论文题目：Clustering by Passing Messages Between Data Points
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
Brendan J. Frey*, Delbert Dueck
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 identifycities 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
通过识别代表性样本的子集对数据进行聚类，对于处理感觉信号和检测数据中的模式非常重要。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，但只有当初始选择接近一个好的解决方案时，这种方法才有效。我们设计了一种称为“亲和力传播”的方法，它将成对数据点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。我们使用亲和力传播对人脸图像进行聚类，检测微阵列数据中的基因，识别本文中具有代表性的句子，并识别航空旅行能够有效访问的城市。亲和力传播发现的聚类误差比其他方法低得多，而且它所用的时间不到百分之一。
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数据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。当从实际数据点中选择中心时，我们称之为“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-centers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。然而，只有当集群数量较少且至少有一个随机初始化接近一个好的解决方案的可能性很高时，这种方法才有效。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范例的方法。通过将每个数据点视为网络中的一个节点，我们设计了一种沿着网络边缘递归传输实值消息的方法，直到出现一组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其示例的当前亲和力，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。
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 xiand xk, s(i,k) = −||xi− xk||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.
亲和力传播将数据点之间的实值相似度集合作为输入，其中相似度s（i，k）表示索引为k的数据点适合作为数据点i的示例的程度。 当目标是最小化平方误差时，将每个相似度设置为负平方误差（欧式距离）：对于点xiand xk，s（i，k）=-|| xi- xk ||| 2。 实际上，当优化标准更为通用时，可以应用此处描述的方法。 稍后，我们描述任务，其中，图像对，微阵列测量对，英语句子对和城市对都具有相似性。 当依赖于样本的概率模型可用时，可以将s（i，k）设置为数据点i的对数似然，前提是其样本为点k。 可替代地，在适当时，可以手动设置相似性。
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).亲和力传播不是要求预先指定簇的数量，而是将每个数据点k的实数s（k，k）作为输入，以便更可能选择具有s（k，k）的值的数据点。 作为示例。 这些值称为“首选项”。 所确定的样本数量（集群数量）受输入首选项值的影响，但也来自消息传递过程。 如果先验，所有数据点都适合作为示例，则应将首选项设置为一个公共值-可以更改此值以产生不同数量的聚类。 共享值可以是输入相似度的中位数（导致中等数量的集群）或最小值（导致少数数量的集群）。
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) a n d 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
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.
数据点之间交换两种消息，每种消息都考虑了不同的竞争。 可以在任何阶段组合消息，以确定哪些点是示例，对于其他每个点，则确定消息属于哪个示例。 从数据点i发送到候选示例点k的“责任” r（i，k）反映了考虑到的其他潜在示例，积累的证据表明适合点k用作点i的示例。 点i（图1B）。 从候选示例点k发送到点i的“可用性” a（i，k）反映了累积的证据，表明考虑到其他点的支持，点i选择点k作为其示例是多么合适。 该点k应该是一个示例（图1C）。 r（i，k）和n d a（i，k）可以看作是对数概率比。 首先，将可用性初始化为零：a（i，k）=0。然后，使用规则
计算责任 ′Þsði; k′Þgð1Þ在第一次迭代中，由于可用性为零，因此将r（i，k）设置为点i与点k之间的输入相似度作为示例，减去点之间的最大相似度 我和其他候选人范例。 此竞争性更新是由数据驱动的，并且未考虑有多少其他点对每个候选示例都有利。 在以后的迭代中，当某些点有效分配给其他示例时，其可用性将降至零以下，这是下面的更新规则所规定的。 这些负可用性将降低上述规则中某些输入相似度s（i，k’）的有效值，从而从竞争中移除相应的候选样例。 对于k = i，责任r（k，k）设置为输入优先级，即选择点k作为示例s（k，k），减去点i与所有其他候选示例之间的最大相似度。 这种“自我负责”反映了积累的证据，证明了k点是一个样例，这是基于其输入偏好受到对它的分配不恰当的程度而改变的。
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:
尽管上述职责更新允许所有候选示例争夺数据点的所有权，但以下可用性更新从数据点收集证据，以确定每个候选示例是否会成为一个好的示例：
The availability a(i,k) is set to the selfresponsibility 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 selfresponsibility 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:
This message reflects accumulated evidence that
point k is an exemplar, based on the positive
responsibilities sent to candidate exemplar k
from other points.
