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We investigate the problem of estimating a smooth invertible transformation fwhen observing independent samples X_1, ..., X_n ~ P \circ f, where P is aknown measure. We focus on the two dimensional case where P and f are definedon R^2. We present a flexible class of smooth invertible transformations in twodimensions with variational equations for optimizing over the classes, thenstudy the problem of estimating the transformation f by penalized maximumlikelihood estimation. We apply our methodology to the case when P \circ f hasa density with respect to Lebesgue measure on R^2 and demonstrate improvementsover kernel density estimation on three examples.

Two Dimensional Density Estimation using Smooth Invertible Transformations