and the noise, respectively, captured along direction $v$. In Table \[tab:signalnoise\] we present many well known approaches that could be cast in this framework. Principal Component Analysis (PCA) finds the directions of maximal variance without any particular noise model. The recently proposed Vanishing Component Analysis (VCA) [@livni2013vanishing] finds the directions on which the projections vanish so it can be thought as swapping the roles of signal and noise in PCA. Fisher LDA maximizes the variability in the class means while minimizing the within class variance. Sliced Inverse Regression first whitens $x$, and then uses the second moment matrix of the conditional whitened means as the signal and, like PCA, has no particular noise model. Finally, oriented PCA [@diamantaras1996principal; @platt2010translingual] is a very general framework in which the noise matrix can be the correlation matrix of any type of noise $z$ meaningful to the task at hand.

By closely examining the signal and noise matrices, it is clear that each method can be further distinguished according to two other capabilities: whether it is possible to extract many directions, and whether the directions are discriminative. For example, PCA and VCA can extract many directions but these are not discriminative. In contrast, Fisher 