I am looking for the best way to approximate the Mahalanobis distance by the standardized Euclidean distance, which would reduce the number of the required multiplications. The easiest way is the diagonalization of the inverse covariance matrix (concentration matrix) by zeroing the elements outside the main diagonal.
Is there a better approximation which slightly modifies the values on the covariance matrix diagonal to make up for zeroing those elements?
For example: assuming $ \Sigma^{-1} = \begin{bmatrix} 1 & 0.2\\ 0.2 & 0.9 \end{bmatrix}$ what should be the best approximation $ \Sigma^{-1} \approx \begin{bmatrix} a & 0\\ 0 & b \end{bmatrix}$?
