Efficient algorithms for certifying lower bounds on the discrepancy of random matrices
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We initiate the study of the algorithmic problem of certifying lower bounds on the discrepancy of random matrices: given an input matrix A ∈ℝ^m × n, output a value that is a lower bound on 𝖽𝗂𝗌𝖼(A) = min_x ∈{± 1}^n ||Ax||_∞ for every A, but is close to the typical value of 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 𝖽𝗂𝗌𝖼 (A) = Θ (√(n)2^-n/m) with high probability. Our algorithm certifies that 𝖽𝗂𝗌𝖼(A) ≥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, …, a_n, certify the non-existence of a perfect partition, i.e. certify that 𝖽𝗂𝗌𝖼 (A) ≥ 1 for A = (a_1, …, a_n). Under the scaling b = α n, it is known that the probability of the existence of a perfect partition undergoes a phase transition from 1 to 0 at α = 1; our algorithm certifies the non-existence of perfect partitions for some α = O(n). We also give efficient non-deterministic algorithms with significantly improved guarantees. Our algorithms involve a reduction to the Shortest Vector Problem.
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