AI 中文总结
研究从噪声数据中精确恢复线性逆问题，核心方法是通过平均和噪声感知自适应布雷格曼 - 卡兹马尔兹方法，包括用块平均代替块和、引入噪声感知加权、自适应步长插值等，证明收敛性与批量大小等的关系，通过实验验证方法有效性。
AI 中文摘要
自适应布雷格曼 - 卡兹马尔兹方法即使在查询的每个测量值都被噪声破坏时，也能恢复线性逆问题的精确无噪声解，前提是噪声是新的、独立的且均值为零。为并行硬件提出了块版本，但更大的块是否实际收敛更快尚待解决。我们表明答案取决于块的使用方式：用块平均代替块和可将分析归结为单个半正定矩阵，由此证明保证收敛性随批量大小单调提高，总增益由系统矩阵的稳定秩决定。然后通过引入噪声感知加权来处理异质噪声，该加权降低不可靠测量的权重，并证明只要噪声与行范数不成比例，它就严格优于均匀加权，而这种情况在实际中基本不成立。这两种改进是兼容的且益处可结合。最后，我们表明自适应步长在快速初始阶段和缓慢消失的尾部之间自动插值，将误差精确归零，并且解释了如何在不知道真实解的情况下估计其超参数。在稀疏异质噪声下的数值实验说明了这些发现，并证实启发式估计产生有效的步长。
英文摘要
The adaptive Bregman-Kaczmarz method recovers the exact, noise-free solution of a linear inverse problem even when every measurement it queries is corrupted, provided the corruption is fresh, independent and zero-mean. A block version was proposed for parallel hardware, but whether larger blocks actually converge faster was left open. We show the answer hinges on how the block is used: replacing the block sum with a block average collapses the analysis onto a single positive-semidefinite matrix, through which we prove that the guaranteed convergence improves monotonically with the batch size, with a total gain governed by the stable rank of the system matrix. We then address heterogeneous noise by introducing a noise-aware weighting that down-weights unreliable measurements, and prove it is strictly better than uniform weighting whenever the noise is not proportional to the row norms - a condition that essentially never holds in practice. The two improvements are compatible and their benefits combine. Finally, we show the adaptive step size interpolates automatically between a fast initial phase and a slowly vanishing tail that carries the error exactly to zero, and we explain how its hyperparameters can be estimated without knowing the true solution. Numerical experiments under sparse, heterogeneous noise illustrate these findings and confirm that the heuristic estimates produce effective step sizes.
Comments33 pages, 6 figures
