The Many-to-Many Mapping Between the Concordance Correlation Coefficient and the Mean Square Error

We derive the mapping between two of the most pervasive utility functions, the mean square error ($MSE$) and the concordance correlation coefficient (CCC, $\rho_c$). Despite its drawbacks, $MSE$ is one of the most popular performance metrics (and a loss function); along with lately $\rho_c$ in many of the sequence prediction challenges. Despite the ever-growing simultaneous usage, e.g., inter-rater agreement, assay validation, a mapping between the two metrics is missing, till date. While minimisation of $L_p$ norm of the errors or of its positive powers (e.g., $MSE$) is aimed at $\rho_c$ maximisation, we reason the often-witnessed ineffectiveness of this popular loss function with graphical illustrations. The discovered formula uncovers not only the counterintuitive revelation that `$MSE_1<MSE_2$' does not imply `$\rho_{c_1}>\rho_{c_2}$', but also provides the precise range for the $\rho_c$ metric for a given $MSE$. We discover the conditions for $\rho_c$ optimisation for a given $MSE$; and as a logical next step, for a given set of errors. We generalise and discover the conditions for any given $L_p$ norm, for an even p. We present newly discovered, albeit apparent, mathematical paradoxes. The study inspires and anticipates a growing use of $\rho_c$-inspired loss functions e.g., $\left|\frac{MSE}{\sigma_{XY}}\right|$, replacing the traditional $L_p$-norm loss functions in multivariate regressions.
Comment: Why this discovery, or the mapping formulation is important: MSE1<MSE2 does not necessarily mean CCC1>CCC2. In other words, MSE minimisation does not necessarily guarantee CCC maximisation