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QiShunwang

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【论文翻译】Clustering by Passing Messages Between Data Points

2020/8/20 12:40:28   来源:

【论文题目】:Clustering by Passing Messages Between Data Points
【论文来源】:Clustering by Passing Messages Between Data Points
【翻译人】:BDML@CQUT实验室

Clustering by Passing Messages Between Data Points

Brendan J. Frey*, Delbert Dueck

通过在数据点之间传递消息进行聚类

Abstract

Clustering data by identifying a subset of representative examples is important for processing sensory signals and detecting patterns in data. Such “exemplars” can be found by randomly choosing an initial subset of data points and then iteratively refining it, but this works well only if that initial choice is close to a good solution. We devised a method called “affinity propagation,” which takes as input measures of similarity between pairs of data points. Real-valued messages are exchanged between data points until a high-quality set of exemplars and corresponding clusters gradually emerges. We used affinity propagation to cluster images of faces, detect genes in microarray data, identify representative sentences in this manuscript, and identify cities that are efficiently accessed by airline travel. Affinity propagation found clusters with much lower error than other methods, and it did so in less than one-hundredth the amount of time.

摘要

通过识别一个有代表性的例子子集来聚类数据对于处理感觉信号和检测数据中的模式是很重要的。这样的“范例”可以通过随机选择数据点的初始子集,然后迭代地细化它来找到,但是只有当初始选择接近一个好的解决方案时,这种方法才能很好地工作。我们设计了一种称为“亲和传播”的方法,它将数据点对之间的相似性作为输入度量。实值消息在数据点之间交换,直到一组高质量的示例和相应的集群逐渐出现。我们使用亲和传播对人脸图像进行聚类,检测微阵列数据中的基因,识别手稿中的代表性句子,以及识别航空旅行可有效访问的城市。亲和力传播发现聚类误差比其他方法要小得多,而且只花了不到百分之一的时间。

正文

Clustering data based on a measure of similarity is a critical step in scientific data analysis and in engineering systems. A common approach is to use data to learn a set of centers such that the sum of squared errors between data points and their nearest centers is small. When the centers are selected from actual data points, they are called “exemplars.” The popular k-centers clustering technique (1) begins with an initial set of randomly selected exemplars and iteratively refines this set so as to decrease the sum of squared errors. k-centers clustering is quite sensitive to the initial selection of exemplars, so it is usually rerun many times with different initializations in an attempt to find a good solution. However,this works well only when the number of clusters is small and chances are good that at least one random initialization is close to a good solution. We take a quite different approach and introduce a method that simultaneously considers all data points as potential exemplars. By viewing each data point as a node in a network, we devised a method that recursively transmits real-valued messages along edges of the network until a good set of exemplars and corresponding clusters emerges.As described later, messages are updated on the basis of simple formulas that search for minima of an appropriately chosen energy function. At any point in time, the magnitude of each message reflects the current affinity that one data point has for choosing another data point as its exemplar, so we call our method “affinity propagation.” Figure 1A illustrates how clusters gradually emerge during the message-passing procedure.
基于相似性度量的聚类是科学数据分析和工程系统中的关键步骤。一种常见的方法是使用数据来学习一组中心,这样数据点与其最近的中心之间的平方误差之和很小。当这些中心是从实际数据点中选择的时,称之为“样本”。流行的k中心聚类技术(1)首先从随机选择的样本集开始,然后迭代地细化该集合,以减少误差平方和。k-中心聚类对样本的初始选择非常敏感,因此通常会用不同的初始化方法多次重新运行,试图找到一个好的解决方案。但是,只有当集群的数量很小,并且至少有一个随机初始化很有可能接近一个好的解决方案时,这种方法才能很好地工作。我们采用了一种完全不同的方法,并引入了一种同时将所有数据点视为潜在样本的方法。通过将每个数据点视为网络中的一个节点,我们设计了一种方法,该方法沿着网络的边缘递归地传递实值信息,直到出现一组良好的样本和相应的簇。如后文所述,在简单公式的基础上更新消息,这些公式搜索适当选择的能量函数的最小值。在任何时间点,每个消息的大小都反映了一个数据点选择另一个数据点作为样本的当前亲和力,因此我们将我们的方法称为“亲和力传播”。图1A说明了在消息传递过程中集群是如何逐渐出现的。

