MAP IT visualizes representations by taking a fundamentally different approach to dimensionality reduction. MAP IT aligns distributions over discrete marginal probabilities in the input space versus the target space, thus capturing information in wider local regions, as opposed to current methods which align based on pairwise probabilities between states only. The MAP IT theory reveals that alignment based on a projective divergence avoids normalization of weights (to obtain true probabilities) entirely, and further reveals a dual viewpoint via continuous densities and kernel smoothing. MAP IT is shown to produce visualizations which capture class structure better than the current state of the art.