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Randomly initialized wide neural networks transition to linear functions ofweights as the width grows, in a ball of radius $O(1)$ around initialization. Anecessary condition for this result is that all layers of the network are wideenough, i.e., all widths tend to infinity. However, the transition to linearitybreaks down when this infinite width assumption is violated. In this work weshow that linear networks with a bottleneck layer learn bilinear functions ofthe weights, in a ball of radius $O(1)$ around initialization. In general, for$B-1$ bottleneck layers, the network is a degree $B$ multilinear function ofweights. Importantly, the degree only depends on the number of bottlenecks andnot the total depth of the network.

A note on Linear Bottleneck networks and their Transition to Multilinearity