Chin-Wei Huang

One way to design such maps begins with the Knothe-Rosenblatt rearrangement. Its triangular structure reduces multivariate transport to a sequence of monotone one-dimensional transformations and gives a tractable triangular Jacobian. I used this structure in Neural Autoregressive Flows, replacing affine transformations with expressive monotone neural networks, and proved the first universal approximation theorem of normalizing flows.

A distinct route comes from Brenier's theorem: the optimal transport map is the gradient of a convex potential. Convex Potential Flows use this structure to obtain invertibility through strong convexity, inversion through convex conjugacy, and scalable likelihood gradients through convex optimization.

Augmentation is orthogonal to either choice of transport map. By introducing auxiliary variables, an invertible model can define a more flexible marginal distribution while retaining a tractable variational likelihood bound. The resulting position-auxiliary-variable construction is analogous to phase-space dynamics, and I showed that Augmented Normalizing Flows can approximate a Hamiltonian ODE as a universal transport map.

This connection has since been used in molecular modeling. Timewarp treats velocity-like variables as non-physical auxiliaries in a learned time-coarsened dynamics model, while scalable Boltzmann generators combine augmented coupling flows with graph neural networks for equilibrium sampling and free-energy calculations in large materials systems.

Score-based models formulated through stochastic differential equations emerged as a promising approach to generative modeling, but initially lacked a likelihood-based theoretical treatment. I derived a continuous-time variational framework that unifies diffusion models with hierarchical variational autoencoders and continuous normalizing flows. Within this framework, I proved that minimizing the score-matching loss is equivalent to maximizing a lower bound on the likelihood of the learned reverse SDE. This result applies to a family of reverse processes whose deterministic limit is the probability-flow ODE.

We then extended this variational framework to general Riemannian manifolds, proving its equivalence to Riemannian score matching and deriving the associated probability-flow ODE. We also developed scalable methods for estimating Riemannian divergence without the closest-point projections required by earlier approaches. The resulting models apply across spheres, tori, hyperbolic spaces, and rotation groups, including protein and RNA backbone angles.