  We study the problem of estimating a temporally varying coefficient and
varying structure (VCVS) graphical model underlying nonstationary time series
data, such as social states of interacting individuals or microarray expression
profiles of gene networks, as opposed to i.i.d. data from an invariant model
widely considered in current literature of structural estimation. In
particular, we consider the scenario in which the model evolves in a piece-wise
constant fashion. We propose a procedure that minimizes the so-called TESLA
loss (i.e., temporally smoothed L1 regularized regression), which allows
jointly estimating the partition boundaries of the VCVS model and the
coefficient of the sparse precision matrix on each block of the partition. A
highly scalable proximal gradient method is proposed to solve the resultant
convex optimization problem; and the conditions for sparsistent estimation and
the convergence rate of both the partition boundaries and the network structure
are established for the first time for such estimators.
