Automatic Local Inverse Calculation for Change of Variables

Inversion is a fundamental operation that arises frequently in probabilistic inference and computer graphics. For example, inversion is used to decrease variance and to enable differentiation in variational inference (e.g., reparameterization trick) and in differentiable rendering (e.g., to integrate over object boundaries). Existing approaches to inversion limit the class of functions inverted, for example, to affine functions, or require a user-specified inverse. We study when a local inverse—an inverse that is valid in a neighborhood of a point—exists. We provide an algorithm to approximate the local inverse and give the convergence rate of the solver. We present LIN, a system that automatically computes the local inverse of a function using a fixed-point solver. We implement LIN in Python and use it to automatically compute the local inverse of affine, polar, and hyperbolic changes of variables arising in image stylization.