Generating DDPM-based Samples from Tilted Distributions

Himadri Mandal; Dhruman Gupta; Rushil Gupta; Sarvesh Ravichandran Iyer; Agniv Bandyopadhyay; Achal Bassamboo; Varun Gupta; Sandeep Juneja

cs.LGApr 3, 2026

Generating DDPM-based Samples from Tilted Distributions

Authors: Himadri Mandal, Dhruman Gupta, Rushil Gupta, Sarvesh Ravichandran Iyer, Agniv Bandyopadhyay, Achal Bassamboo, Varun Gupta, Sandeep Juneja

Organizations: Indian Statistical Institute, Kolkata. · Ashoka University. · Tata Institute of Fundamental Research · Northwestern University ·

Abstract

Given $n$ independent samples from a $d$ -dimensional probability distribution, our aim is to generate diffusion-based samples from a distribution obtained by tilting the original, where the degree of tilt is parametrized by $θ\in \mathbb{R}^d$ . We define a plug-in estimator and show that it is minimax-optimal. We develop Wasserstein bounds between the distribution of the plug-in estimator and the true distribution as a function of $n$ and $θ$ , illustrating regimes where the output and the desired true distribution are close. Further, under some assumptions, we prove the TV-accuracy of running Diffusion on these tilted samples. Our theoretical results are supported by extensive simulations. Applications of our work include finance, weather and climate modelling, and many other domains, where the aim may be to generate samples from a tilted distribution that satisfies practically motivated moment constraints.

Figures & tables

Figure 1. The left plot shows the empirical sliced Wasserstein distance ( $\mathcal{W}_{2}$ ) between the reweighed estimator and the true tilted distribution. The right plot shows the theoretical bound derived in Theorem 2 as a function of sample size $N$ . As the theoretical bound vanishes, the empirical error decreases correspondingly.

Figure 2. Samples generated by twisting a bounded in 50 dimensions by $\vartheta=\theta\cdot(1,\ldots,1)$ , for $\theta$ = 1.0, 2.0, 2.5 using reweighed sampling, weighted diffusion, DPS, and LGD-MC. We see that our method performs as well as the empirical samples, outperforming the guidance methods.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3. DDPM samples from $P$ . Daily temperature fields (May–June, India, $5^{\circ}\!\times\!5^{\circ}$ ).

Figure 4. DDPM samples from $P_{\theta}$ . Reweighted training targets the hotter, rarer slice with $\mathbb{E}_{P_{\theta}}[g]=\mathbb{E}_{P}[g]+1$ .

Diffusion Sampling

Diffusion sampling studies how to turn a trained diffusion model into high-quality samples efficiently and reliably.

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Internal Guidance as a Sampling Method for Tilted Diffusion Models

Curtis McDonald, Peter Bartlett

Simons Institute, University of California Berkeley · Google DeepMind

Diffusion Model Guidance Diffusion Sampling Diffusion Models

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Reward guidance methods sample an exponential tilt of a base density using a diffusion model trained with only samples from the base distribution. Existing methods use what we call external guidance: additive terms in the drift or multiplicative terms on the transition probabilities. We propose internal guidance which leaves the output of the base denoiser unmodified and instead changes the point at which it is evaluated. For discrete distributions embedded in the hypercube

\{-1,1\}^{d}

, the denoiser of any linearly tilted density is exactly the base denoiser evaluated at a shifted and rescaled input. In practice the base denoiser is a neural network trained on noised samples of the base distribution. The required evaluation points for sampling the tilted density would be out-of-distribution for basic noising of the base distribution. We modify the training algorithm with random diagonal rescalings of the data, so that the denoiser network learns both in- and out-of-distribution evaluations of the denoiser relative to the base distribution. A single denoiser network samples binary reward tilts, class labels, and infilling with no fine-tuning and no external guidance term.

Abstract

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