Over the past decade deep learning has revolutionized the field of computer vision, with convolutional neural network models proving to be very effective for image classification benchmarks. However, a fundamental theoretical questions remain answered: why can they solve discrete image classification tasks that involve feature extraction? We address this question in this paper by introducing a novel mathematical model for image classification, based on feature extraction, that can be used to generate images resembling real-world datasets. We show that convolutional neural network classifiers can solve these image classification tasks with zero error. In our proof, we construct piecewise linear functions that detect the presence of features, and show that they can be realized by a convolutional network.

翻译：过去十年间，深度学习彻底革新了计算机视觉领域，其中卷积神经网络模型在图像分类基准测试中展现出卓越有效性。然而，一个基础性的理论问题仍未得到解答：为何它们能够解决涉及特征提取的离散图像分类任务？本文通过引入一种基于特征提取的新型图像分类数学模型来探讨该问题，该模型可生成模拟真实世界数据集的图像。我们证明卷积神经网络分类器能够以零误差解决这些图像分类任务。在证明过程中，我们构造了可检测特征存在的分段线性函数，并证明这些函数可由卷积网络实现。