标题： 高维Bayes最优推理与M估计之间的等价性 显示英文标题

标题： An equivalence between high dimensional Bayes optimal inference and M-estimation

摘要： 当从带有噪声的测量值中恢复未知信号时，执行最优贝叶斯MMSE（最小均方误差）推断的计算难度通常需要使用最大后验概率（MAP）推断，这是正则化M估计的一个特例，作为替代方法。 然而，在高维情况下，当未知信号分量的数量与测量值的数量相近时，MAP是次优的。 在这项工作中，我们证明了当信号分布和与噪声相关的似然函数均为对数凹函数时，可以通过另一种M估计过程渐近实现最优的MMSE性能。 该过程涉及最小化凸损失函数和正则化函数，这些函数是非线性平滑版本的广泛使用的MAP优化问题。 我们的发现为线性测量和加性噪声条件下的最近找到的最优M估计器提供了新的启发式推导和解释，并进一步将这些结果扩展到非线性测量和非加性噪声的情况。 我们通过数值演示了我们的最优M估计器相对于MAP的优越性能。 总体而言，我们工作的核心在于揭示了两个看似非常不同的计算问题之间的惊人等价性：即支撑MMSE推断的高维贝叶斯积分问题和支撑M估计的高维凸优化问题。 本质上，我们展示了前者的困难积分可以通过解决后者更简单的优化问题来计算。 显示英文摘要

摘要： When recovering an unknown signal from noisy measurements, the computational difficulty of performing optimal Bayesian MMSE (minimum mean squared error) inference often necessitates the use of maximum a posteriori (MAP) inference, a special case of regularized M-estimation, as a surrogate. However, MAP is suboptimal in high dimensions, when the number of unknown signal components is similar to the number of measurements. In this work we demonstrate, when the signal distribution and the likelihood function associated with the noise are both log-concave, that optimal MMSE performance is asymptotically achievable via another M-estimation procedure. This procedure involves minimizing convex loss and regularizer functions that are nonlinearly smoothed versions of the widely applied MAP optimization problem. Our findings provide a new heuristic derivation and interpretation for recent optimal M-estimators found in the setting of linear measurements and additive noise, and further extend these results to nonlinear measurements with non-additive noise. We numerically demonstrate superior performance of our optimal M-estimators relative to MAP. Overall, at the heart of our work is the revelation of a remarkable equivalence between two seemingly very different computational problems: namely that of high dimensional Bayesian integration underlying MMSE inference, and high dimensional convex optimization underlying M-estimation. In essence we show that the former difficult integral may be computed by solving the latter, simpler optimization problem.