Compatible Matrices of Spearman's Rank Correlation

TL;DR

This paper resolves a long-standing open problem by demonstrating that Spearman's rho matrices are not always compatible with linear correlation matrices in dimensions 12 or higher, contrasting with previous results for lower dimensions.

Contribution

The paper proves that Spearman's rho matrices and linear correlation matrices are not equivalent in dimensions 12 or higher, providing new insights into their compatibility.

Findings

Spearman's rho matrices are not compatible with linear correlation matrices in dimensions ≥12.

Equivalence between Spearman's rho and linear correlation matrices holds only up to dimension 9.

The study links the problem to the existence of certain random vectors under linear projection constraints.

Abstract

In this paper, we provide a negative answer to a long-standing open problem on the compatibility of Spearman's rho matrices. Following an equivalence of Spearman's rho matrices and linear correlation matrices for dimensions up to 9 in the literature, we show non-equivalence for dimensions 12 or higher. In particular, we connect this problem with the existence of a random vector under some linear projection restrictions in two characterization results.