8 research outputs found
State-Space Network Topology Identification from Partial Observations
In this work, we explore the state-space formulation of a network process to
recover, from partial observations, the underlying network topology that drives
its dynamics. To do so, we employ subspace techniques borrowed from system
identification literature and extend them to the network topology
identification problem. This approach provides a unified view of the
traditional network control theory and signal processing on graphs. In
addition, it provides theoretical guarantees for the recovery of the
topological structure of a deterministic continuous-time linear dynamical
system from input-output observations even though the input and state
interaction networks might be different. The derived mathematical analysis is
accompanied by an algorithm for identifying, from data, a network topology
consistent with the dynamics of the system and conforms to the prior
information about the underlying structure. The proposed algorithm relies on
alternating projections and is provably convergent. Numerical results
corroborate the theoretical findings and the applicability of the proposed
algorithm.Comment: 13 pages, 3 appendix page
Does the -norm Learn a Sparse Graph under Laplacian Constrained Graphical Models?
We consider the problem of learning a sparse graph under Laplacian
constrained Gaussian graphical models. This problem can be formulated as a
penalized maximum likelihood estimation of the precision matrix under Laplacian
structural constraints. Like in the classical graphical lasso problem, recent
works made use of the -norm regularization with the goal of promoting
sparsity in Laplacian structural precision matrix estimation. However, we find
that the widely used -norm is not effective in imposing a sparse
solution in this problem. Through empirical evidence, we observe that the
number of nonzero graph weights grows with the increase of the regularization
parameter. From a theoretical perspective, we prove that a large regularization
parameter will surprisingly lead to a fully connected graph. To address this
issue, we propose a nonconvex estimation method by solving a sequence of
weighted -norm penalized sub-problems and prove that the statistical
error of the proposed estimator matches the minimax lower bound. To solve each
sub-problem, we develop a projected gradient descent algorithm that enjoys a
linear convergence rate. Numerical experiments involving synthetic and
real-world data sets from the recent COVID-19 pandemic and financial stock
markets demonstrate the effectiveness of the proposed method. An open source
package containing the code for all the experiments is available
at https://github.com/mirca/sparseGraph