半监督KMeans

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；反过来说，KMeans算法是把无监督学习转化成一系列监督学习的迭代过程。

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。

符号约定

• $D_l$ 有标签数据集
• $D_u$ 无标签数据集
• $(x_i,c_i)$ 数据点
• $\gamma$ 分类器/聚类器
• $C_l,C_u$: 类标签集
• ${\mu }_c$: $c$类聚类中心

算法

输入点集$D_l=\{(x_i,c_i)\},D_u=\{x_i^{\prime }\}$​
输出分类器（或聚类中心）

令类标签集$C_l=\{c_i\},C_u=C\setminus C_l$, 下述迭代不改变$\gamma (x_i)=c_i,(x_i,c_i)\in D_l$。

1. 根据$D_u$，随机地初始化聚类中心$\{{\mu }_c,c\in C_u\}$​​​​​​​​；如下初始化聚类中心$\{{\mu }_c,c\in C_l\}$:
   $${\mu }_c=\frac{1}{♯\{x\in D_l|\gamma (x)=c\}}\sum _{\gamma (x)=c,x\in D_l}x,c\in C_l;$$
2. 设置$D_u$分类结果$\gamma (x)=\arg {\min }_{c\in C}\Vert x-{\mu }_c\Vert ,x\in D_u$；(不要设置$D_l$的分类结果)
3. 更新中心${\mu }_c=\frac{1}{N_c}{\sum }_{\gamma (x)=c}x$；（注意求和包括$D_l,D_u$中的x）
4. 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$C_l$不包括所有类别，那么首先给$C_u$指定聚类中心，如在$D_u$中随机选择，然后$D_l$中每个类的中心作为$C_l$的聚类中心（退化为一个有监督的分类算法）。

代码

#!/usr/bin/env python


"""
Semi K-Means
"""

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.utils.validation import check_is_fitted

class SupervisedKMeans(ClassifierMixin, KMeans):
    classes = None
    def fit(self, X, y):
        if self.classes is None:
            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_
        self.n_classes = len(self.classes)
        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, Xl, yl, Xu):
        """To fit the semisupervised model
        
        Args:
            Xl (array): input variables with labels
            yl (array): labels
            Xu (array): input variables without labels
        
        Returns:
            the model
        """
        classes0 = np.unique(yl)
        if not hasattr(self, 'classes') or self.classes is None:
            self.classes = np.arange(self.n_clusters)
        else:
            assert all(c in self.classes for c in classes0), 'yl has an element not in `classes`!'

        X = np.row_stack((Xl, Xu))
        
        n1 = self.n_clusters - len(classes0)
        mu0 = SupervisedKMeans().fit(Xl, yl).centers_
        if n1:
            centers, indices = kmeans_plusplus(Xu, n_clusters=n1)
            self.cluster_centers_ = np.row_stack((centers, mu0))
        else:
            self.cluster_centers_ = mu0

        return self._fit(Xl,yl, Xu, self.cluster_centers_, self.classes)

    def _fit(self, Xl, yl, Xu, cluster_centers, classes):
        X = np.row_stack((Xl, Xu))

        for _ in range(self.max_iter):
            ED = euclidean_distances(Xu, cluster_centers)
            yu = [classes[k] for k in np.argmin(ED, axis=1)]
            y = np.concatenate((yl, yu))
            cluster_centers = np.array([np.mean(X[y==c], axis=0) for c in classes])
        self.labels_ = y
        self.cluster_centers_ = cluster_centers
        return self

    def partial_fit(self, *args, **kwargs):
        check_is_fitted(self, ('cluster_centers_',))
        return self._fit(Xl,yl, Xu, self.cluster_centers_, self.classes)
        

运行

#!/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
from semi_kmeans import *


digists = datasets.load_digits()
X_train, X_test, y_train, y_test = train_test_split(digists.data, digists.target, test_size=0.5)

X_labeled, X_unlabeled, y_labeled, _ = train_test_split(X_train, y_train, test_size=0.95)

if __name__ == '__main__':
    
    km = SemiKMeans(n_clusters=10)
    km.fit(X_labeled, y_labeled, X_unlabeled) # y_test0 is unknown
    skm = SupervisedKMeans()
    skm.fit(X_labeled, y_labeled)
    print(f"""
    # clusters: 10
    # samples: {X_labeled.shape[0]} + {X_unlabeled.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