Convex Clustering(凸聚类)

原文链接：What is convex clustering？

1.解决问题：

把n个p维观察点 x1,x2,...xn∈Rnx_1, x_2, ... x_n \in{\Bbb R^n}x1,x2,...xn∈Rn 划分到k个簇中，每个簇中的相似度，要比簇间相似度高。

2.Convex Clustering的解决方案：

Instead of directly assigning each observation to a cluster, it assigns each observation to a point called the “cluster centroid”. Two observations are then said to belong to the same cluster if they share the same cluster centroid.
简而言之：convex clustering是把每一个观察点划分到一个簇中心中。
Convex clustering computes the cluster centroids by solving a minimizing a convex function, hence its name.
如何计算簇中心？通过最小化convex funtion实现，也就是convex名字的由来。

3. 两个优点

有全局最优点，有完备的理论支持
聚类的结果k可以任意指定，从n到1都可以
（Convex clustering can give not just a single clustering solution but an entire path of clustering solutions, ranging from each observation being in its own cluster, to all observations belonging to a single cluster. The user can choose how many clusters they want to see, and the algorithm can give a solution that closely matches that. ）

4.数学实现

其中，uju_juj 是聚类中心cluster centroid，xjx_jxj 是观察点，q是常数项，λ\lambdaλ 是可调超参数，
wijw_{ij}wij 是pair-specific weighs

The first term encourages the cluster centroids to be near to their assigned points, while the second term encourages the cluster centroids to be near to each other.
第一项，衡量观察点和聚类中心的距离，第二项考虑的是距离中心之间的距离
Let’s look at what happens for extreme values of \lambda to see this. When \lambda = 0, there is no penalty at all for cluster centroids being far from each other. Hence, the optimal solution is uj=xju_j = x_juj=xj for all j, i.e. each point is in its own cluster. On the other hand, when λ=∞\lambda = \inftyλ=∞, any difference in cluster centroids is penalized infinitely. This means that the optimal solution is ui=uju_i = u_jui=uj for all i and j, i.e. all points are in the same cluster.
λ\lambdaλ 为0，所以点各自为一类；反之λ\lambdaλ 正无穷则所有点都是一类。