Simplicial and Minimal-Variance Distances in Multivariate Data Analysis

Polynomial whitening for high-dimensional data

The inverse square root of a covariance matrix is often desirable for performing data whitening in the process of applying many common multivariate data analysis methods. Direct calculation of the inverse square root is not available when the covariance matrix is either singular or nearly singular, as often occurs in high dimensions. We develop new methods, which we broadly call polynomial whitening, to construct a low-degree polynomial in the empirical covariance matrix which has similar properties to the true inverse square root of the covariance matrix (should it exist). Our method does not suffer in singular or near-singular settings, and is computationally tractable in high dimensions. We demonstrate that our construction of low-degree polynomials provides a good substitute for high-dimensional inverse square root covariance matrices, in both $$d < N$$ d < N and $$d \ge N$$ d ≥ N cases. We offer examples on data whitening, outlier detection and principal component analysis to demonstrate the performance of the proposed method.