Closed-form Jensen–Shannon divergence for Gaussian distributions (arithmetic mean)

Derive a closed-form expression for the Jensen–Shannon divergence JS(P||Q) defined by JS(P||Q) = (1/2) KL(P || (P+Q)/2) + (1/2) KL(Q || (P+Q)/2) between two multivariate Gaussian distributions P = N(m0, C0) and Q = N(m1, C1) on R^n, where m0, m1 ∈ R^n and C0, C1 ∈ Sym^{++}(n).

Background

The Jensen–Shannon divergence (JSD) is a widely used symmetric measure derived from the Kullback–Leibler divergence by mixing the two distributions using the arithmetic mean. For Gaussian distributions in finite dimensions, the arithmetic mixture (P+Q)/2 is not Gaussian, which complicates derivations and has prevented a closed-form expression from being established.

In contrast, the geometric mean leads to a Gaussian mixture and admits closed-form formulas, which the paper leverages to develop the Geometric Jensen–Shannon divergence. The explicit absence of a closed-form for the standard (arithmetic-mean) JSD for Gaussians motivates this open problem.