Generative Modeling by Value-Driven Transport

Generative Modeling by Value-Driven Transport reframes sample generation as a discrete-time stochastic control problem for Wasserstein-2 optimal transport, then solves the resulting constrained linear program with a primal-dual algorithm that learns a scalar value function rather than a transport policy directly. The key novelty is the value-driven transport (VDT) policy, π_h(x)=x-(H+1)^{-1}∇V_h(x), which is derived from the LP dual and provably recovers the optimal straight-line interpolation paths of dynamic OT; the final-time dual variable also acts as a discriminator enforcing the target distribution. Training is simulation-free: each minibatch is initialized with an approximate coupling, particle clouds are refined by Wasserstein gradient descent, and a single neural network parameterizes the value function, using no learned trajectory simulator. On 2D benchmarks including two moons, s-curve, 8 Gaussians, and moons-to-8-Gaussians, OT-initialized VDT matched or approached state-of-the-art flow and Schrödinger bridge baselines, with 10-step generation essentially matching 100-step generation; for example, on moons-to-8-Gaussians it reduced Wasserstein-2 error from 1.365 to 0.626 when moving from 1 to 10 steps, while preserving path energy near the oracle. On MNIST, the same framework handled paired deblurring, unpaired EMNIST-to-MNIST translation, reverse generation, and classifier-free guidance, showing that OT-style straight-path geometry can support conditional and bidirectional generation within one unified control-based model.