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Affinity propagation takes as input a collection of real-valued similarities between data points, where the similarity s(i,k) indicates how well the data point with index k is suited to be the exemplar for data point i. When the goal is to minimize squared error, each similarity is set to a negative squared error (Euclidean distance): For points xix_{i} and xkx_{k}, s(i,k) =−xixk2\left \| x_{i} -x_{k}\right \|^{2}. Indeed, the method described here can be applied when the optimization criterion is much more general. Later, we describe tasks where similarities are derived for pairs of images, pairs of microarray measurements, pairs of English sentences, and pairs of cities. When an exemplar-dependent probability model is available, s(i,k) can be set to the log-likelihood of data point i given that its exemplar is point k. Alternatively, when appropriate, similarities may be set by hand.

亲和力的传播将数据点间的实值相似性的集合作为输入,其中,相似度s(i,k)表示索引为k的数据点很适合作为数据点i的样本。当目标是最小化平方误差时,将每个相似度被设置为一个负的平方误差(欧氏距离):对于点xix_{i}xkx_{k}, s(i,k) =−xixk2\left \| x_{i} -x_{k}\right \|^{2},实际上,这里描述的方法可以在优化准则更一般的情况下应用。随后,我们描述的任务中,相似性是由成对的图像、成对的微阵列测量、成对的英语句子和成对的城市得出的。当一个样本相关的概率模型可用时,s(i,k)可以被设为数据点i的对数似然,假设它的样本是点k。或者,在适当的时候,可以手工设置相似性。

Rather than requiring that the number of clusters be prespecified, affinity propagation takes as input a real number s(k,k) for each data point k so that data points with larger values of s(k,k) are more likely to be chosen as exemplars. These values are referred to as “preferences.” The number of identified exemplars (number of clusters) is influenced by the values of the input preferences, but also emerges from the message-passing procedure. If a priori, all data points are equally suitable as exemplars, the preferences should be set to a common value—this value can be varied to produce different numbers of clusters. The shared value could be the median of the input similarities (resulting in a moderate number of clusters) or their minimum (resulting in a small number of clusters).

亲和力传播是对每个数据点k取一个实数s(k,k)作为输入,这样s(k,k)值越大的数据点更有可能被选择为样本,而不是预先指定聚类的数量。这些值称为“首选项”。识别出的样本的数量(聚类的数量)受到输入偏好值的影响,但也会从消息传递过程中产生。如果先验条件下,所有数据点都同样适合作为样本,那么偏好应该设置为一个同样的值——这个值可以被改变以产生不同数量的聚类。共享值可以是输入相似性的中值(导致聚类数量适中),也可以是它们的最小值(导致聚类数量较少)。

There are two kinds of message exchanged between data points, and each takes into account a different kind of competition. Messages can be combined at any stage to decide which points are exemplars and, for every other point, which exemplar it belongs to. The“responsibility” r(i,k), sent from data point i to candidate exemplar point k, reflects the accumulated evidence for how well-suited point k is to serve as the exemplar for point i, taking into account other potential exemplars for point i (Fig. 1B). The “availability” a(i,k), sent from candidate exemplar point k to point i,reflects the accumulated evidence for how appropriate it would be for point i to choose point k as its exemplar, taking into account the support from other points that point k should be an exemplar (Fig. 1C). r(i,k) and a(i,k) can be viewed as log-probability ratios. To begin with, the availabilities are initialized to zero:a(i,k) = 0. Then, the responsibilities are computed using the rule

