Fusion and inference from multiple and massive disparate data sources—the requirement for our most challenging data analysis problems and the goal of our most ambitious statistical pattern recognition methodologies—has many and varied aspects which are currently the target of intense research and development. One aspect of the overall challenge is manifold matching—identifying embeddings of multiple disparate data spaces into the same low-dimensional space where joint inference can be pursued. We investigate this manifold matching task from the perspective of jointly optimizing the fidelity of the embeddings and their commensurability with one another, with a specific statistical inference exploitation task in mind. Our results demonstrate when and why our joint optimization methodology is superior to either version of separate optimization. The methodology is illustrated with simulations and an application in document matching.