Neural networks have been shown to significantly outperform kernel methods (including neural tangent kernels) in problems such as image classification. Most theoretical explanations of this performance gap use a complex or stylized hypothesis class to explain this gap, which leads to a disconnect between the theory and practice. In this talk, I will demonstrate a simple hypothesis class inspired from natural images which explains this performance gap based on finding a sparse signal in the presence of noise, suggesting an improved denoising ability or more generally improved ability to discard irrelevant features using neural networks rather than kernels. Specifically, we show that, for a simple data distribution with sparse signal amidst high-variance noise, a convolutional neural network trained using stochastic gradient descent learns to threshold out the noise and find the signal. On the other hand, the corresponding neural tangent kernel, with a fixed set of predetermined features, is unable to adapt to the signal in this manner. This is joint work with Stefani Karp, Ezra Winston and Yuanzhi Li.