The Pairwise Gaussian Random Field for High-Dimensional Data Imputation

Abstract

In this paper, we consider the problem of imputation (recovering missing values) in very high-dimensional data with an arbitrary covariance structure. The modern solution to this problem is the Gaussian Markov random field (GMRF). The problem with applying a GMRF to very high-dimensional data imputation is that while the GMRF model itself can be useful even for data having tens of thousands of dimensions, utilizing a GMRF requires access to a sparsified, inverse covariance matrix for the data. Computing this matrix using even state-of-the-art methods is very costly, as it typically requires first estimating the covariance matrix from the data (at a O(nm2) cost for m dimensions and n data points) and then performing a regularized inversion of the estimated covariance matrix, which is also very expensive. This is impractical for even moderately-sized, high-dimensional data sets. In this paper, we propose a very simple alternative to the GMRF called the pair wise Gaussian random field or PGRF for short. The PGRF is a graphical, factor-based model. Unlike traditional Gaussian or GMRF models, a PGRF does not require a covariance or correlation matrix as input. Instead, a PGRF takes as input a set of p (dimension, dimension) pairs for which the user suspects there might be a strong correlation or anti-correlation. This set of pairs defines the graphical structure of the model, with a simple Gaussian factor associated with each of the p (dimension, dimension) pairs. Using this structure, it is easy to perform simultaneous inference and imputation of the model. The key benefit of the approach is that the time required for the PGRF to perform inference is approximately linear with respect to p, where p will typically be much smaller than the number of entries in a m×m covariance or precision matrix.