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Here, we study another case, where each layer itself is a vector valued Gaussian process, and our aim is similarly to understand the limiting behaviour of the prior as depth grows.\n  Previous GP work has established that for the RBF kernel and a certain range of bandwidths $r$, the prior degenerates in the limit, converging to the set of constant functions -- which is not useful as a probabilistic model. In this paper we establish several new results. First, we identify a sharp bandwidth threshold $r_c(d) = \\Theta(\\sqrt{d})$ above which the limit is degenerate, strengthening the earlier bounds. Seco","title":"How Deep Are Deep GPs, Really? 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First, we identify a sharp bandwidth threshold $r_c(d) = \\Theta(\\sqrt{d})$ above which the limit is degenerate, strengthening the earlier bounds. Seco","title":"How Deep Are Deep GPs, Really? 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