As computing the inverse second-order derivatives is the most computation-intensive operation, we will focus on it. In Section 3.1, we use the trick of least square to compute the We can leverage the Neumann series to compute the matrix inverse. B.1 Proof of the Approximation by Implicit Gradients Here, we provide the proof for J. B.2 Proof of Theorem 3.1 Before we prove our main theorem, we prove several essential lemmas as below. Using Assumption 3.4 and 3.5 directly lead to r By Assumption 3.4, we have r By Lemma B.1 and Lemma B.2, we have r If Assumption 3.4 and 3.5 hold, then the The linear model we use is a matrix that maps the input data into a vector. LeNet model is a convolutional neural network with 4 convolutional layers and 1 fully connected layer.

Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on anisotropic Besov spaces. The anisotropic Besov space is characterized by direction-dependent smoothness and includes several function classes that have been investigated thus far. We demonstrate that the approximation error and estimation error of deep learning only depend on the average value of the smoothness parameters in all directions. Consequently, the curse of dimensionality can be avoided if the smoothness of the target function is highly anisotropic.