In statistics and machine learning, k-means clustering is a method of cluster analysis which aims topartition n observations into k clusters in which each observation belongs to the cluster with the nearestmean. It is similar to the expectation-maximization algorithm for mixtures of Gaussians in that they both attempt to find the centers of natural clusters in the data as well as in the iterative refinement approach employed by both algorithms.

Description

Given a set of observations (x₁, x₂, …, x_n), where each observation is a d-dimensional real vector, k-means clustering aims to partition the n observations into k sets (k < n) S = {S₁, S₂, …, S_k} so as to minimize the within-cluster sum of squares (WCSS):

where μ_i is the mean of points in S_i.

Algorithms

Regarding computational complexity, the k-means clustering problem is:

• NP-hard in general Euclidean space d even for 2 clusters ^[4][5]
• NP-hard for a general number of clusters k even in the plane ^[6]
• If k and d are fixed, the problem can be exactly solved in time O(n^dk+1 log n), where n is the number of entities to be clustered ^[7]

Thus, a variety of heuristic algorithms are generally used.

所以注意到Algorithm是一個(gè)典型的NP問題，所以通常我們尋找使用的是啟發(fā)式方法。

Standard algorithm

The most common algorithm uses an iterative refinement technique.最常用的一個(gè)技巧是迭代求精。

Due to its ubiquity it is often called the k-means algorithm; it is also referred to as , particularly in the computer science community.

Given an initial set of k means m₁⁽¹⁾,…,m_k⁽¹⁾, which may be specified randomly or by some heuristic, the algorithm proceeds by alternating between two steps:[8]

Assignment step: Assign each observation to the cluster with the closest mean (i.e. partition the observations according to the Voronoi diagram generated by the means（這里等價(jià)于把原空間根據(jù)Voronoi 圖劃分為k個(gè)，此處的范數(shù)指的是2范數(shù)，即歐幾里得距離，和Voronoi圖對應(yīng)）).

 

Update step: Calculate the new means to be the centroid of the observations in the cluster.

重新計(jì)算means

The algorithm is deemed to have converged when the assignments no longer change.

整個(gè)算法的流程就是如上圖所示

As it is a heuristic algorithm, there is no guarantee that it will converge to the global optimum, and the result may depend on the initial clusters. As the algorithm is usually very fast, it is common to run it multiple times with different starting conditions. However, in the worst case, k-means can be very slow to converge: in particular it has been shown that there exist certain point sets, even in 2 dimensions, on whichk-means takes exponential time, that is 2^Ω(n), to converge^[9][10]. These point sets do not seem to arise in practice: this is corroborated by the fact that the smoothed running time of k-means is polynomial^[11].

最壞的時(shí)間復(fù)雜度是O(2^Ω(n)),但是在實(shí)踐中，一般表現(xiàn)是一個(gè)多項(xiàng)式算法。

The "assignment" step is also referred to as expectation step, the "update step" as maximization step, making this algorithm a variant of the generalized expectation-maximization algorithm.

Variations

• The expectation-maximization algorithm (EM algorithm) maintains probabilistic assignments to clusters, instead of deterministic assignments, and multivariate Gaussian distributions instead of means.
• k-means++ seeks to choose better starting clusters.
• The filtering algorithm uses kd-trees to speed up each k-means step.^[12]
• Some methods attempt to speed up each k-means step using coresets^[13] or the triangle inequality.^[14]
• Escape local optima by swapping points between clusters.^[15]

Discussion

k-means clustering result for the Iris flower data set and actual species visualized using ELKI. Cluster means are marked using larger, semi-transparent symbols.

k-means clustering and EM clustering on an artificial dataset ("mouse"). The tendency of k-means to produce equi-sized clusters leads to bad results, while EM benefits from the Gaussian distribution present in the data set

The two key features of k-means which make it efficient are often regarded as its biggest drawbacks:

• Euclidean distance is used as a metric and variance is used as a measure of cluster scatter.
• The number of clusters k is an input parameter: an inappropriate choice of k may yield poor results. That is why, when performing k-means, it is important to run diagnostic checks for determining the number of clusters in the data set.

A key limitation of k-means is its cluster model. The concept is based on spherical clusters that are separable in a way so that the mean value converges towards the cluster center. The clusters are expected to be of similar size, so that the assignment to the nearest cluster center is the correct assignment. When for example applying k-means with a value of k = 3 onto the well-known Iris flower data set, the result often fails to separate the three Iris species contained in the data set. With k = 2, the two visible clusters (one containing two species) will be discovered, whereas withk = 3 one of the two clusters will be split into two even parts. In fact, k = 2 is more appropriate for this data set, despite the data set containing 3 classes. As with any other clustering algorithm, the k-means result relies on the data set to satisfy the assumptions made by the clustering algorithms. It works very well on some data sets, while failing miserably on others.

The result of k-means can also be seen as the Voronoi cells of the cluster means. Since data is split halfway between cluster means, this can lead to suboptimal splits as can be seen in the "mouse" example. The Gaussian models used by the Expectation-maximization algorithm (which can be seen as a generalization of k-means) are more flexible here by having both variances and covariances. The EM result is thus able to accommodate clusters of variable size much better than k-means as well as correlated clusters (not in this example).

這篇是概念介紹篇，以后出代碼和一個(gè)K均值優(yōu)化的論文

Fast Hierarchical Clustering Algorithm Using Locality-Sensitive Hashing