Stay on path: PCA along graph paths
This work addresses the challenge of incorporating network prior knowledge into PCA for improved statistical efficiency, representing an incremental advancement in constrained PCA methods.
The paper tackles the problem of performing sparse PCA with graph-structured constraints, where the support set must lie along a path in a directed acyclic graph, aiming to reduce the number of observations needed for recovery. It introduces a non-convex estimator and algorithms, showing that side information can improve statistical complexity, with empirical validation on synthetic and real datasets.
We introduce a variant of (sparse) PCA in which the set of feasible support sets is determined by a graph. In particular, we consider the following setting: given a directed acyclic graph $G$ on $p$ vertices corresponding to variables, the non-zero entries of the extracted principal component must coincide with vertices lying along a path in $G$. From a statistical perspective, information on the underlying network may potentially reduce the number of observations required to recover the population principal component. We consider the canonical estimator which optimally exploits the prior knowledge by solving a non-convex quadratic maximization on the empirical covariance. We introduce a simple network and analyze the estimator under the spiked covariance model. We show that side information potentially improves the statistical complexity. We propose two algorithms to approximate the solution of the constrained quadratic maximization, and recover a component with the desired properties. We empirically evaluate our schemes on synthetic and real datasets.