6 research outputs found
The average number of distinct sites visited by a random walker on random graphs
We study the linear large behavior of the average number of distinct
sites visited by a random walker after steps on a large random
graph. An expression for the graph topology dependent prefactor in is proposed. We use generating function techniques to relate this prefactor
to the graph adjacency matrix and then devise message-passing equations to
calculate its value. Numerical simulations are performed to evaluate the
agreement between the message passing predictions and random walk simulations
on random graphs. Scaling with system size and average graph connectivity are
also analysed.Comment: 22 pages, 4 figure
Mat\'ern Gaussian Processes on Graphs
Gaussian processes are a versatile framework for learning unknown functions
in a manner that permits one to utilize prior information about their
properties. Although many different Gaussian process models are readily
available when the input space is Euclidean, the choice is much more limited
for Gaussian processes whose input space is an undirected graph. In this work,
we leverage the stochastic partial differential equation characterization of
Mat\'ern Gaussian processes - a widely-used model class in the Euclidean
setting - to study their analog for undirected graphs. We show that the
resulting Gaussian processes inherit various attractive properties of their
Euclidean and Riemannian analogs and provide techniques that allow them to be
trained using standard methods, such as inducing points. This enables graph
Mat\'ern Gaussian processes to be employed in mini-batch and non-conjugate
settings, thereby making them more accessible to practitioners and easier to
deploy within larger learning frameworks
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
Random Walk Kernels and Learning Curves for Gaussian Process Regression on Random Graphs
We consider learning on graphs, guided by kernels that encode similarity between vertices. Our focus is on random walk kernels, the analogues of squared exponential kernels in Euclidean spaces. We show that on large, locally treelike graphs these have some counter-intuitive properties, specifically in the limit of large kernel lengthscales. We consider using these kernels as covariance functions of Gaussian processes. In this situation one typically scales the prior globally to normalise the average of the prior variance across vertices. We demonstrate that, in contrast to the Euclidean case, this generically leads to significant variation in the prior variance across vertices, which is undesirable from a probabilistic modelling point of view. We suggest the random walk kernel should be normalised locally, so that each vertex has the same prior variance, and analyse the consequences of this by studying learning curves for Gaussian process regression. Numerical calculations as well as novel theoretical predictions for the learning curves using belief propagation show that one obtains distinctly different probabilistic models depending on the choice of normalisation. Our method for predicting the learning curves using belief propagation is significantly more accurate than previous approximations and should become exact in the limit of large random graphs