1 research outputs found
Replica theory for learning curves for Gaussian processes on random graphs
Statistical physics approaches can be used to derive accurate predictions for
the performance of inference methods learning from potentially noisy data, as
quantified by the learning curve defined as the average error versus number of
training examples. We analyse a challenging problem in the area of
non-parametric inference where an effectively infinite number of parameters has
to be learned, specifically Gaussian process regression. When the inputs are
vertices on a random graph and the outputs noisy function values, we show that
replica techniques can be used to obtain exact performance predictions in the
limit of large graphs. The covariance of the Gaussian process prior is defined
by a random walk kernel, the discrete analogue of squared exponential kernels
on continuous spaces. Conventionally this kernel is normalised only globally,
so that the prior variance can differ between vertices; as a more principled
alternative we consider local normalisation, where the prior variance is
uniform