Energy-based models for discrete domains, such as graphs, explicitly capture relative likelihoods, naturally enabling composable probabilistic inference tasks like conditional generation or enforcing constraints at test-time. However, discrete energy-based models typically struggle with efficient and high-quality sampling, as off-support regions often contain spurious local minima, trapping samplers and causing training instabilities. This has historically resulted in a fidelity gap relative to discrete diffusion models. We introduce Graph Energy Matching (GEM), a generative framework for graphs that closes this fidelity gap. Motivated by the transport map optimization perspective of the Jordan-Kinderlehrer-Otto (JKO) scheme, GEM learns a permutation-invariant potential energy that simultaneously provides transport-aligned guidance from noise toward data and refines samples within regions of high data likelihood. Further, we introduce a sampling protocol that leverages an energy-based switch to seamlessly bridge: (i) rapid, gradient-guided transport toward high-probability regions to (ii) a mixing regime for exploration of the learned graph distribution. On molecular graph benchmarks, GEM matches or exceeds strong discrete diffusion baselines. Beyond sample quality, explicit modeling of relative likelihood enables targeted exploration at inference time, facilitating compositional generation, property-constrained sampling, and geodesic interpolation between graphs.

We present a theoretical framework that reinterprets Population Annealing (PA) through the lens of the discrete-time Schrödinger Bridge (SB) problem. We demonstrate that the heuristic reweighting step in PA is derived by analytically solving the Schrödinger system without iterative computation via instantaneous projection. In addition, we identify the thermodynamic work as the optimal control potential that solves the global variational problem on path space. This perspective unifies non-equilibrium thermodynamics with the geometric framework of optimal transport, interpreting the Jarzynski equality as a consistency condition within the Donsker-Varadhan variational principle, and elucidates the thermodynamic optimality of PA.

Specifically, SegRefiner takes coarse masks as inputs and refines them using a discrete diffusion process.

To leverage the complementary merits of both worlds, we propose BetterDepth to achieve geometrically correct affine-invariant MDE while capturing fine details.

Ensuring robust 3D object detection and localization is crucial for many applications in robotics and autonomous driving.

If we consider for example the image pictured in Figure 1 on the left, we can easily describe its content by'what' we see - the building, sky and a flag.

We provide more visualization results in Figure 1, Figure 1, Figure 1, and Figure 1. As mentioned in the limitation section of the main text, our method can generate realistic textures in most regions. However, it may restore incorrect small characters as shown in Figure 1, which is highly ill-posed. Compared with the Uformer, it shows consistent improvements in perceptual quality. Learning to see in the dark. We compare the PSNR-oriented methods and our method.