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In the first iteration, because the availabilities are zero, r(i,k) is set to the input similarity between point i and point k as its exemplar,minus the largest of the similarities between point i and other candidate exemplars. This competitive update is data-driven and does not take into account how many other points favor each candidate exemplar. In later iterations, when some points are effectively assigned to other exemplars, their availabilities will drop below zero as prescribed by the update rule below. These negative availabilities will decrease the effective values of some of the input similarities s(i,k′) in the above rule, removing the corresponding candidate exemplars from competition. For k = i, the responsibility r(k,k) is set to the input preference that point k be chosen as an exemplar, s(k,k), minus the largest of the similarities between point i and all other candidate exemplars. This “self-responsibility” reflects accumulated evidence that point k is an exemplar, based on its input preference tempered by how ill-suited it is to be assigned to another exemplar.
数据点之间交换的消息有两种,每一种都考虑到竞争的不同类型。消息可以在任何阶段进行组合,以决定哪些点是样本,对于每个其他的点,它属于哪些样本。“责任”r(i,k),从数据点i发送到候选样本点k,反映了积累的证据,表明k点是多么适合作为第i点的样本,考虑到第i点的其他潜在样本(图1B)。“可用性”a(i,k),从候选样本点k发送到点i,反映了积累的证据,说明点i选择点k作为它的样本是多么合适,考虑到其他点的支持,认为点k应该是一个样本(图1C)。r(i,k) 和 a(i,k)可以看成是对数概率比。首先,可用性被初始化为零:a(i,k) = 0。然后,使用规则计算职责
在这里插入图片描述

在第一次迭代中,由于可用性为0,r(i,k)设为作为样本的点i和点k之间的输入相似度,减去点i和其他候选样本之间的相似度的最大值。这个竞争性的更新是数据驱动的,并且没有考虑有多少其他点有利于每个候选样本。在以后的迭代中,当一些点被有效地分配给其他样本时,它们的可用性将会降到0以下,这是下面的更新规则所规定的。这些负可用性会降低上述规则中某些输入相似点s(i,k’)的有效值,使相应的候选样本从竞争中消失。对于k = i,责任r(k,k)被设为点k被选为样本的输入偏好s(k,k),减去点i和所有候选样本之间最大的相似性。这种“自我责任”反映了k点是一个样本的积累证据,基于它的输入偏好再加上它有多不适合被分配到另一个范例。

Whereas the above responsibility update lets all candidate exemplars compete for ownership of a data point, the following availability update gathers evidence from data points as to whether each candidate exemplar would make a good exemplar:
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The availability a(i,k) is set to the self-responsibility r(k,k) plus the sum of the positive responsibilities candidate exemplar k receives from other points. Only the positive portions of incoming responsibilities are added, because it is only necessary for a good exemplar to explain some data points well (positive responsibilities),regardless of how poorly it explains other data points (negative responsibilities). If the self-responsibility r(k,k) is negative (indicating that point k is currently better suited as belonging to another exemplar rather than being an exemplar itself), the availability of point k as an exemplar can be increased if some other points have positive responsibilities for point k being their exemplar. To limit the influence of strong incoming positive responsibilities, the total sum is thresholded so that it cannot go above zero. The “self-availability” a(k,k) is updated differently:
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This message reflects accumulated evidence that point k is an exemplar, based on the positive responsibilities sent to candidate exemplar k from other points.

上面的职责更新让所有候选范例为数据点的所有权而竞争,以下可用性更新从数据点收集证据,以确定每个候选范例是否会成为一个好范例:在这里插入图片描述

可用性a(i,k)被设为自我责任r(k,k)加上候选范例k从其他点接收到的正责任的总和。
只增加传入责任的积极部分,因为一个好的范例只需要很好地解释一些数据点(积极的责任),而不管它对其他数据点的解释有多差(消极的责任)。如果自我责任r (k, k)是负的(表明点k是目前适合属于另一个样本,而不是作为一个样本本身),如果其他点对作为范例的点有积极的责任,那么k点作为范例的可用性就会增加。为了限制传入的积极责任的影响,总和是有阈值的,因此不能超过零。“自我可用性”a(k,k)的更新方式不同:
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这条消息反映了基于从其他点发送给候选范例k的积极责任所积累的证据,证明点k是一个范例。

