针对高维变量多仿射关系问题，提出高效收敛的估计算法。
Maximum likelihood inference for high-dimensional problems with multiaffine variable relations
- 设计交替重加权最小二乘算法，利用变量间多仿射结构提升求解效率。
- 在广义正态分布下证明算法收敛，实测具超线性收敛速度。
- 可计算估计方差，适用于图模型等复杂统计推断场景。
高维连续变量模型的最大似然估计因概率分布复杂且变量间存在多重依赖而极具挑战，常需依赖网格搜索、蒙特卡洛采样或特定问题算法。本文研究变量由多仿射表达式关联的推断问题，提出一种新的交替重加权最小二乘（AIRLS）算法，并证明其在广义正态分布下的收敛性。同时提供一种高效计算估计量方差的方法，并展示其在图模型中的应用。数值实验表明，相比现有方法，该算法在可扩展性、抗噪鲁棒性及收敛速度方面均有显著提升，归因于观察到的超线性收敛率。
原文摘要 · Abstract (English)
Maximum Likelihood Estimation of continuous variable models can be very challenging in high dimensions, due to potentially complex probability distributions. The existence of multiple interdependencies among variables can make it very difficult to establish convergence guarantees. This leads to a wide use of brute-force methods, such as grid searching and Monte-Carlo sampling and, when applicable, complex and problem-specific algorithms. In this paper, we consider inference problems where the variables are related by multiaffine expressions. We propose a novel Alternating and Iteratively-Reweighted Least Squares (AIRLS) algorithm, and prove its convergence for problems with Generalized Normal Distributions. We also provide an efficient method to compute the variance of the estimates obtained using AIRLS. Finally, we show how the method can be applied to graphical statistical models. We perform numerical experiments on several inference problems, showing significantly better performance than state-of-the-art approaches in terms of scalability, robustness to noise, and convergence speed due to an empirically observed super-linear convergence rate.
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