算法原理：
由于傳統(tǒng)KMeans算法的聚類結(jié)果易受初始聚類中心點(diǎn)選擇的影響，因此在傳統(tǒng)的KMeans算法的基礎(chǔ)上進(jìn)行改進(jìn)。
二分KMeans(Bisecting KMeans)算法的主要思想是：首先將所有點(diǎn)作為一個(gè)簇，然后將該簇一分為二。之后選擇能最大限度降低聚類代價(jià)函數(shù)（誤差平方和SSE）的簇劃分為兩個(gè)簇。以此進(jìn)行下去，直到簇的數(shù)目等于給定的數(shù)目K為止。

代碼實(shí)現(xiàn)：
基于DataFrame

def bi_kmeans(data: 'DataFrame', K: int) -> 'DataFrame':
    """二分KMeans
    data: data['embedding'] 輸入x的向量
    K: 聚類類別數(shù)
    """

    def sse(error):
        """計(jì)算誤差平方和"""
        return np.square(np.linalg.norm(error))

    def euclidean_dist(v1, v2):
        """計(jì)算兩個(gè)向量間的歐氏距離"""
        return np.linalg.norm(v1 - v2)

    def dist_from_center(label, embedding, cluster_center, offset=0):
        """計(jì)算每個(gè)元素與其類別中心的歐氏距離
        offset: label的偏移量
        return: float"""
        center = cluster_center[label - offset]
        dist = euclidean_dist(embedding, center)
        return dist
    
    # 初始化類的中心
    cluster_center = [np.mean(data.embedding)]
    # 初始化每個(gè)item的label和到聚類中心的距離
    data['label'] = 0
    data['dist_from_center'] = data.apply(lambda x: dist_from_center(x['label'], x['embedding'], cluster_center), axis=1)

    # 當(dāng)前k小于給定K值時(shí)
    k = 1
    while k < K :
        print('Current Cluster Number: {}  >>>'.format(k))

        # 計(jì)算當(dāng)前sse
        total_sse = sse(data.dist_from_center)
        sharp_drop = 0

        # 遍歷當(dāng)前每個(gè)簇，將其一分為二，計(jì)算新的sse
        keep_i = -1
        for i in range(k):
            # 第i簇?cái)?shù)據(jù)
            group_i = data[data.label == i]
            if group_i.shape[0] > 2:
                pre_sse = sse(group_i.dist_from_center)
                # 二分當(dāng)前簇
                bi_kmeans = KMeans(n_clusters=2).fit(group_i.embedding.tolist())
                # 更新label和dist
                group_i['label'] = bi_kmeans.labels_
                new_center = bi_kmeans.cluster_centers_
                group_i['dist_from_center'] = group_i.apply(lambda x: dist_from_center(x['label'], x['embedding'], new_center), axis=1)
                # 計(jì)算當(dāng)前sse
                post_sse = sse(group_i.dist_from_center)
                # sse下降程度
                drop = pre_sse - post_sse
                # 保留最大下降ssd的i
                if drop > sharp_drop:
                    keep_i = i
                    sharp_drop = drop

        # 選出待二分的數(shù)據(jù)
        group_i = data[data.label == keep_i]
        group_i_index = data[data.label == keep_i].index
        # 二分，更新label
        bi_kmeans = KMeans(n_clusters=2).fit(group_i.embedding.tolist())
        group_i['label'] = bi_kmeans.labels_ + k
        data.loc[group_i_index, 'label'] = bi_kmeans.labels_ + k
        new_center = bi_kmeans.cluster_centers_
        # 更新距中心距離
        data.loc[group_i_index, 'dist_from_center'] = group_i.apply(lambda x: dist_from_center(x['label'], x['embedding'], new_center, offset=k), axis=1)
        # 更新超出k的label為原有l(wèi)abel
        k_plus_1_index = data[data.label == k + 1].index
        data.loc[k_plus_1_index, 'label'] = keep_i
        # 更新類別數(shù)
        k += 1
    return data