Is K-means some kind of EM algrithom?

[JiaRu2016]

[JiaRu2016]

https://stats.stackexchange.com/questions/76866/clustering-with-k-means-and-em-how-are-they-related

summary of @Anony-mousse answer:

1. K-means is not an "algorithm", is a question. We solve it using the Lloyd/Forgy algorithm for k-means, the Hartigan-Wong method etc.
2. there is also no "EM algorithm", it's a general scheme of repeatedly expecting the likelyhoods and maximizing the model

Recall LiHang's book
K-means problem: choose the K centroid to minimize inertia sum of Euclidean distance of sample point to its center
$$ \min_{c1, c2, ..., ck} \sum | x_i - c_{x_i} |^2 $$

We use Lloyd’s algorithm to solve it. It's "heuristic" algorithm 1. "hard" assign each sample point to one centroid and 2. update centroid.
Local optima, depend on choice of initial centroid

EM
E step:
$$ Q(\theta, \theta^{(t)} ) = E_{Z|Y, \theta} [ \log P(Y,Z | \theta^{(t)} ) ] $$

M step: maximize $Q(\theta, \theta^{(t)})$

最重要的区别：EM是对隐变量（属于哪个中心）求期望，而 K-means Lloyd's Algrithom 直接以概率1指定到某个中心。

[JiaRu2016]

知乎：请问如何用数学方法证明K-means是EM算法的特例

主要参考思路，实际未验证。
为原始的k-means聚类问题添加一个”生成模型“，再去证明：该生成模型的EM算法解 == 原始k-means的启发式算法解