The above update rules require only simple,local computations that are easily implemented (2), and messages need only be exchanged between pairs of points with known similarities. At any point during affinity propagation, availabilities and responsibilities can be combined to identify exemplars. For point i, the value of k that maximizes a(i,k) + r(i,k) either identifies point i as an exemplar if k = i, or identifies the data point that is the exemplar for point i. The message-passing procedure may be terminated after a fixed number of iterations, after changes in the messages fall below a threshold, or after the local decisions stay constant for some number of iterations. When updating the messages,it is important that they be damped to avoid numerical oscillations that arise in some circumstances. Each message is set to l times its value from the previous iteration plus 1 – λ times its prescribed updated value, where the damping factor l is between 0 and 1. In all of our experiments (3), we used a default damping factor of λ = 0.5, and each iteration of affinity propagation consisted of (i) updating all responsibilities given the availabilities, (ii) updating all availabilities given the responsibilities, and (iii) combining availabilities and responsibilities to monitor the exemplar decisions and terminate the algorithm when these decisions did not change for 10 iterations.
上面的更新规则只需要简单的、容易实现的本地计算(2),并且只需要在已知相似点对之间交换消息。在亲和传播期间的任何点,可用性和职责都可以组合起来识别样本。对于点i,使a(i,k) + r(i,k)最大化的k的值,如果k = i,可以将点i识别为一个样本,或者识别为点i的样本数据点。消息传递过程可以在固定次数的迭代之后终止,也可以在消息中的更改低于阈值之后终止,或者在本地决策一定次数的迭代中保持不变之后终止。在更新信息时,重要的是要对它们进行阻尼,以避免在某些情况下出现的数值振荡。每条消息被设置为λ 乘以它在以前的迭代中得到的值加上1 -λ 乘以它所规定的更新值,其中阻尼因子λ 在0到1之间。(3)我们所有的实验中,我们使用一个默认的阻尼因子λ = 0.5,和每次迭代的亲和力传播包括(i)更新所有责任考虑到可用性,(2)更新所有可用性的责任,和(3)结合可用性和责任监督样本决定,终止算法当10次迭代中这些决定不改变。

Figure 1A shows the dynamics of affinity propagation applied to 25 two-dimensional data points (3), using negative squared error as the similarity. One advantage of affinity propagation is that the number of exemplars need not be specified beforehand. Instead, the appropriate number of exemplars emerges from the message passing method and depends on the input exemplar preferences. This enables automatic model selection, based on a prior specification of how preferable each point is as an exemplar. Figure 1D shows the effect of the value of the common input preference on the number of clusters. This relation is nearly identical to the relation found by exactly minimizing the squared error (2).
图1A显示了应用于25个二维数据点(3)的亲和传播动力学,使用负平方误差作为相似性。关联传播的一个优点是不需要预先指定范例的数量。相反,从信息传递方法中会出现适当数目的范例,并取决于输入范例的偏好。这使得自动的模型选择成为可能,基于对每个点作为范例的可取程度的预先说明。图1D显示了公共输入偏好的值对集群数量的影响。这个关系与精确地最小化平方误差(2)所发现的关系几乎相同。

We next studied the problem of clustering images of faces using the standard optimization criterion of squared error. We used both affinity propagation and k-centers clustering to identify exemplars among 900 grayscale images extracted from the Olivetti face database (3). Affinity propagation found exemplars with much lower squared error than the best of 100 runs of k-centers clustering (Fig. 2A), which took about the same amount of computer time.We asked whether a huge number of random restarts of k-centers clustering could achieve the same squared error. Figure 2B shows the error achieved by one run of affinity propagation and the distribution of errors achieved by 10,000 runs of k-centers clustering, plotted against the number of clusters. Affinity propagation uniformly achieved much lower error in more than two orders of magnitude less time. Another popular optimization criterion is the sum of absolute pixel differences (which better tolerates outlying pixel intensities), so we repeated the above procedure using this error measure. Affinity propagation again uniformly achieved lower error (Fig. 2C).
接下来,我们利用平方误差的标准优化准则研究了人脸图像聚类问题。我们使用亲和传播和k-中心聚类来识别从Olivetti人脸数据库(3)中提取的900幅灰度图像样本。亲和力传播发现样本的平方误差远低于k-中心聚类(图2A)100次中最好的一次,后者花费了大约相同的计算机时间。我们询问大量随机重启k-中心聚类是否可以获得相同的平方误差。图2B显示了一次亲和力传播所获得的误差,以及10000次k-中心聚类所获得的误差分布,并根据聚类的数量绘制。亲和传播在两个数量级以上的时间内均匀地获得了更低的误差。另一个流行的优化标准是绝对像素差的总和(它可以更好地容忍离群的像素强度),因此我们使用这个误差度量重复上述过程。亲和传播再次均匀地获得较低的误差(图2C)。

