  Even the best scientific equipment can only partially observe reality.
Recorded data is often lower-dimensional, e.g., two-dimensional pictures of the
three-dimensional world. Combining data from multiple experiments then results
in a marginal density. This work shows how to transport such lower-dimensional
marginal densities into a more informative, higher-dimensional joint space by
leveraging time-delayed measurements from an observation process. This can
augment the information from scientific equipment to construct a more coherent
view. Classical transportation algorithms can be used when the source and
target dimensions match. Our approach allows the transport of samples between
spaces of different dimensions by exploiting information from the sample
collection process. We reconstruct the surface of an implant from partial
recordings of bacteria moving on it and construct a joint space for satellites
orbiting the Earth by combining one-dimensional, time-delayed altitude
measurements.
