Algorithmic Thresholds for Refuting Random Polynomial Systems
Algorithmic Thresholds for Refuting Random Polynomial Systems
复制标题
反驳随机多项式系统的算法阈值
DOI：
10.1137/1.9781611977073.49
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Kothari, Pravesh K.
中科院分区：
文献类型：
--
作者：
Hsieh, Tim;Kothari, Pravesh K.
考虑一个n维变元多项式方程组{pi（x）=bi}i≤ m，D ≥ 2，且每个pi和bi的系数都是随机选取的，且独立于某个连续分布.我们研究的基本问题，确定最小的算法阈值，有效的算法可以findrefutations（即证书的不满足性），这样的系统。这种设置推广的问题，如反驳随机SAT的情况下，低秩矩阵传感和证明伪随机Goldreich的候选generatorsand generalizations.We证明，对于每一个随机，（n+m）O（d）-时间的正则平方和（SoS）松弛反驳这样的系统，以高概率每当。我们证明了一个下界的restrictedlow-degrees polynomialmodel的计算表明，这种权衡之间的SoS度和方程的数量几乎是紧的所有。我们还证实了这个下界的预测，在有限的设置显示一个下界的规范度-4平方和松弛反驳随机二次多项式。总之，我们的结果提供了证据的算法阈值的问题在所有δ的时间算法。我们的上限依赖于建立一个尖锐的界限上的最小integerd，使degreed-D多项式组合的输入pis生成的所有degreed-d多项式在理想的pis生成。我们的下界实际上适用于区分随机多项式系统的更简单的问题，如上所述，从多项式系统的分布与“种植”的解决方案。我们对种植分布的选择稍微（也必然）微妙：事实证明，当verm ≥ n（n）时，二次系统的自然且经过充分研究的种植分布（在机器学习中被研究为矩阵感知问题）很容易区分-n小于我们上面的上限阈值。因此，我们的设置提供了一个例子，反驳比搜索在自然种植模型。
Consider a system ofmpolynomial equations {pi(x) =bi}i≤mof degreeD≥ 2 inn-dimensional variablex∊ ℝnsuch that each coefficient of everypiandbis are chosen at random and independently from some continuous distribution. We study the basic question of determining the smallestm–thealgorithmic threshold–for which efficient algorithms can findrefutations(i.e. certificates of unsatisfiability) for such systems. This setting generalizes problems such as refuting random SAT instances, low-rank matrix sensing and certifying pseudo-randomness of Goldreich's candidate generators and generalizations.We show that for everyd∊ ℕ, the (n+m)O(d)-time canonical sum-of-squares (SoS) relaxation refutes such a system with high probability whenever . We prove a lower bound in the restrictedlow-degree polynomialmodel of computation which suggests that this trade-off between SoS degree and the number of equations is nearly tight for alld. We also confirm the predictions of this lower bound in a limited setting by showing a lower bound on the canonical degree-4 sum-of-squares relaxation for refuting random quadratic polynomials. Together, our results provide evidence for an algorithmic threshold for the problem at -time algorithms for allδ.Our upper-bound relies on establishing a sharp bound on the smallest integerdsuch that degreed–Dpolynomial combinations of the inputpis generate all degree-dpolynomials in the ideal generated by thepis. Our lower bound actually holds for the easier problem of distinguishing random polynomial systems as above from a distribution on polynomial systems with a “planted” solution. Our choice of planted distribution is slightly (and necessarily) subtle: it turns out that the natural and well-studied planted distribution for quadratic systems (studied as thematrix sensingproblem in machine learning) is easily distinguishable wheneverm≥Õ(n)–a factornsmaller than the threshold in our upper bound above. Thus, our setting provides an example where refutation is harder than search in the natural planted model.
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影响因子：
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作者：
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通讯作者：
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DOI：
10.1145/3178538
发表时间：
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期刊：
ACM Transactions on Algorithms (TALG)
影响因子：
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DOI：
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发表时间：
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期刊：
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作者：
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影响因子：
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作者：
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通讯作者：
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DOI：
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发表时间：
2019-11
期刊：
Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
影响因子：
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