格量化器的优化与识别
Optimization and Identification of Lattice Quantizers
AI总结：
提出一种基于随机梯度下降的格量化器数值优化算法，并借助 theta image 识别精确格形，在至16维中得到已知最佳或更优格。
摘要：
本文采用一种新的数值优化算法来设计具有最小归一化二阶矩的格。该算法从随机下三角生成矩阵出发，应用随机梯度下降，将所有元素沿负梯度方向更新，这使其成为迄今为止为该目的提出的最高效算法。文中引入了 theta 级数的图形化表示，称为 theta image，并证明它是将数值格表示转换为其底层精确形式的强大工具。作为概念验证，本文在最高至16维的维度中设计了优化格。在所有维度中，该算法都收敛到此前已知最佳格或更优的格。文中推测15维层叠格的对偶格在其维度中是最优的，并计算了其精确归一化二阶矩。
英文摘要：
Lattices with minimal normalized second moments are designed using a new numerical optimization algorithm. Starting from a random lower-triangular generator matrix and applying stochastic gradient descent, all elements are updated towards the negative gradient, which makes it the most efficient algorithm proposed so far for this purpose. A graphical illustration of the theta series, called theta image, is introduced and shown to be a powerful tool for converting numerical lattice representations into their underlying exact forms. As a proof of concept, optimized lattices are designed in dimensions up to 16. In all dimensions, the algorithm converges to either the previously best known lattice or a better one. The dual of the 15-dimensional laminated lattice is conjectured to be optimal in its dimension and its exact normalized second moment is computed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
