Graph-Guided Banding of the Covariance Matrix

ABSTRACT

Regularization has become a primary tool for developing reliable estimators of the covariance matrix in high-dimensional settings. To curb the curse of dimensionality, numerous methods assume that the population covariance (or inverse covariance) matrix is sparse, while making no particular structural assumptions on the desired pattern of sparsity. A highly-related, yet complementary, literature studies the specific setting in which the measured variables have a known ordering, in which case a banded population matrix is often assumed. While the banded approach is conceptually and computationally easier than asking for “patternless sparsity,” it is only applicable in very specific situations (such as when data are measured over time or one-dimensional space). This work proposes a generalization of the notion of bandedness that greatly expands the range of problems in which banded estimators apply. We develop convex regularizers occupying the broad middle ground between the former approach of “patternless sparsity” and the latter reliance on having a known ordering. Our framework defines bandedness with respect to a known graph on the measured variables. Such a graph is available in diverse situations, and we provide a theoretical, computational, and applied treatment of two new estimators. An R package, called ggb, implements these new methods. Supplementary materials for this article are available online.

KEYWORDS:

• Covariance estimation
• High-dimensional
• Hierarchical group lasso
• Network

Funding

The author thanks Christian Müller for a useful conversation and acknowledges the support of the NSF grant DMS-1405746 and NSF CAREER Award DMS-1748166.

Notes

1 Available at http://statweb.stanford.edu/~tibs/ElemStatLearn/datasets/zip.digits/