High-Performance Variance-Covariance Matrix Construction Using an Uncentered Gram Formulation

TL;DR

This paper introduces a novel method for constructing covariance matrices efficiently by leveraging an uncentered Gram formulation, significantly reducing computational complexity and improving runtime performance in high-dimensional data analysis.

Contribution

The authors extend Reichel's variance measure to covariance matrices using an uncentered Gram approach, eliminating the need for explicit centering and enabling faster computations.

Findings

Significant runtime improvements in Python benchmarks

Effective use of BLAS optimizations enhances performance

Optional use of RXTX routines further accelerates computation

Abstract

Reichel (2025) defined the bariance as a pairwise-difference measure that can be rewritten in linear time using only scalar sums. We extend this idea to the covariance matrix by showing that the standard matrix expression involving the uncentered Gram matrix and a correction term is algebraically identical to the pairwise-difference definition while avoiding explicit centering. The computation then reduces to one outer product of dimension p-by-p and a single subtraction. Benchmarks in Python show clear runtime gains, especially when BLAS optimizations are absent. Optionally faster Gram-matrix routines such as RXTX (Rybin et al., 2025) further reduce overall cost.