的部分内容用作随机性时，具有适当秘密泄露的植入团问题变体可能比标准植入团问题更具实用性。（摘要因arXiv限制而缩短）