We present the first iterative spectral algorithm to find near-optimal solutions for a random quadratic objective over the discrete hypercube, resolving a conjecture of Subag [Subag, Communications on Pure and Applied Mathematics, 74(5), 2021]. The algorithm is a randomized Hessian ascent in the solid cube, with the objective modified by subtracting an instance-independent potential function [Chen et al., Communications on Pure and Applied Mathematics, 76(7), 2023]. Using tools from free probability theory, we construct an approximate projector into the top eigenspaces of the Hessian, which serves as the covariance matrix for the random increments. With high probability, the iterates' empirical distribution approximates the solution to the primal version of the Auffinger-Chen SDE [Auffinger et al., Communications in Mathematical Physics, 335, 2015]. The per-iterate change in the modified objective is bounded via a Taylor expansion, where the derivatives are controlled through Gaussian concentration bounds and smoothness properties of a semiconcave regularization of the Fenchel-Legendre dual to the Parisi PDE. These results lay the groundwork for (possibly) demonstrating low-degree sum-of-squares certificates over high-entropy step distributions for a relaxed version of the Parisi formula [Open Question 1.8, arXiv:2401.14383].

翻译：我们提出了首个迭代谱算法，用于在离散超立方体上寻找随机二次目标函数的近最优解，解决了Subag的一个猜想[Subag, Communications on Pure and Applied Mathematics, 74(5), 2021]。该算法是固体立方体中的随机化Hessian上升法，其目标函数通过减去一个与实例无关的势函数进行了修正[Chen et al., Communications on Pure and Applied Mathematics, 76(7), 2023]。利用自由概率论的工具，我们构造了一个近似投影算子，将其嵌入Hessian的顶部特征空间，该算子作为随机增量的协方差矩阵。以高概率而言，迭代值的经验分布近似于Auffinger-Chen SDE原始版本的解[Auffinger et al., Communications in Mathematical Physics, 335, 2015]。修正目标函数每次迭代的变化通过泰勒展开进行界定，其中导数通过高斯集中界和Parisi PDE的Fenchel-Legendre对偶的半凹正则化的光滑性性质加以控制。这些结果为（可能）证明低次平方和证书在高熵步分布上关于Parisi公式的松弛版本奠定了基础[开放问题1.8, arXiv:2401.14383]。