Is there a multi-Gaussian version of the Mahalanobis distance ? Let's say we are in an $N$-dimensional space and that we have a large set of data. The distribution of this $N$-dimensional point cloud can be modeled by a multivariate Gaussian mixture model (estimated using the EM algorithm). 
Now, given two data points $x$ and $y$ from my dataset, which distance can-I use to evaluate the closeness of these two points given the global distribution ? 
In the case of a single Gaussian, I think that the Mahalanobis distance would be a natural choice, but when the global distribution is known to be non-Gaussian and a GMM is used to model the data, we end-up with $K$ different covariance matrices (one for each Gaussian). 
Should I compute $K$ Mahlanobis distances (one for each covariance matrix) then do a weighted sum using the posterior probabilities over the Gaussians ? Something like this maybe :
\begin{equation}
dist(x,y)=\sum_{i=1}^K \frac{\gamma_i(x)+\gamma_i(y)}{2} \sqrt{(x-y)^T\Sigma_i^{-1}(x-y)}
\end{equation}
where $\Sigma_i$ is the covariance matrix of the $i$-th Gaussian and $\gamma_i(x)$ is the posterior probability of the vector $x$ with respect to the Gaussian $i$. 
Any suggestions ? Thanks !
 A: After a little research, I found what I was looking for, a paper called "Deriving cluster analytic distance functions from gaussian mixture models" which proposes an extension of the Mahalanobis distance in the context of multi-modal data (a GMM representation) using Fisher Kernel method and other techniques. 
A: I know this is several years ago, but I wanted to point out that it is possible (given some data) to estimate the Kullback-leibler divergence between two GMMs.  Depending on your goal, this may prove a very elegant approach.
See:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwja2cjTu7DaAhWng1QKHRfwCYgQFgg_MAE&url=https%3A%2F%2Flabrosa.ee.columbia.edu%2F~dpwe%2Fpubs%2FJenECJ07-gmmdist.pdf&usg=AOvVaw3x4mQMf0wrdHZMMP_nwu_v
and:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwja2cjTu7DaAhWng1QKHRfwCYgQFgguMAA&url=https%3A%2F%2Fpdfs.semanticscholar.org%2F4f8d%2Feabc58014eae708c3e6ee27114535325067b.pdf&usg=AOvVaw0W11eUEeCobIk3zNa5TQzy
