This paper introduces a class of mixed-integer formulations for trained ReLU neural networks.

Neural Information Processing Systems http://nips.cc/

It is commonplace to encounter heterogeneous data, of which some aspects of the data distribution may vary but the underlying causal mechanisms remain constant. When data are divided into distinct environments according to the heterogeneity, recent invariant learning methods have proposed to learn robust and invariant models using this environment partition.

Nonlinear independent component analysis (ICA) is fundamental in unsupervised learning.

We develop a diffusion-based sampler for target distributions known up to a normalising constant. To this end, we rely on the well-known diffusion path that smoothly interpolates between a (simple) base distribution and the target distribution, widely used in diffusion models. Our approach is based on a practical implementation of diffusion-annealed Langevin Monte Carlo, which approximates the diffusion path with convergence guarantees. We tackle the score estimation problem by developing an efficient sequential Monte Carlo sampler that evolves auxiliary variables from conditional distributions along the path, which provides principled score estimates for time-varying distributions. We further develop novel control variate schedules that minimise the variance of these score estimates. Finally, we provide theoretical guarantees and empirically demonstrate the effectiveness of our method on several synthetic and real-world datasets.

Diffusion models have become a central tool in deep generative modeling, but standard formulations rely on a single network and a single diffusion schedule to transform a simple prior, typically a standard normal distribution, into the target data distribution. As a result, the model must simultaneously represent the global structure of the distribution and its fine-scale local variations, which becomes difficult when these scales are strongly mismatched. This issue arises both in natural images, where coarse manifold-level structure and fine textures coexist, and in low-dimensional distributions with highly concentrated local structure. To address this issue, we propose Residual Prior Diffusion (RPD), a two-stage framework in which a coarse prior model first captures the large-scale structure of the data distribution, and a diffusion model is then trained to represent the residual between the prior and the target data distribution. We formulate RPD as an explicit probabilistic model with a tractable evidence lower bound, whose optimization reduces to the familiar objectives of noise prediction or velocity prediction. We further introduce auxiliary variables that leverage information from the prior model and theoretically analyze how they reduce the difficulty of the prediction problem in RPD. Experiments on synthetic datasets with fine-grained local structure show that standard diffusion models fail to capture local details, whereas RPD accurately captures fine-scale detail while preserving the large-scale structure of the distribution. On natural image generation tasks, RPD achieved generation quality that matched or exceeded that of representative diffusion-based baselines and it maintained strong performance even with a small number of inference steps.