I'm a biologist trying to understand math. It would help me out no end if somebody could clarify my misunderstandings. Question 1 is causing me the most grief.
The image below is taken from the paper 'Fast Gauss Transforms based on a High Order Singular Value Decomposition for Nonlinear Filtering' by Mittelman and Miller, 2007. My questions refer to Equation 6, which shows a weighted sum of Gaussian kernels.
1) In the expressions for $s$ and $\xi$, why isn't $\Lambda^{1/2}$ inverted, i.e. $\Lambda^{-1/2}$? My reasoning is this: if $\Sigma = V^T \Lambda V$ then $\Sigma^{-1} = V^T \Lambda^{-1} V = V^T \Lambda^{-1/2} \Lambda^{-1/2} V $ (note that V is orthogonal so that $V^{-1} = V^T$). The transformation of $x$ and $f(x)$ in $s$ and $\xi$, respectively, looks to me like a whitening transformation (specifically Mahalanobis whitening), where a vector of random variables with a known covariance matrix is transformed into a set of new variables whose covariance is the identity matrix, meaning that they are uncorrelated and each have variance 1. However, this would require us to transform $x$ and $f(x)$ by pre-multiplying them with $\Lambda^{-1/2} V$, as derived above, not $\Lambda^{1/2} V$, to normalize the distance (i.e., compute the Mahalonobis distance)
2) Why is $\begin{vmatrix} \Sigma \end{vmatrix} ^{1/2}$ not included in the normalizing term $\kappa$ given that it forms a part of the normalization constant for a multivariate gaussian?
3) Is the factorization of the covariance matrix simply eigendecomposition? I normally see this written as $\Sigma = V \Lambda V^T$ not $\Sigma = V^T \Lambda V$ so I wasn't sure.
Thank you so much in advance!
