Critical points via monodromy and local methods

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• 作者：Abraham Martí ; n del Campo^a ; ¹ ; ^{abraham.mc@cimat.mx" class="auth_mail" title="E-mail the corresponding author} ; Jose Israel Rodriguez^b ; ¹ ; ² ; ^{JoIsRo@UChicago.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Critical points ; Monodromy ; Homotopy continuation ; Gradient descent homotopy ; Maximum likelihood degree ; Euclidean distance ; Trace test
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P3
• 页码：559-574
• 全文大小：719 K

文摘

A fundamental problem in many areas of applied mathematics and statistics is to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the function restricted to the model.

We compile ideas arising from numerical algebraic geometry to compute these critical points. Our method consists of using numerical homotopy continuation and a monodromy action on the total critical space to compute all of the complex critical points. To illustrate the relevance of our method, we apply it to the Euclidean distance function to compute ED-degrees and the likelihood function to compute maximum likelihood degrees.