We perform detailed theoretical analysis of an expectation-maximization-based algorithm recently proposed in for solving a variation of the 3D registration problem, named multi-model 3D registration. Despite having shown superior empirical results, did not theoretically justify the conditions under which the EM approach converges to the ground truth. In this project, we aim to close this gap by establishing such conditions. In particular, the analysis revolves around the usage of probabilistic tail bounds that are developed and applied in various instances throughout the course. The problem studied in this project stands as another example, different from those seen in the course, in which tail-bounds help advance our algorithmic understanding in a probabilistic way. We provide self-contained background materials on 3D Registration

翻译：我们对近期提出的一种基于期望最大化算法进行了详细的理论分析，该算法旨在解决三维配准问题的一个变体——多模型三维配准问题。尽管该算法已展现出卓越的实证结果，但尚未从理论上论证期望最大化方法收敛到真实解的条件。本项目旨在通过建立此类条件来填补这一理论空白。具体而言，分析围绕概率尾界的使用展开，这些尾界在分析过程中被多次推导和应用。本研究探讨的问题作为课程中未涉及的另一个典型案例，展示了尾界如何以概率化方式推进我们对算法的理论理解。我们提供了关于三维配准问题的自包含背景材料。