通过倾斜传输利用去噪预言机进行后验采样
Posterior Sampling with Denoising Oracles via Tilted Transport
AI总结：
本文提出倾斜传输技术，利用去噪预言机和对数似然的二次结构将线性逆问题的后验采样转化为更易采样的强对数凹后验，并验证了其在高维模型上的有效性。
摘要：
基于分数的扩散模型通过学习数据集中的去噪预言机（或分数），在各种领域中显著推进了高维数据生成。从贝叶斯的角度来看，它们提供了对数据先验的现实建模，并有助于通过后验采样解决逆问题。尽管最近为此目的开发了许多启发式方法，但它们缺乏许多科学应用中所需的定量保证。在这项工作中，我们引入了倾斜传输技术，该技术利用线性逆问题中对数似然的二次结构，结合先验去噪预言机，将原始后验采样问题转化为一个新的“增强”后验，该后验被证明更容易采样。我们量化了该增强后验强对数凹的条件，突出了对测量矩阵条件数和信噪比的依赖。所得到的后验采样方案被证明能够通过直接分析达到预测用于采样Ising模型的计算阈值[Kunisky'23]，并在高维高斯混合模型和标量场φ⁴模型上得到进一步验证。
英文摘要：
Score-based diffusion models have significantly advanced high-dimensional data generation across various domains, by learning a denoising oracle (or score) from datasets. From a Bayesian perspective, they offer a realistic modeling of data priors and facilitate solving inverse problems through posterior sampling. Although many heuristic methods have been developed recently for this purpose, they lack the quantitative guarantees needed in many scientific applications. In this work, we introduce the \textit{tilted transport} technique, which leverages the quadratic structure of the log-likelihood in linear inverse problems in combination with the prior denoising oracle to transform the original posterior sampling problem into a new `boosted' posterior that is provably easier to sample from. We quantify the conditions under which this boosted posterior is strongly log-concave, highlighting the dependencies on the condition number of the measurement matrix and the signal-to-noise ratio. The resulting posterior sampling scheme is shown to reach the computational threshold predicted for sampling Ising models [Kunisky'23] with a direct analysis, and is further validated on high-dimensional Gaussian mixture models and scalar field $φ^4$ models.
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