Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks
This provides a theoretical foundation for handling complex data distributions in machine learning, though it appears incremental as it builds on existing approximation bounds.
The paper tackles the problem of approximating score functions and their derivatives for data distributions with low-dimensional structure and unbounded support, achieving error bounds that avoid the curse of dimensionality and extend to any derivative order.
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.