论文标题
在最好的晶格量化器上
On the best lattice quantizers
论文作者
论文摘要
晶格量化器近似于任意实现的源矢量，并从特定的离散晶格中取出向量。量化误差是源向量和晶格向量之间的差异。扎米尔（Zamir）和费德尔（Feder）在经典的1996年论文中表明，全球最佳晶格量化器（最小化均方根误差）具有白色量化误差：对于均匀分布的源，误差的协方差是身份矩阵，乘以正真实因子。我们概括了定理，表明相同的属性（i）对于任何均方根误差都无法通过发电机矩阵的小扰动而减少的任何晶格，而（ii）对于（i）的意义上，晶格的最佳产物本身是本地最佳的。我们通过证明对产品晶格的发电机矩阵的任何下层或上层修饰来降低NSM的任何下层或上层修饰，从而在任何维度的归一化第二矩（NSM）上得出了上限。使用这些工具并采用当前最著名的晶格量化器来构建产品晶格，我们在13至15、15、17至23和25至48方面构建了改进的晶格量化器。在某些维度中，这些是第一个报告的晶格，其归一化的第二瞬间低于最著名的上限。
A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization error: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any lattice whose mean square error cannot be decreased by a small perturbation of the generator matrix, and (ii) for an optimal product of lattices that are themselves locally optimal in the sense of (i). We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the best known upper bound.
