抗噪性群体测试：局限性与构建
Noise-Resilient Group Testing: Limitations and Constructions
摘要：
我们研究组合组测试计划学习$d$稀疏布尔向量使用高度不可靠的析取测量。我们考虑了一个对抗性噪声模型，它只限制了错误观测的数量，并证明了该模型中的任何抗噪声方案只能近似地重建稀疏向量。在积极的一面，我们把这个障碍，我们的优势，并表明，近似重建（在一个令人满意的近似程度），使我们能够打破$\tildeΩ(d^2 \log n)$的信息理论下限，这是已知的精确重建的$d$ -稀疏向量的长度$n$通过非自适应测量，由一个乘法因子$\tildeΩ(d)$。具体而言，我们给出了简单的随机结构的非自适应测量方案，与$m=O(d \log n)$测量，允许有效的重建$d$稀疏向量高达$O(d)$误报，即使在存在的$δm$误报和$O(m/d)$误报内的测量结果，对于任何常数$δ< 1$。我们表明，从理论上讲，这些参数中没有一个可以在不显著影响其他参数的情况下得到实质性改善。此外，我们得到几个明确的结构，特别是一个匹配的随机权衡，但使用$m = O(d^{1+o(1)} \log n)$测量。我们还得到明确的建设，允许快速重建的时间$\poly(m)$，这将是次线性的$n$足够稀疏的向量。在我们的构造中使用的主要工具是随机性压缩器和提取器的列表解码视图。
英文摘要：
We study combinatorial group testing schemes for learning $d$-sparse Boolean vectors using highly unreliable disjunctive measurements. We consider an adversarial noise model that only limits the number of false observations, and show that any noise-resilient scheme in this model can only approximately reconstruct the sparse vector. On the positive side, we take this barrier to our advantage and show that approximate reconstruction (within a satisfactory degree of approximation) allows us to break the information theoretic lower bound of $\tildeΩ(d^2 \log n)$ that is known for exact reconstruction of $d$-sparse vectors of length $n$ via non-adaptive measurements, by a multiplicative factor $\tildeΩ(d)$. Specifically, we give simple randomized constructions of non-adaptive measurement schemes, with $m=O(d \log n)$ measurements, that allow efficient reconstruction of $d$-sparse vectors up to $O(d)$ false positives even in the presence of $δm$ false positives and $O(m/d)$ false negatives within the measurement outcomes, for any constant $δ< 1$. We show that, information theoretically, none of these parameters can be substantially improved without dramatically affecting the others. Furthermore, we obtain several explicit constructions, in particular one matching the randomized trade-off but using $m = O(d^{1+o(1)} \log n)$ measurements. We also obtain explicit constructions that allow fast reconstruction in time $\poly(m)$, which would be sublinear in $n$ for sufficiently sparse vectors. The main tool used in our construction is the list-decoding view of randomness condensers and extractors.
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