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Statistical physics approaches can be used to derive accurate predictions forthe performance of inference methods learning from potentially noisy data, asquantified by the learning curve defined as the average error versus number oftraining examples. We analyse a challenging problem in the area ofnon-parametric inference where an effectively infinite number of parameters hasto be learned, specifically Gaussian process regression. When the inputs arevertices on a random graph and the outputs noisy function values, we show thatreplica techniques can be used to obtain exact performance predictions in thelimit of large graphs. The covariance of the Gaussian process prior is definedby a random walk kernel, the discrete analogue of squared exponential kernelson continuous spaces. Conventionally this kernel is normalised only globally,so that the prior variance can differ between vertices; as a more principledalternative we consider local normalisation, where the prior variance isuniform.

Replica theory for learning curves for Gaussian processes on random graphs