标题： 一条细线：总最小二乘线拟合作为QCQP优化 显示英文标题

标题： A Fine Line: Total Least-Squares Line Fitting as QCQP Optimization

摘要： 本说明使用总最小二乘（TLS）线拟合问题作为画布，以探索一些现代优化工具。 贡献旨在具有教学性质。 TLS问题与机器人技术和计算机视觉中的重要问题有很多数学相似之处，但更容易可视化和理解。 我们演示如何将这个问题转化为二次约束二次规划（QCQP），以便可以将其表示为特征值问题或半定规划（SDP）。 然后我们转向更具有挑战性的情况，其中使用Geman-McClure代价函数和M估计来拒绝异常数据点。 使用Black-Rangarajan对偶性，我们表明这也可以表示为QCQP并作为SDP求解；然而，当数据量很大时，SDP可能会很慢，因此我们展示了如何为更快的方法（如迭代加权最小二乘法（IRLS））构建最优性证书。 显示英文摘要

摘要： This note uses the Total Least-Squares (TLS) line-fitting problem as a canvas to explore some modern optimization tools. The contribution is meant to be tutorial in nature. The TLS problem has a lot of mathematical similarities to important problems in robotics and computer vision but is easier to visualize and understand. We demonstrate how to turn this problem into a Quadratically Constrained Quadratic Program (QCQP) so that it can be cast either as an eigenproblem or a Semi-Definite Program (SDP). We then turn to the more challenging situation where a Geman-McClure cost function and M-estimation are used to reject outlier datapoints. Using Black-Rangarajan duality, we show this can also be cast as a QCQP and solved as an SDP; however, with a lot of data the SDP can be slow and as such we show how we can construct a certificate of optimality for a faster method such as Iteratively Reweighted Least-Squares (IRLS).