We describe a measure quantization procedure i.e., an algorithm which finds the best approximation of a target probability law (and more generally signed finite variation measure) by a sum of $Q$ Dirac masses ($Q$ being the quantization parameter). The procedure is implemented by minimizing the statistical distance between the original measure and its quantized version; the distance is built from a negative definite kernel and, if necessary, can be computed on the fly and feed to a stochastic optimization algorithm (such as SGD, Adam, ...). We investigate theoretically the fundamental questions of existence of the optimal measure quantizer and identify what are the required kernel properties that guarantee suitable behavior. We propose two best linear unbiased (BLUE) estimators for the squared statistical distance and use them in an unbiased procedure, called HEMQ, to find the optimal quantization. We test HEMQ on several databases: multi-dimensional Gaussian mixtures, Wiener space cubature, Italian wine cultivars and the MNIST image database. The results indicate that the HEMQ algorithm is robust and versatile and, for the class of Huber-energy kernels, matches the expected intuitive behavior.

翻译：本文描述了一种测度量化过程，即一种寻找目标概率律（更一般地，带符号有限变差测度）最佳逼近的算法，该逼近表示为$Q$个狄拉克质量之和（$Q$为量化参数）。该过程通过最小化原始测度与其量化版本之间的统计距离来实现；该距离由一个负定核构建，且必要时可动态计算并馈入随机优化算法（如SGD、Adam等）。我们从理论上研究了最优测度量化器存在性这一基本问题，并确定了保证算法良好行为所需的核性质。我们提出了两个平方统计距离的最佳线性无偏（BLUE）估计量，并将其用于一个称为HEMQ的无偏过程以寻找最优量化。我们在多个数据库上测试了HEMQ：多维高斯混合模型、维纳空间求积、意大利葡萄酒品种数据集以及MNIST图像数据库。结果表明，HEMQ算法具有鲁棒性和通用性，并且对于Huber能量核类，其表现符合预期的直观行为。