A noise-corrected Langevin algorithm and sampling by half-denoising
<|reference_start|>A noise-corrected Langevin algorithm and sampling by half-denoising: The Langevin algorithm is a classic method for sampling from a given pdf in a real space. In its basic version, it only requires knowledge of the gradient of the log-density, also called the score function. However, in deep learning, it is often easier to learn the so-called "noisy score function", i.e. the gradient of the log-density of noisy data, more precisely when Gaussian noise is added to the data. Such an estimate is biased and complicates the use of the Langevin method. Here, we propose a noise-corrected version of the Langevin algorithm, where the bias due to noisy data is removed, at least regarding first-order terms. Unlike diffusion models, our algorithm needs to know the noisy score function for one single noise level only. We further propose a simple special case which has an interesting intuitive interpretation of iteratively adding noise the data and then attempting to remove half of that noise.<|reference_end|>
@article{hyvärinen2024a,
    title={A noise-corrected Langevin algorithm and sampling by half-denoising},
    author={Aapo Hyv"arinen},
    journal={arXiv preprint arXiv:2410.05837},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.05837},
    primaryClass={cs.LG stat.ML}
}