We present a theory for simultaneous approximation of the score function and its derivatives, enabling the handling of data distributions with low-dimensional structure and unbounded support. Our approximation error bounds match those in the literature while relying on assumptions that relax the usual bounded support requirement. Crucially, our bounds are free from the curse of dimensionality. Moreover, we establish approximation guarantees for derivatives of any prescribed order, extending beyond the commonly considered first-order setting.

翻译：本文提出了一种同时逼近评分函数及其导数的理论，使得处理具有低维结构及无界支撑的数据分布成为可能。我们的逼近误差界与文献中的结果相当，同时依赖于放松了通常有界支撑要求的假设。关键的是，我们的误差界避免了维数灾难。此外，我们为任意指定阶数的导数建立了逼近保证，从而超越了通常仅考虑一阶导数的研究框架。