Learning of continuous exponential family distributions with unbounded support remains an important area of research for both theory and applications in high-dimensional statistics. In recent years, score matching has become a widely used method for learning exponential families with continuous variables due to its computational ease when compared against maximum likelihood estimation. However, theoretical understanding of the statistical properties of score matching is still lacking. In this work, we provide a non-asymptotic sample complexity analysis for learning the structure of exponential families of polynomials with score matching. The derived sample bounds show a polynomial dependence on the model dimension. These bounds are the first of its kind, as all prior work has shown only asymptotic bounds on the sample complexity.

翻译：具有无界支撑的连续指数族分布的学习，对于高维统计的理论和应用而言仍是一个重要的研究领域。近年来，得分匹配因其相较于最大似然估计在计算上的简易性，已成为学习连续变量指数族分布的常用方法。然而，对得分匹配统计特性的理论理解仍显不足。本文中，我们针对利用得分匹配学习多项式指数族结构的问题，进行了非渐近的样本复杂度分析。导出的样本界显示其与模型维度呈多项式依赖关系。这些界是该领域的首个成果，因为此前所有工作仅展示了样本复杂度的渐近界。