Many tasks require the identification of exemplars among sparsely related data, i.e., where most similarities are either unknown or large and negative. To examine affinity propagation in this context, we addressed the task of clustering putative exons to find genes, using the sparse similarity matrix derived from microarray data and reported in (4). In that work, 75,066 segments of DNA (60 bases long) corresponding to putative exons were mined from the genome of mouse chromosome 1. Their transcription levels were measured across 12 tissue samples, and the similarity between every pair of putative exons (data points) was computed. The measure of similarity between putative exons was based on their proximity in the genome and the degree of coordination of their transcription levels across the 12 tissues. To account for putative exons that are not exons (e.g., introns), we included an additional artificial exemplar and determined the similarity of each other data point to this “non-exon exemplar” using statistics taken over the entire data set. The resulting 75,067 × 75,067 similarity matrix (3) consisted of 99.73% similarities with values of −∞, corresponding to distant DNA segments that could not possibly be part of the same gene. We applied affinity propagation to this similarity matrix, but because messages need not be exchanged between point i and k if s(i,k) = −∞, each iteration of affinity propagation required exchanging messages between only a tiny subset (0.27% or 15million) of data point pairs.
许多任务需要在稀疏相关的数据中识别样本,也就是说,在这些数据中,大多数相似性要么是未知的,要么是大而负面的。为了研究这种背景下的亲和力传播,我们利用从微阵列数据和报告(4)中导出的稀疏相似性矩阵,聚类假定的外显子找到基因。在这项研究中,我们从小鼠1号染色体的基因组中提取了75066段DNA片段(60碱基长),这些片段与假定的外显子相对应。在12个组织样本中测量了它们的转录水平,并计算了每对假定外显子(数据点)之间的相似性。假设外显子之间的相似性是基于它们在基因组中的接近程度以及它们在12种组织中转录水平的协调程度。为了解释不是外显子的假定外显子(例如内含子),我们加入了一个额外的人工样本,并使用对整个数据集的统计数据来确定其他数据点与这个“非外显样本”的相似性。由此得到的75067×75067相似矩阵(3)与-∞值的相似度为99.73%,对应于不可能是同一基因一部分的远缘DNA片段。我们将相似性传播应用到这个相似性矩阵中,但是由于如果s(i,k)=-∞时,不需要在点i和k之间交换消息,所以亲合传播的每次迭代只需要在数据点对的一小部分(0.27%或1500万)之间交换消息。

Figure 3A illustrates the identification of gene clusters and the assignment of some data points to the nonexon exemplar. The reconstruction errors for affinity propagation and k-centers clustering are compared in Fig. 3B.For each number of clusters, affinity propagation was run once and took 6 min, whereas k-centers clustering was run 10,000 times and took 208 hours. To address the question of how well these methods perform in detecting bona fide gene segments, Fig. 3C plots the truepositive (TP) rate against the false-positive (FP) rate, using the labels provided in the RefSeq database (5). Affinity propagation achieved significantly higher TP rates, especially at low FP rates, which are most important to biologists. At a FP rate of 3%, affinity propagation achieved a TP rate of 39%, whereas the best k-centers clustering result was 17%. For comparison, at the same FP rate, the best TP rate for hierarchical agglomerative clustering (2) was 19%, and the engineering tool described in (4), which accounts for additional biological knowledge, achieved a TP rate of 43%.
图3A说明了基因聚类的识别和将一些数据点的分配给非外显样本。图3B中比较了亲和传播和k-中心聚类的重建误差。对于每个聚类数,亲和力传播运行一次,花费6分钟;而k-中心聚类则运行10000次,花费208小时。为了解决这些方法在检测真实基因片段方面的表现如何的问题,图3C使用RefSeq数据库(5)中提供的标签绘制了真阳性(TP)率和假阳性(FP)率。亲和力繁殖获得了显著较高的TP率,尤其是在低FP率下,这对生物学家来说是最重要的。当FP率为3%时,亲和繁殖的TP率为39%,而k-中心聚类的最好结果为17%。作为比较,在相同的FP率下,层次聚集聚类(2)的最佳TP率为19%,而(4)中描述的工程工具(它解释了额外的生物学知识)实现了43%的TP率。

