ESL作业笔记2.1.

本文章基本是对https://stats.stackexchange.com/questions/130998/explanation-of-formula-for-median-closest-point-to-origin-of-n-samples-from-unit 的翻译。
解决问题为Element of Statistical Learning (ESL, https://web.stanford.edu/~hastie/ElemStatLearn/)课后习题2.1.

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题目简介

Suppose each of K-classes has an associated target tk, which is a vector of all zeros, except a one in the kth position. Show that classifying to the largest element of y^ amounts to choosing the closest target, mink||tk−y^||2, if the elements of y^ sum to one.

解题

||tk−y^||2=∑i=1Ky^2i−y^2k+(1−y^k)2=∑i=1Ky^2i+1−2y^kargmink||tk−y^||2=argmin(−2y^k)=argmaxk(y^k)