Implicit Neural Optimal Transport via Fixed-Point Optimization
Authors: Yesom Park, Eric Gelphman, Stanley Osher, Samy Wu Fung
Organizations: Department of Mathematics, University of California, Los Angeles · Department of Applied Mathematics and Statistics, Colorado School of Mines
We propose an implicit neural formulation of optimal transport that eliminates adversarial min--max optimization and multi-network architectures commonly used in existing approaches. Our key idea is to parameterize a single potential in the Kantorovich dual and reformulate the associated c-transform as a proximal fixed-point problem. This yields a stable single-network framework in which dual feasibility is enforced exactly through proximal optimality conditions rather than adversarial training. Despite the inner fixed-point computation, gradients can be computed without differentiating through the fixed-point iterations, enabling efficient training without requiring implicit differentiation. We further establish convergence of stochastic gradient descent. The resulting framework is efficient, scalable, and broadly applicable: it simultaneously recovers forward and backward transport maps and naturally extends to class-conditional settings. Experiments on high-dimensional Gaussian benchmarks, physical datasets, and image translation tasks demonstrate strong transport accuracy together with improved training stability and favorable computational and memory efficiency.
This paper shows that the semi-dual formulation of the optimal transport problem has a degenerate saddle-point structure, and that its numerical solution is equivalent to solving a constrained optimization problem. We derive necessary and sufficient conditions for the convergence of Monge maps without requiring optimality of the dual potential. This analysis helps explain why, in practice, numerical algorithms often require more iterations to update the transport map than the potential.
Hard thresholds, quantization, and discrete routing can produce training losses with flat regions and jumps, where ordinary gradients vanish or are undefined. We introduce PolyStep, a forward-only optimizer that evaluates rotated polytope probes and moves parameter blocks along weighted averages of the probe directions. We derive the weights from one-sided entropic transport and use its uncoupled softmax solution in our primary experiments. Our analysis explains when variation among probe costs produces motion and when that motion decreases the loss. On a regular simplex, nonconstant costs always give a nonzero direction. For monotone ridge losses, the softmax update cannot increase the loss at any positive temperature; a perturbation bound gives sufficient conditions for descent near curved jumps. For bounded measurable losses, we randomize the probe radii and identify an exact smoothing whose gradient equals the expected linear cost-weighted direction up to scale. This identity yields a stationarity bound for an idealized fixed-temperature variant: under regularity and sampling assumptions stronger than those met by our trained configurations, the bound has an O(T−1/2) term and a persistent bias floor. We evaluate the practical method on networks with hard operations, discrete optimization, and policy search. On MNIST with hard-threshold spiking neurons, PolyStep reaches 93.0±0.2%, compared with 79.6±5.2% for the best-tuned gradient-free baseline at matched evaluations. These gains come with a query cost proportional to the search dimension per fresh step, which limits the number of updates available at a fixed budget.
Optimal transport (OT) provides a principled framework for mapping between probability distributions. Despite extensive progress, applying OT to large-scale data remains computationally demanding, and the resulting pointwise transport plans are often difficult to interpret. We introduce Optimal Mixture Transport (OMT), a scalable framework that shifts the transport paradigm from individual samples to mixtures of subpopulations, reformulating the transport problem as a strictly biconvex optimization with a unique global minimizer. We further establish theoretical guarantees on the stability of the OMT map, showing that bounded perturbations of the underlying distributions lead to bounded changes in the transport plan. By formulating subpopulations as exponential-family distributions, OMT decouples computational complexity from the sample size, scaling solely with the number of mixture components. We demonstrate the effectiveness and practicality of OMT on a wide range of synthetic benchmarks and real-world datasets, including image data and large-scale single-cell RNA sequencing measurements.