Estimating Networks With Jumps

TL;DR

This paper introduces a novel method for estimating time-varying graphical models with piece-wise constant structures, using a temporally smoothed L1 regularization approach, and provides theoretical guarantees for its accuracy and convergence.

Contribution

It proposes a scalable convex optimization procedure for jointly estimating change points and network structures in nonstationary time series, with proven consistency and convergence rates.

Findings

Successfully estimates piece-wise constant network structures.

Provides theoretical guarantees for sparsistent estimation.

Develops a scalable proximal gradient algorithm.

Abstract

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.