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We derive a cost functional for estimating the relationship between highdimensional observations and the low-dimensional process that generated them with no input-output examples. Limiting our search to invertible observation functions confers numerous benefits, including a compact representation and no suboptimal local minima. Our approximation algorithms for optimizing this cost functional are fast and give diagnostic bounds on the quality of their solution. Our method can be viewed as a manifold learning algorithm that utilizes a prior on the low-dimensional manifold coordinates. The benefits of taking advantage of such priors in manifold learning and searching for the inverse observation functions in system identification are demonstrated empirically by learning to track moving targets from raw measurements in a sensor network setting and in an RFID tracking experiment.