Score-Based Variational Inference for Inverse Problems
Zhipeng Xue, Penghao Cai, Xiaojun Yuan, Xiqi Gao
TL;DR
This paper introduces reverse mean propagation (RMP), a novel framework for directly estimating the posterior mean in inverse problems using score-based diffusion, leading to improved reconstruction accuracy and efficiency.
Contribution
The paper develops RMP, a new method that directly targets the posterior mean in diffusion-based inverse problem solutions, with a theoretical foundation and practical algorithm.
Findings
RMP accurately estimates the posterior mean in inverse problems.
The proposed algorithm outperforms state-of-the-art methods in reconstruction quality.
RMP achieves lower computational complexity while maintaining high performance.
Abstract
Existing diffusion-based methods for inverse problems sample from the posterior using score functions and accept the generated random samples as solutions. In applications that posterior mean is preferred, we have to generate multiple samples from the posterior which is time-consuming. In this work, by analyzing the probability density evolution of the conditional reverse diffusion process, we prove that the posterior mean can be achieved by tracking the mean of each reverse diffusion step. Based on that, we establish a framework termed reverse mean propagation (RMP) that targets the posterior mean directly. We show that RMP can be implemented by solving a variational inference problem, which can be further decomposed as minimizing a reverse KL divergence at each reverse step. We further develop an algorithm that optimizes the reverse KL divergence with natural gradient descent using…
Peer Reviews
Decision·Submitted to ICLR 2025
The topic of solving inverse problems with score-based models is interesting.
* The posterior mean is not the only relevant quantity; access to samples from the posterior distribution also enables uncertainty quantification. This work provides only an approximation to the mean, sacrificing sample generation for computational efficiency, which may not even be fully realized unless unjustified approximations are made (see more below). * While RMP aims to reduce complexity relative to sampling methods, it still requires nested optimization loops and numerous neura
The paper focuses on predicting $\mathbb{E}({\bf x} _0|{\bf y})$ rather than sampling from the distribution $p({\bf x} _0|{\bf y})$, which sets it apart from most existing works. The Gaussian VI approach for learning the reverse conditional mean is innovative, and the experimental results presented are also compelling. The writing is clear, and the mathematical derivations are solid. Overall, this paper is of high quality and meets the standards of the ICLR conference.
The introduction section would benefit from a concise summary of the main contributions. Additionally, a detailed literature review of existing methods for inverse problems, particularly those related to VI, should be included to highlight the novelty of this approach. In the experiments, the proposed method RMP is compared with DPS, MCG, DDRM, and $\Pi$GDM. While RMP demonstrates strong performance against these baselines, it is important to note that most of them (DPS, MCG, and $\Pi$GDM) do n
S1: The paper proposes a novel framework, Reverse Mean Propagation, which significantly reduces the complexity of estimating the posterior mean compared to other methods that rely on generating samples from the posterior distribution. S2: The details of each component of the RMP algorithm are well presented, including the reverse mean updates and estimation using stochastic natural gradient descent. The discussion on estimating the trace of the Hessian matrix $\nabla^2_{x_k} \log p_k$ is also c
W1: In line 148, the authors discuss the variance preserving (VP) diffusion model and the variance exploding (VE) diffusion model, mentioning their different training approaches. It would be helpful to explain that the VP scheme, like the VE scheme, is also equivalent to learning the score functions of perturbed data distributions[1]. Clarifying this point could provide more comprehensive background information. W2: In Table 2, the FID results appear inconsistent with the previously reported re
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Taxonomy
TopicsStatistical Methods and Inference
MethodsNatural Gradient Descent · Diffusion · Variational Inference