However, this means that if one were to use this approximation then the method would be optimizing a different objective function. While section 3.4 explains that this fast approximation is only used to consider the candidates for inclusion and that the actual inclusion does evaluate the true changes in the objective function, it is unclear how good this approximation is at selecting good candidates. In other words, what rejection rates are usually observed during optimization?

We present a generalization of independent component analysis (ICA), where instead of looking for a linear transform that makes the data components independent, we look for a transform that makes the data components well fit by a tree-structured graphical model. Treating the problem as a semiparametric statistical problem, we show that the optimal transform is found by minimizing a contrast function based on mutual information, a function that directly extends the contrast function used for classical ICA. We provide two approximations of this contrast function, one using kernel density estimation, and another using kernel generalized variance. This tree-dependent component analysis framework leads naturally to an efficient general multivariate density estimation technique where only bivariate density estimation needs to be performed.