Affinity propagation’s ability to operate on the basis of nonstandard optimization criteria makes it suitable for exploratory data analysis using unusual measures of similarity. Unlike metricspace clustering techniques such as k-means clustering (1), affinity propagation can be applied to problems where the data do not lie in a continuous space. Indeed, it can be applied to problems where the similarities are not symmetric [i.e., s(i,k) ≠ s(k,i)] and to problems where the similarities do not satisfy the triangle inequality [i.e., s(i,k) < s(i,j) + s( j,k)]. To identify a small number of sentences in a draft of this manuscript that summarize other sentences, we treated each sentence as a “bag of words” (6) and computed the similarity of sentence i to sentence k based on the cost of encoding the words in sentence i using the words in sentence k. We found that 97% of the resulting similarities (2, 3) were not symmetric. The preferences were adjusted to identify (using l = 0.8) different numbers of representative exemplar sentences (2), and the solution with four sentences is shown in Fig. 4A.

亲和力传播基于非标准优化准则的操作能力使得它适合于使用不寻常的相似性度量进行探索性数据分析。不像度量空间聚类的技术,例如k-均值聚类(1),亲和传播可以应用于数据不在连续空间中的问题。实际上,它可以被应用于相似性不对称的问题[即s(i,k)≠s(k,i)]和相似性不满足三角形不等式[即s(i,k) < s(i,j) + s( j,k)]的问题。为了找出本文草稿中总结其他句子的少量句子,我们将每个句子视为一个“单词包”(6),并根据k句子中单词的编码成本计算出句子与k句子的相似度。我们发现97%的相似性(2,3)不是对称的。调整偏好以识别(使用l=0.8)不同数量的代表性例句(2),四个句子的解决方案如图4A所示。

