We initiate the study of the algorithmic problem of certifying lower bounds on the discrepancy of random matrices: given an input matrix $A \in \mathbb{R}^{m \times n}$, output a value that is a lower bound on $\mathsf{disc}(A) = \min_{x \in \{\pm 1\}^n} ||Ax||_\infty$ for every $A$, but is close to the typical value of $\mathsf{disc}(A)$ with high probability over the choice of a random $A$. This problem is important because of its connections to conjecturally-hard average-case problems such as negatively-spiked PCA, the number-balancing problem and refuting random constraint satisfaction problems. We give the first polynomial-time algorithms with non-trivial guarantees for two main settings. First, when the entries of $A$ are i.i.d. standard Gaussians, it is known that $\mathsf{disc} (A) = \Theta (\sqrt{n}2^{-n/m})$ with high probability. Our algorithm certifies that $\mathsf{disc}(A) \geq \exp(- O(n^2/m))$ with high probability. As an application, this formally refutes a conjecture of Bandeira, Kunisky, and Wein on the computational hardness of the detection problem in the negatively-spiked Wishart model. Second, we consider the integer partitioning problem: given $n$ uniformly random $b$-bit integers $a_1, \ldots, a_n$, certify the non-existence of a perfect partition, i.e. certify that $\mathsf{disc} (A) \geq 1$ for $A = (a_1, \ldots, a_n)$. Under the scaling $b = \alpha n$, it is known that the probability of the existence of a perfect partition undergoes a phase transition from 1 to 0 at $\alpha = 1$; our algorithm certifies the non-existence of perfect partitions for some $\alpha = O(n)$. We also give efficient non-deterministic algorithms with significantly improved guarantees. Our algorithms involve a reduction to the Shortest Vector Problem.

翻译：我们首次系统研究随机矩阵差异下界验证的算法问题：给定输入矩阵$A \in \mathbb{R}^{m \times n}$，对任意$A$输出其差异$\mathsf{disc}(A) = \min_{x \in \{\pm 1\}^n} ||Ax||_\infty$的下界，且对于随机选取的$A$，该输出以高概率接近$\mathsf{disc}(A)$的典型值。该问题的重要性在于其与负尖峰主成分分析、数值平衡问题及随机约束满足问题的反驳等推测难解的平均情形的联系。我们在两个主要场景下首次给出具有非平凡保证的多项式时间算法。首先，当$A$的条目为独立同分布标准高斯变量时，已知$\mathsf{disc}(A) = \Theta (\sqrt{n}2^{-n/m})$以高概率成立。我们的算法以高概率验证$\mathsf{disc}(A) \geq \exp(- O(n^2/m))$。作为应用，该结果形式化反驳了Bandeira、Kunisky和Wein关于负尖峰Wishart模型中检测问题计算难解性的猜想。其次，我们考虑整数划分问题：给定$n$个均匀随机的$b$比特整数$a_1, \ldots, a_n$，验证完美划分的不存在性，即对$A = (a_1, \ldots, a_n)$验证$\mathsf{disc}(A) \geq 1$。在标度$b = \alpha n$下，已知完美划分存在概率在$\alpha = 1$处经历从1到0的相变；我们的算法对某些$\alpha = O(n)$验证了完美划分的不存在性。我们还给出具有显著改进保证的高效非确定性算法。我们的算法涉及归约到最短向量问题。