For a large class of feature maps we provide a tight asymptotic characterisation of the test error associated with learning the readout layer, in the high-dimensional limit where the input dimension, hidden layer widths, and number of training samples are proportionally large. This characterization is formulated in terms of the population covariance of the features. Our work is partially motivated by the problem of learning with Gaussian rainbow neural networks, namely deep non-linear fully-connected networks with random but structured weights, whose row-wise covariances are further allowed to depend on the weights of previous layers. For such networks we also derive a closed-form formula for the feature covariance in terms of the weight matrices. We further find that in some cases our results can capture feature maps learned by deep, finite-width neural networks trained under gradient descent.

We introduce rainbow networks as a probabilistic model of trained deep neural networks. The model cascades random feature maps whose weight distributions are learned. It assumes that dependencies between weights at different layers are reduced to rotations which align the input activations. Neuron weights within a layer are independent after this alignment. Their activations define kernels which become deterministic in the infinite-width limit. This is verified numerically for ResNets trained on the ImageNet dataset. We also show that the learned weight distributions have low-rank covariances. Rainbow networks thus alternate between linear dimension reductions and non-linear high-dimensional embeddings with white random features. Gaussian rainbow networks are defined with Gaussian weight distributions. These models are validated numerically on image classification on the CIFAR-10 dataset, with wavelet scattering networks. We further show that during training, SGD updates the weight covariances while mostly preserving the Gaussian initialization.