用近似算子加速贝叶斯反问题采样，显著提升计算效率。
Latent-IMH: Efficient Bayesian Inference for Inverse Problems with Approximate Operators
- 先用低成本近似算子生成潜在变量，再用精确算子修正。
- 在多个模型上比NUTS快数个数量级，且理论保证收敛性。
- 适合高维反问题中计算昂贵的观测算子场景。
我们研究贝叶斯线性反问题中后验分布的采样，其中将参数映射到可观测值的算子 $A$ 计算成本高昂。许多应用中，$A$ 可分解为便于构造低成本近似 $ ilde{A}$ 的形式。本文提出基于独立梅特罗波利斯-哈斯廷斯（IMH）采样的 Latent-IMH 方法：首先利用近似算子 $ ilde{A}$ 生成中间潜变量，再通过精确算子 $A$ 进行精修。其主要优势在于将大部分计算开销转移至离线阶段。我们通过 KL 散度与混合时间界对方法性能进行理论分析。在多个模型问题上的数值实验表明，在合理假设下，该方法在计算效率上优于当前最优的 No-U-Turn Sampler（NUTS），某些情况下速度提升达数个数量级。
原文摘要 · Abstract (English)
We study sampling from posterior distributions in Bayesian linear inverse problems where $A$, the parameters to observables operator, is computationally expensive. In many applications, $A$ can be factored in a manner that facilitates the construction of a cost-effective approximation $\tilde{A}$. In this framework, we introduce Latent-IMH, a sampling method based on the Metropolis-Hastings independence (IMH) sampler. Latent-IMH first generates intermediate latent variables using the approximate $\tilde{A}$, and then refines them using the exact $A$. Its primary benefit is that it shifts the computational cost to an offline phase. We theoretically analyze the performance of Latent-IMH using KL divergence and mixing time bounds. Using numerical experiments on several model problems, we show that, under reasonable assumptions, it outperforms state-of-the-art methods such as the No-U-Turn sampler (NUTS) in computational efficiency. In some cases, Latent-IMH can be orders of magnitude faster than existing schemes.
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