Entropic Riemannian Neural Optimal Transport

Many machine learning problems involve data supported on curved spaces such as spheres, rotation groups, hyperbolic spaces, and general Riemannian manifolds, where Euclidean geometry can distort distances, averages, and the resulting optimal transport (OT) problem. Existing manifold OT methods have pursued amortized out-of-sample maps, while entropic regularization has made discrete OT more scalable, but these advantages have remained largely disjoint. We propose Entropic Riemannian Neural Optimal Transport (Entropic RNOT), a unified framework that combines intrinsic entropic OT with amortized out-of-sample evaluation on Riemannian manifolds. Our method learns a single target-side Schrödinger potential through a neural pullback parameterization, recovers the induced Gibbs coupling, and uses the resulting conditional laws to construct intrinsic transport surrogates. These include barycentric projections on Cartan-Hadamard manifolds and heat-smoothed conditional surrogates on stochastically complete manifolds, the latter turning possibly atomic target laws into absolutely continuous ones. For fixed regularization $\varepsilon>0$, we prove that the proposed hypothesis class recovers the entropic optimal coupling in strong probabilistic metrics. As consequences, barycentric surrogates converge in $L^2$, while heat-smoothed surrogates are stable at fixed heat time and asymptotically unbiased as the heat time vanishes. The guarantees hold for compactly supported data on possibly noncompact manifolds. Empirically, our method matches or improves over Euclidean, tangent-space, and log-Euclidean baselines on benchmarks over $\mathbb{S}^2$, $\mathrm{SO}(3)$, $\mathrm{SPD}(3)$, $\mathrm{SE}(3)$, and $\mathbb{H}^2$, scales favorably relative to discrete manifold Sinkhorn, and in a protein-ligand docking application, refines poses on $\mathrm{SE}(3)$ without retraining or per-instance optimization.

Riemannian Neural Optimal Transport

Computational optimal transport (OT) offers a principled framework for generative modeling. Neural OT methods, which use neural networks to learn an OT map (or potential) from data in an amortized way, can be evaluated out of sample after training, but existing approaches are tailored to Euclidean geometry. Extending neural OT to high-dimensional Riemannian manifolds remains an open challenge. In this paper, we prove that any method for OT on manifolds that produces discrete approximations of transport maps necessarily suffers from the curse of dimensionality: achieving a fixed accuracy requires a number of parameters that grows exponentially with the manifold dimension. Motivated by this limitation, we introduce Riemannian Neural OT (RNOT) maps, which are continuous neural-network parameterizations of OT maps on manifolds that avoid discretization and incorporate geometric structure by construction. Under mild regularity assumptions, we prove that RNOT maps approximate Riemannian OT maps with sub-exponential complexity in the dimension. Experiments on synthetic and real datasets demonstrate improved scalability and competitive performance relative to discretization-based baselines.

NeuralSurv: Deep Survival Analysis with Bayesian Uncertainty Quantification

We introduce NeuralSurv, the first deep survival model to incorporate Bayesian uncertainty quantification. Our non-parametric, architecture-agnostic framework flexibly captures time-varying covariate-risk relationships in continuous time via a novel two-stage data-augmentation scheme, for which we establish theoretical guarantees. For efficient posterior inference, we introduce a mean-field variational algorithm with coordinate-ascent updates that scale linearly in model size. By locally linearizing the Bayesian neural network, we obtain full conjugacy and derive all coordinate updates in closed form. In experiments, NeuralSurv delivers superior calibration compared to state-of-the-art deep survival models, while matching or exceeding their discriminative performance across both synthetic benchmarks and real-world datasets. Our results demonstrate the value of Bayesian principles in data-scarce regimes by enhancing model calibration and providing robust, well-calibrated uncertainty estimates for the survival function.

Diffusion Models for Inverse Problems in the Exponential Family

Diffusion models have emerged as powerful tools for solving inverse problems, yet prior work has primarily focused on observations with Gaussian measurement noise, restricting their use in real-world scenarios. This limitation persists due to the intractability of the likelihood score, which until now has only been approximated in the simpler case of Gaussian likelihoods. In this work, we extend diffusion models to handle inverse problems where the observations follow a distribution from the exponential family, such as a Poisson or a Binomial distribution. By leveraging the conjugacy properties of exponential family distributions, we introduce the evidence trick, a method that provides a tractable approximation to the likelihood score. In our experiments, we demonstrate that our methodology effectively performs Bayesian inference on spatially inhomogeneous Poisson processes with intensities as intricate as ImageNet images. Furthermore, we demonstrate the real-world impact of our methodology by showing that it performs competitively with the current state-of-the-art in predicting malaria prevalence estimates in Sub-Saharan Africa.