<>半监督KMeans

KMeans是无监督的。当然也可以是有监督的。有监督形式非常简单。就是根据labels计算聚类中心即可。相当于无监督KMeans的半步迭代。

本文贡献的是半监督KMeans。半监督KMeans可以充分利用已知的labels信息。在机器学习里,有利于将人类知识和机器从数据发现的知识相互融合。

<>算法

输入点集 D 0 = { ( x i , c i ) } , D 1 = { x i ′ } D_0=\{(x_i,c_i)\}, D_1=\{x_i'\}
D0​={(xi​,ci​)},D1​={xi′​}​
输出分类器(或聚类中心)

令 C 0 = { c i } , C 1 = C ∖ C 0 C_0=\{c_i\}, C_1=C\setminus C_0 C0​={ci​},C1​=C
∖C0​, 下述迭代不改变 γ ( x i ) = c i , ( x i , c i ) ∈ D 0
\gamma(x_i)=c_i,(x_i,c_i)\in D_0γ(xi​)=ci​,(xi​,ci​)∈D0​。

* 根据 D 1 D_1 D1​,初始化聚类中心 { μ c , c ∈ C 1 } \{\mu_c,c\in C_1\} {μc​,c∈C1​}
​​​​​​​​,再根据 D 2 D_2 D2​,初始化聚类中心 { μ c , c ∈ C 1 } \{\mu_c,c\in C_1\} {μc​,c∈C1​
};如下初始化聚类中心 { μ c , c ∈ C 0 } \{\mu_c,c\in C_0\} {μc​,c∈C0​}:
μ c = 1 ♯ { x ∈ D 0 ∣ γ ( x ) = c } ∑ γ ( x ) = c , x ∈ D 0 x , c ∈ C 0 ;
\mu_c=\frac{1}{\sharp\{x\in D_0|\gamma(x)=c\}}\sum_{\gamma(x)=c,x\in D_0}x,c\in
C_0;μc​=♯{x∈D0​∣γ(x)=c}1​γ(x)=c,x∈D0​∑​x,c∈C0​;
* 重置分类结果 γ ( x ) = arg ⁡ min ⁡ c ∈ C ∥ x − μ c ∥ , x ∈ D 1
\gamma(x)=\arg\min_{c\in C} \|x-\mu_c\|,x\in D_1γ(x)=argminc∈C​∥x−μc​∥,x∈D1​;
* 更新中心 μ c = 1 N c ∑ γ ( x ) = c x \mu_c=\frac{1}{N_c}\sum_{\gamma(x)=c}x μc​=
Nc​1​∑γ(x)=c​x;(包括 D 0 , D 1 D_0,D_1 D0​,D1​中的x)
* 重复 2-3 直到收敛;
和无监督的KMeans相比,这里唯一复杂的是初始化。如果 C 0 C_0 C0​不包括所有类别,那么首先给 C 1 C_1 C1​指定聚类中心,如在 D 1
D_1D1​中随机选择,然后 D 0 D_0 D0​中每个类的中心作为 C 0 C_0 C0​的聚类中心(退化为一个有监督的分类算法)。

<>代码
#!/usr/bin/env python import numpy as np from sklearn.base import BaseEstimator
, ClassifierMixin from sklearn.cluster import KMeans, kmeans_plusplus from
sklearn.metrics.pairwise import euclidean_distances from sklearn.model_selection
import train_test_split from sklearn import datasets digists = datasets.
load_digits() X_train, X_test, y_train, y_test = train_test_split(digists.data,
digists.target, test_size=0.5) X_train0, X_train1, y_train0, _ =
train_test_split(X_train, y_train, test_size=0.95) class SupervisedKMeans(
ClassifierMixin, KMeans): def fit(self, X, y): self.classes = np.unique(y) self.
centers_= np.array([np.mean(X[y==c], axis=0) for c in self.classes]) self.
cluster_centers_= self.centers_ return self def predict(self, X): ed =
euclidean_distances(X, self.cluster_centers_) return [self.classes[k] for k in
np.argmin(ed, axis=1)] def score(self, X, y): y_ = self.predict(X) return np.
mean(y == y_) class SemiKMeans(SupervisedKMeans): def fit(self, X0, y0, X1):
"""To fit the semisupervised model Args: X0 (array): input variables with
labels y0 (array): labels X1 (array): input variables without labels Returns:
the model """ classes0 = np.unique(y0) classes1 = np.setdiff1d(np.arange(self.
n_clusters), classes0) self.classes = np.concatenate((classes0, classes1)) X =
np.row_stack((X0, X1)) n1 = len(classes1) mu0 = SupervisedKMeans().fit(X0, y0).
centers_if n1: centers, indices = kmeans_plusplus(X1, n_clusters=n1) self.
cluster_centers_= np.row_stack((centers, mu0)) else: self.cluster_centers_ = mu0
for _ in range(30): ED = euclidean_distances(X1, self.cluster_centers_) y1 = [
self.classes[k] for k in np.argmin(ED, axis=1)] y = np.concatenate((y0, y1))
self.cluster_centers_ = np.array([np.mean(X[y==c], axis=0) for c in self.classes
]) return self if __name__ == '__main__': km = SemiKMeans(n_clusters=10) km.fit(
X_train0, y_train0, X_train1) # y_test0 is unknown skm = SupervisedKMeans(
n_clusters=10) skm.fit(X_train0, y_train0) print(f""" # clusters: 10 # samples:
{X_train0.shape[0]} + {X_train1.shape[0]} SemiKMeans: {km.score(X_test, y_test)}
SupervisedKMeans:{skm.score(X_test, y_test)} """) # clusters: 10 # samples: 44
+ 854 SemiKMeans: 0.7975528364849833 SupervisedKMeans: 0.7675194660734149

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