Projected Energy Matching for Generative 3D Priors

Energy Matching has emerged as a powerful generative framework that combines flow model efficiency with the explicit likelihood of Energy-Based Models (EBMs) via a single, time-independent scalar potential.

However, directly training this potential on high-dimensional 3D data remains computationally challenging.

While distilling a pre-trained flow model circumvents some of the initial training costs, we demonstrate that velocity fields inevitably contain non-conservative rotational artifacts (curl).

Forcing a strictly conservative scalar potential to match this unconstrained field creates a "structural conflict", which degrades generation quality and mode coverage.

To solve this, we propose Projected Energy Matching, a scalable framework that resolves these structural and computational bottlenecks.

We introduce Helmholtz Distillation, a structural relaxation that leverages a Hutchinson trace estimator to explicitly absorb rotational noise into an auxiliary residual network.

We subsequently refine this landscape using Negative Caching, a memory-efficient strategy that reuses negative samples across micro-batches, rendering sampling tractable during contrastive training with gradient accumulation.

We deploy our method as an unconditional prior for real-world medical CT inverse problems, specifically sparse-view reconstruction.

Ultimately, our amortized pipeline reduces total compute to a small fraction of that required by standard energy matching, while achieving high-fidelity reconstructions and successfully resolving severe measurement artifacts.