A comment on "Laplacian linear discriminant analysis"

The recent paper by Tang et al. [1] has proven that Fisher’s criterion function is equal to a novel linear discriminant criterion function. The novel function is then related to spectral decomposition of the Laplacian of a graph. The equivalence between the two functions is based on Theorem 1 given in the beginning of Ref. [1]. Although some mathematical analysis is established in their paper, we will show in this comment that Theorem 1 that is fundamental for the paper [1] is not true in general case. Theorem 1 given in Ref. [1] is stated as follows.