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- W4309137000 abstract "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." @default.
- W4309137000 created "2022-11-23" @default.
- W4309137000 creator A5075103678 @default.
- W4309137000 date "2022-11-14" @default.
- W4309137000 modified "2023-10-17" @default.
- W4309137000 title "Efficient algorithms for certifying lower bounds on the discrepancy of random matrices" @default.
- W4309137000 doi "https://doi.org/10.48550/arxiv.2211.07503" @default.
- W4309137000 hasPublicationYear "2022" @default.
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