Whitening transformation for skewness? Let $X$ be an $(m,n)$-matrix interpreted as a two dimensional array with each column representing $m$ samples from a random variable, with known covariance matrx $M$ and mean equal to $0$, it is possible, under mild conditions ($M$ must be positive semidefinite), to transform $X$ into a (possibly not uniquely defined) matrix $X'$, having covariance the identity matrix $I$. This transformation is generally called a whitening transformation. 
My question is: does it exist an analog transformation on the matrix $X$ which produces a matrix $X''$ whose sample skewness of each column is equal to zero and such that the mean and the covariance matrix of $X$ remain unchanged?
 A: A generalization of whitening is Gaussianization (Chen & Gopinath, 2001). It can be used to turn continuous distributions into standard normal distributions. If $F$ is the CDF of $X$ and $\Phi$ is the CDF of a standard normal, then
$$Y = \Phi^{-1}(F(X))$$
has standard normal distribution. If $X$ is assumed to be Gaussian distributed, $\Phi^{-1} \circ F$ is just a linear whitening transformation.
You could try to design an $F$ which additionally captures some form of skewness and then reintroduce the mean and covariance if desired, i.e.,
$$Y = A\Phi^{-1}(F(X)) + b,$$
where $b$ is the mean you are trying to preserve and $A$ is such that $M = AA^\top$.
Here is a simple example where the data was generated by sampling independently from a log-normal distribution before applying an affine transformation (see here for Python code).

Data is on the left (black), skewness removed on the right (red). In practice you will not know the underlying distribution or even just the affine transformation and will have to estimate it. Unlike for the Gaussian distribution, this is generally not possible in closed form. That is, before you can remove the skewness, you will probably have to run some kind of optimization (e.g., ICA).
