Image classifiers based on convolutional neural networks are defined, and the rate of convergence of the misclassification risk of the estimates towards the optimal misclassification risk is analyzed. Under suitable assumptions on the smoothness and structure of a posteriori probability, the rate of convergence is shown which is independent of the dimension of the image. This proves that in image classification, it is possible to circumvent the curse of dimensionality by convolutional neural networks. Furthermore, the obtained result gives an indication why convolutional neural networks are able to outperform the standard feedforward neural networks in image classification. Our classifiers are compared with various other classification methods using simulated data. Furthermore, the performance of our estimates is also tested on real images.

定义了基于卷积神经网络的图像分类器，分析了估计的误分类风险向最优误分类风险的收敛速度。在对后验概率的平滑度和结构的适当假设下，显示了与图像尺寸无关的收敛速度。这证明在图像分类中，可以通过卷积神经网络规避维数灾难。此外，获得的结果表明为什么卷积神经网络能够在图像分类中优于标准前馈神经网络。我们的分类器使用模拟数据与其他各种分类方法进行比较。此外，我们的估计性能也在真实图像上进行了测试。