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.

翻译：本文提出一种新的数值优化算法，用于设计具有最小归一化二阶矩的格点。算法以随机下三角生成矩阵为起点，通过随机梯度下降法更新所有矩阵元素，使其沿负梯度方向演进，成为目前该领域最高效的算法。引入一种名为“theta图像”的theta级数图形化表征工具，论证其可将数值格点表征高效转化为潜在精确形式。作为概念验证，我们在最高16维空间内设计了优化格点。在所有维度上，算法均收敛于此前已知最优格点或更优格点。推测15维叠层格点的对偶格点在其维度空间中具有最优性。