A theory of representation learning in deep neural networks gives a deep generalisation of kernel methods

The successes of modern deep neural networks (DNNs) are founded on their ability to transform inputs across multiple layers to build good high-level representations. It is therefore critical to understand this process of representation learning. However, we cannot use standard theoretical approaches involving infinite width limits, as they eliminate representation learning. We therefore develop a new infinite width limit, the representation learning limit, that exhibits representation learning mirroring that in finite-width networks, yet at the same time, remains extremely tractable. For instance, the representation learning limit gives exactly multivariate Gaussian posteriors in deep Gaussian processes with a wide range of kernels, including all isotropic (distance-dependent) kernels. We derive an elegant objective that describes how each network layer learns representations that interpolate between input and output. Finally, we use this limit and objective to develop a flexible, deep generalisation of kernel methods, that we call deep kernel machines (DKMs). We show that DKMs can be scaled to large datasets using methods inspired by inducing point methods from the Gaussian process literature, and we show that DKMs exhibit superior performance to other kernel-based approaches.