We also applied affinity propagation to explore the problem of identifying a restricted number of Canadian and American cities that are most easily accessible by large subsets of other cities, in terms of estimated commercial airline travel time. Each data point was a city, and the similarity s(i,k) was set to the negative time it takes to travel from city i to city k by airline, including estimated stopover delays (3). Due to headwinds, the transit time was in many cases different depending on the direction of travel, so that 36% of the similarities were asymmetric. Further, for 97% of city pairs i and k, there was a third city j such that the triangle inequality was violated, because the trip from i to k included a long stopover delay n city j so it took longer than the sum of the durations of the trips from i to j and j to k. When the number of “most accessible cities” was constrained to be seven (by adjusting the input preference appropriately), the cities shown in Fig. 4, B to E, were identified. It is interesting that several major cities were not selected, either because heavy international travel makes them inappropriate as easily accessible domestic destinations (e.g., New YorkCity, Los Angeles) or because their neighborhoods can be more efficiently accessed through other destinations (e.g., Atlanta, Philadelphia, and Minneapolis account for Chicago’s destinations, while avoiding potential airport delays).
我们还应用亲和力传播来探索确定有限数量的加拿大和美国城市的问题,这些城市最容易被其他大城市的子集所访问,根据估计的商业航空旅行时间。每个数据点都是一个城市,相似性s(i,k)被设置为乘飞机从城市i到城市k所需的负时间,包括估计的中途停留延误(3)。由于逆风,在许多情况下,运输时间依赖于旅行方向,因此36%的相似性是不对称的。此外,对于97%的城市对i和k,有第三个城市j违反了三角不等式,因为从i到k的行程包括在城市j的长的中途停留延迟,所以它比从i到j和j到k的旅行时间总和要长。当“最容易到达的城市”的数量被限制为7(通过适当地调整输入偏好),图4中B到E所示的城市被识别。有趣的是,几个主要城市没有被选中,要么是因为大量的国际旅行使得它们不适合作为容易到达的国内目的地(如纽约市、洛杉矶),要么是因为它们的社区可以通过其他目的地更有效地到达(如亚特兰大、费城,明尼阿波利斯是芝加哥的目的地,同时避免了潜在的机场延误)。
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Affinity propagation can be viewed as a method that searches for minima of an energy function (7) that depends on a set of N hidden labels, c1,…,cN, corresponding to the N data points. Each label indicates the exemplar to which the point belongs, so that s(i,ci) is the similarity of data point i to its exemplar. ci = i is a special case indicating that point i is itself an exemplar, so that s(i,ci) is the input preference for point i. Not all configurations of the labels are valid; a configuration c is valid when for every point i, if some other point i′ has chosen i as its exemplar (i.e., ci′ = i), then i must be an exemplar (i.e., ci = i). The energy of a valid configuration is E(c)=i=1Ns(i,ci).E(c)=-\sum_{i=1}^{N}s(i,c_{i}). Exactly minimizing the energy is computationally intractable, because a special case of this minimization problem is the NP-hard k-median problem (8). However, the update rules for affinity propagation correspond to fixed-point recursions for minimizing a Bethe free-energy (9) approximation. Affinity propagation is most easily derived as an instance of the max-sum algorithm in a factor graph (10) describing the constraintson the labels and the energy function (2).
亲和传播可以看作是一种搜索能量函数(7)的极小值的方法,该能量函数依赖于与N个数据点相对应的一组N个隐藏标签c1,…,cN。每个标签都表示该点所属的样本,因此s(i,Ci)是数据点i与其范例的相似性。Ci=i是一个特例,表明i点本身就是一个样本,因此s(i,Ci)是点i的输入偏好。并非所有标签的配置都有效;配置c是有效的,当对于每个点i,如果其他点 i’ 选择i作为其样本(即Ci=i),那么i必须是一个范例(即Ci=i)。有效配置的能量为E(c)=i=1Ns(i,ci).E(c)=-\sum_{i=1}^{N}s(i,c_{i}).精确地最小化能量在计算上是困难的,因为这个最小化问题的一个特例是NP难k-中值问题(8)。然而,亲合传播的更新规则对应于最小化Bethe自由能(9)近似的不动点递归。亲和力传播作为因子图(10)中描述标签约束和能量函数(2)的最大和算法的实例最容易导出。然而,亲合传播的更新规则对应于最小化Bethe自由能(9)近似的不动点递归。亲和力传播作为因子图(10)中描述标签约束和能量函数(2)的最大和算法的实例最容易导出。

In some degenerate cases, the energy function may have multiple minima with corresponding multiple fixed points of the update rules, and these may prevent convergence. For example, if s(1,2) = s(2,1) and s(1,1) = s(2,2), then the solutions c1 = c2 = 1 and c1 = c2 = 2 both achieve the same energy. In this case, affinity propagation may oscillate, with both data points alternating between being exemplars and nonexemplars. In practice, we found that oscillations could always be avoided by adding a tiny amount of noise to the similarities to prevent degenerate situations,or by increasing the damping factor.
在某些退化情形下,能量函数可能具有多个极小值,且更新规则的多个不动点可能会阻止收敛。例如,如果s(1,2)=s(2,1)和s(1,1)=s(2,2),则解c1=c2=1和c1=c2=2都获得相同的能量。在这种情况下,亲和性传播可能会振荡,两个数据点在样本和非样本之间交替。在实践中,我们发现通过在相似点上添加少量的噪声来防止退化情况,或者通过增加阻尼因子来避免振荡。

