A holistic similarity measure for GMMs by embedding it into the manifold of SPD matrices

In this paper, a new similarity measure for comparing two Gaussian Mixture Models (GMMs) is obtained. This is based on an embedding of the manifold of K-component GMMs into the manifold of the symmetric positive definite matrices (SPD). The GMM manifold with the pullback of the induced metric is shown to be isometric to the submanifold with the metric induced by the affine-invariant Riemannian metric (AIRM) on the SPD manifold. We also prove that on the GMM manifold the AIRM is a lower bound for the pullback of the induced metric. This enables to use the AIRM as a similarity measure of GMMs. The effectiveness of this framework is demonstrated through texture recognition experiments on standard machine learning benchmarks.