# WEIRDO: WEak resIdual Regularized DOob's h-transform diffusion alignment

arXiv: https://arxiv.org/abs/2609.39531 · HTML: https://mlsanity.com/paper/2609.39531/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 3 of its 42 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 3 of 3)

- [Understanding Reinforcement Learning-Based Fine-Tuning of Diffusion Models: A Tutorial and Review](https://mlsanity.com/paper/2407.13734/) (2024)
- [Conditional Diffusion Guidance under Hard Constraint: A Stochastic Analysis Approach](https://mlsanity.com/paper/2602.05533/) (2026)
- [Inference-Time Alignment for Diffusion Models via Variationally Stable Doob's Matching](https://mlsanity.com/paper/2601.06514/) (2026)

## Built on by (most cited first, 1 of 1)

- [Radial Tail Repair: Post-Training Correction for Diffusion and Flow Models](https://mlsanity.com/paper/2610.16851/) (2026)

## Abstract (arXiv)

We study the problem of estimating the guidance that steers the distribution learned by a diffusion generative model toward a tilted target q0 propto w,p0 at inference time. Relying on the stochastic optimal control approach, we observe that the exact drift correction is the gradient of the logarithm of Doob's h-function, and we study the problem of estimating it from a sample. In the present paper, we assume that the score of the pretrained model is available, that the tilting weight is bounded and positive, and that the reference distribution has a bounded support, no smoothness of the weight is required. Introducing a penalized least-squares risk in which the penalty is the residual of the space-time harmonicity equation satisfied by the h-function, measured in a dual Sobolev norm, we derive high-probability bounds on the squared error of the resulting guidance estimate. Since the penalty vanishes at the target, the estimator is free of regularization bias, and in favourable scenarios its rate of convergence is faster than the minimax rate of estimating first-order derivatives of a smooth regression function. Assuming that w is bounded and positive with mathbbEp0[w-s] < ∞ for some s ∈ (0,∞], and that the reference data are compactly supported, we prove that the guidance is estimable in squared L2 at rate varepsilonns/(s+4), where varepsilonn = n-2(β-1)/(2(β-1)+d). We also transfer the obtained bounds to the total variation distance between the marginals of the estimated and the exactly guided samplers, and illustrate the performance of the suggested approach with numerical experiments.