Generator-Aware Data Acquisition for Budgeted Distribution Recovery

We study high-dimensional distribution recovery under a limited measurement budget. Target individuals are sampled from an unknown target distribution, and for each sampled individual only one costly subset of variables can be measured. A pretrained generator is available for a fully observed reference population, and the target is modeled as a low-dimensional exponential tilt of the reference law. We propose Generator-Aware Measurement Design (GAMD), which uses the generator to decide which variable subsets are most informative about how the target population differs from the reference. For each candidate subset, GAMD evaluates how much of the full target-shift signal can be recovered from the variables that would be observed; this quantity determines the Fisher information carried by that measurement. GAMD then combines these information values with measurement costs to allocate the budget and refines the target distribution from the acquired partial data. We prove that GAMD asymptotically matches the oracle design despite the unknown target shift and establish finite-budget guarantees. Synthetic and real-data experiments validate the proposed mechanism and demonstrate improved recovery under constrained measurements.