Revisiting the expressiveness of CNNs: a mathematical framework for feature extraction

Vinoth Nandakumar, Arush Tagade, Tongliang Liu

Keywords: convolutional networks, feature extraction, deep learning theory

TL;DR: This paper introduces a novel mathematical framework for image classification, theoretically analyzing how convolutional neural networks extract features from images.

Abstract: 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. Given their widespread adoption, several theoretical works have analyzed their expressiveness, and study the class of piecewise linear functions that they can realize. However, a fundamental theoretical questions remain answered: why are piecewise linear functions effective for feature extraction tasks that arise in image classification? We address this question in this paper by introducing a simplified mathematical model for feature extraction, based on classical template matching algorithms that are commonly used in computer vision. We then prove that convolutional neural network classifiers can solve this class of image classification problems, by constructing piecewise linear functions that detect the presence of features, and showing that they can be realized by convolutional neurons. We also discuss the interpretability of the networks we construct, and compare them with those obtained via gradient-based optimization methods by conducting experiments on simple datasets.

Primary Area: learning theory

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Submission Number: 8698