一个正则化器能否在两种噪声水平下重现后验均值？
Can one regularizer reproduce posterior means at two noise levels?
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中文总结 AI 辅助
本文证明对于紧支撑先验，除点质量外，不存在单一固定正则化器能在两个不同噪声方差下同时重现后验均值，并给出误差下界与数值示例。
中文摘要 AI 辅助
考虑具有固定先验和变化噪声方差的 Gaussian 去噪问题。当先验具有密度时，最大后验估计使用其负对数作为固定正则化器；只有二次数据项的权重随噪声方差变化。我们询问固定正则化器是否也能恢复后验均值。尽管在每个方差下都存在合适的正则化器，但我们证明对于任何紧支撑的先验（点质量除外），没有任何单一正则化器能在两个不同的正方差下同时有效。这一结论甚至对非凸和非光滑的正则化器也成立。由两次去噪器评估计算出的残差，为使用共同正则化器的估计器的均匀逼近误差和超额均方误差提供了正的下界。数值示例说明了这些界限以及跨噪声水平重用正则化器的代价。
英文摘要
Consider Gaussian denoising with a fixed prior and varying noise variance. When the prior has a density, maximum a posteriori estimation uses its negative logarithm as a fixed regularizer; only the weight of the quadratic data term changes with the noise variance. We ask whether a fixed regularizer can also recover the posterior mean. Although a suitable regularizer exists at each variance, we prove that no single regularizer works at two distinct positive variances for any compactly supported prior other than a point mass. This holds even for nonconvex and nonsmooth regularizers. A residual computed from two denoiser evaluations gives positive lower bounds on the uniform approximation error and excess mean squared error of estimators using a common regularizer. Numerical examples illustrate these bounds and the cost of reusing a regularizer across noise levels.
发表机构
- School of Future Technology, Xi’an Jiaotong University（西安交通大学未来技术学院）
机构由 AI 辅助整理，请以论文原文为准。
