论文标题
细线：总平方线拟合作为QCQP优化
A Fine Line: Total Least-Squares Line Fitting as QCQP Optimization
论文作者
论文摘要
本说明使用总体最小二乘（TLS）线路拟合问题作为探索一些现代优化工具的画布。该贡献本质上是教程。 TLS问题与机器人技术和计算机视觉中的重要问题具有很大的数学相似性，但更易于可视化和理解。我们演示了如何将此问题转变为二次限制的二次程序（QCQP），以便可以将其作为本本特征问题或半明确程序（SDP）施放。然后，我们转向更具挑战性的情况，在这种情况下，Geman-McClure成本函数和M估计用于拒绝离群数据点。使用Black-rangarajan二元性，我们表明它也可以作为QCQP施加并以SDP的形式解决。但是，有了大量数据，SDP可能会很慢，因此我们展示了如何为更快的方法（例如迭代重新加权最小二乘（IRLS））构建最佳证书。
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).