Affinity propagation has several advantages over related techniques. Methods such as k-centers clustering (1), k-means clustering(1), and the expectation maximization (EM) algorithm (11) store a relatively small set of estimated cluster centers at each step. These techniques are improved upon by methods that begin with a large number of clusters and then prune them (12), but they still rely on random sampling and make hard pruning decisions that cannot be recovered from. In contrast, by simultaneously considering all data points as candidate centers and gradually identifying clusters, affinity propagation is able to avoid many of the poor solutions caused by unlucky initializations and hard decisions. Markov chain Monte Carlo techniques (13) randomly search for good solutions, but do not share affinity propagation’s advantage of considering many possible solutions all at once.
与相关技术相比,亲和传播有几个优点。像k-中心聚类(1)、k-均值聚类(1)和期望最大化(EM)算法(11)这样的方法在每个步骤存储一个相对较小的估计聚类中心集。这些技术通过从大量的聚类开始,然后对它们进行修剪的方法得到了改进(12),但是它们仍然依赖于随机抽样,并且做出无法恢复的硬修剪决策。相反,通过同时考虑所有数据点作为候选中心并逐步识别簇,亲和传播能够避免由于不吉利的初始化和艰难的决策而导致的许多糟糕的解决方案。马尔可夫链蒙特卡罗技术(13)随机搜索好的解决方案,但不共享亲和传播的优点,即一次考虑许多可能的解决方案。

Hierarchical agglomerative clustering (14) and spectral clustering (15) solve the quite different problem of recursively comparing pairs of points to find partitions of the data. These techniques do not require that all points within a cluster be similar to a single center and are thus not well-suited to many tasks. In particular, two points that should not be in the same cluster may be grouped together by an unfortunate sequence of pairwise groupings.
层次聚集聚类(14)和谱聚类(15)解决了递归比较点对以找到数据分区的完全不同的问题。这些技术并不要求一个聚类内的所有点都类似于一个中心,因此不太适合许多任务。特别是,不应该在同一个聚类中的两个点可能会被一个不幸的成对分组序列组合在一起。
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In (8), it was shown that the related metric k-median problem could be relaxed to form a linear program with a constant factor approximation. There, the input was assumed to be metric, i.e., nonnegative, symmetric, and satisfying the triangle inequality. In contrast, affinity propagation can take as input general nonmetric similarities. Affinity propagation also provides a conceptually new approach that works well in practice. Whereas the linear programming relaxation is hard to solve and sophisticated software packages need to be applied (e.g., CPLEX), affinity propagation makes use of intuitive message updates that can be implemented in a few lines of code (2).
在(8)中,证明了相关的度量k-中值问题可以放宽到常系数近似线性规划。在这里,假设输入是度量的,即非负的,对称的,满足三角不等式的。相反,亲和传播可以将一般非度量相似性作为输入。亲和力传播还提供了一种概念上新的方法,在实践中效果良好。虽然线性规划松弛很难解决,并且需要应用复杂的软件包(例如 ,CPLEX),但亲和传播利用了可以在几行代码中实现的直观消息更新(2)。

Affinity propagation is related in spirit to techniques recently used to obtain record-breaking results in quite different disciplines (16). The approach of recursively propagating messages (17) i n a “loopy graph” has been used to approach Shannon’s limit in error-correcting decoding (18, 19), solve random satisfiability problems with an order-of-magnitude increase in size (20), solve instances of the NP-hard twodimensional phase-unwrapping problem (21), and efficiently estimate depth from pairs of stereo images (22). Yet, to our knowledge, affinity propagation is the first method to make use of this idea to solve the age-old, fundamental problem of clustering data. Because of its simplicity, general applicability, and performance, we believe affinity propagation will prove to be of broad value in science and engineering.
亲和力传播在精神上与最近在不同学科中获得破纪录结果的技术有关(16)。在“循环图”中递归传播消息(17)的方法被用于逼近纠错解码中的香农极限(18,19),解决大小增加一个数量级的随机可满足性问题(20),解决NP难二维相位展开问题(21),并有效地估计立体图像对的深度(22)。然而,据我们所知,亲和力传播是利用这种思想来解决古老的、基本的数据聚类问题的第一种方法。由于它的简单性,普遍适用性和性能,我们相信亲和传播将被证明在科学和工程中具有广泛的价值。
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