  We propose a topological learning algorithm for the estimation of the
conditional dependency structure of large sets of random variables from sparse
and noisy data. The algorithm, named Maximally Filtered Clique Forest (MFCF),
produces a clique forest and an associated Markov Random Field (MRF) by
generalising Prim's minimum spanning tree algorithm. To the best of our
knowledge, the MFCF presents three elements of novelty with respect to existing
structure learning approaches. The first is the repeated application of a local
topological move, the clique expansion, that preserves the decomposability of
the underlying graph. Through this move the decomposability and calculation of
scores is performed incrementally at the variable (rather than edge) level, and
this provides better computational performance and an intuitive application of
multivariate statistical tests. The second is the capability to accommodate a
variety of score functions and, while this paper is focused on multivariate
normal distributions, it can be directly generalised to different types of
statistics. Finally, the third is the variable range of allowed clique sizes
which is an adjustable topological constraint that acts as a topological
penalizer providing a way to tackle sparsity at $l_0$ semi-norm level; this
allows a clean decoupling of structure learning and parameter estimation. The
MFCF produces a representation of the clique forest, together with a perfect
ordering of the cliques and a perfect elimination ordering for the vertices. As
an example we propose an application to covariance selection models and we show
that the MCFC outperforms the Graphical Lasso for a number of classes of
matrices.