可用性a（i，k）设置为自负r（k，k）加上候选样例k从其他点收到的积极责任的总和。 仅添加传入职责的积极部分，因为一个好的榜样只需要很好地解释某些数据点（积极职责），而不管它对其他数据点的解释（负面职责）有多糟糕。 如果自负r（k，k）为负（表明点k当前更适合属于另一个示例而不是示例本身），则如果其他一些点具有正值，则可以增加点k作为示例的可用性。 k点的责任就是他们的榜样。 为了限制强大的积极主动责任的影响，对总和设置阈值，以使总和不能超过零。 “自我可用性” a（k，k）的更新方式有所不同：
该消息基于从其他点发送到候选样例k的积极责任，反映了点k是样例的累积证据。
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. T h e 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 – l 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 l = 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本地计算即可轻松实现（2），并且仅需要在具有相似相似性的成对点之间交换消息。
在亲和力传播过程中的任何时候，都可以将可用性和职责进行组合以识别示例。 对于点i，如果k = i，则最大化a（i，k）+ r（i，k）的k的值要么将点i识别为示例，要么将数据点识别为点i的示例。 消息传递过程可以在固定次数的迭代之后，消息中的变化降至阈值以下之后，或者在本地决策在一定数量的迭代中保持恒定之后终止。 更新消息时，请务必对它们进行阻尼处理，以避免在某些情况下出现数值波动。 每条消息的值均设置为上次迭代的值的l倍加上其规定的更新值的1-1倍，其中阻尼系数l在0到1之间。在我们所有的实验（3）中，我们使用默认的阻尼系数为 l = 0.5，并且亲和力传播的每个迭代包括（i）在给定可用性的情况下更新所有职责，（ii）在给定职责的情况下更新所有可用性，以及（iii）合并可用性和职责以监控示例决策并在出现以下情况时终止算法 这些决定在10次迭代中都没有改变。
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 messagepassing 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）的相似性传播动力学，使用负平方误差作为相似性。亲和力传播的一个优点是不需要预先指定样本的数量。取而代之的是，适当数量的示例从messagepassing方法中产生，并取决于输入示例首选项。这使得自动模型选择成为可能，这是基于对每个点作为一个范例的优先程度的预先说明。图1D显示了公共输入首选项的值对集群数量的影响。这个关系几乎与通过精确最小化平方误差（2）得到的关系相同。
Fig. 1. How affinity propagation works.
(A) Affinity propagation is illustrated for two-dimensional data points, where negative Euclidean distance (squared error) was used to measure similarity. Each point is colored according to the current evidence that it is a cluster center (exemplar). The darkness of the arrow directed from point i to point k corresponds to the strength of the transmitted message that point i belongs to exemplar point k. (B) “Responsibilities” r(i,k) are sent from data points to candidate exemplars and indicate how strongly each data point favors the candidate exemplar over other candidate exemplars. © “Availabilities” a(i,k) a r e s e n t from candidate exemplars to data points and indicate to what degree each candidate exemplar is available as a cluster center for the data point. (D) The effect of the value of the input preference (common for all data points) on the number of identified exemplars (number of clusters) is shown. The value that was used in (A) is also shown, which was computed from the median of the pairwise similarities.
We next studied the problem of clustering images of faces using the standard optimiza-tion 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，我们使用平方误差的标准优化准则研究了人脸图像聚类的问题。 我们使用亲和力传播和k中心聚类来识别从Olivetti人脸数据库提取的900张灰度图像中的样本（3）。亲和力传播发现示例误差比100个k中心聚类的最佳结果的平方误差低得多（图。 2A），这花费了几乎相同的计算机时间。我们询问是否大量随机重启k-centers聚类可以实现相同的平方误差。 图2B显示了通过运行一次亲和力传播获得的误差以及通过运行10,000个k中心聚类获得的误差分布，并针对聚类数进行了绘制。 亲和力传播在不到两个数量级的时间内均匀地实现了低得多的误差。 另一个流行的优化标准是绝对像素差的总和（可以更好地容忍偏远像素强度），因此我们使用此误差度量重复了上述过程。 亲和传播再次均匀地实现了较低的误差（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 a n d r e p o r t e d i n (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 “nonexon 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 15 million) of data point pairs.
许多任务需要在稀疏相关数据中识别示例，即大多数相似性未知或大而为负的情况。 为了在这种情况下检查亲和力传播，我们使用从微阵列数据和稀疏相似矩阵导出的稀疏相似矩阵，解决了将假定外显子聚类以寻找基因的任务（4）。 在这项工作中，从小鼠染色体1的基因组中提取了与假定的外显子相对应的75,066个DNA片段（长60个碱基）。它们的转录水平在12个组织样本中进行了测量，每对假定的外显子之间都具有相似性（数据点） 计算。 推定外显子之间相似性的度量是基于它们在基因组中的接近度以及它们在12个组织中转录水平的协调程度。 为了说明不是外显子的推定外显子（例如内含子），我们加入了一个额外的人工样本，并使用整个数据集的统计数据确定了每个数据点与此“非外显子样本”的相似性。 所得的75,067×75,067相似性矩阵（3）由99.73％的相似性组成，具有-∞值，对应于可能不是同一基因一部分的遥远DNA片段。 我们将亲和力传播应用于此相似性矩阵，但是因为如果s（i，k）=-∞，则无需在点i和k之间交换消息，因此，亲和力传播的每次迭代都仅需要在很小的子集（0.27％或15）之间交换消息 万）的数据点对。
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 kcenters 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说明了基因簇的鉴定以及一些数据点对非外显子样本的分配。 在图3B中比较了亲和力传播和kcenters聚类的重构误差.对于每个数量的群集，亲和力传播运行一次，耗时6分钟，而k中心群集运行10,000次，耗时208小时。 为了解决这些方法在检测真实基因片段中的表现如何的问题，图3C使用RefSeq数据库中提供的标记绘制了真阳性（TP）率与假阳性（FP）率的关系图（5）。 亲和力传播实现了更高的TP速率，尤其是在低FP速率下，这对生物学家而言最重要。 在FP率为3％时，亲和力传播的TP率为39％，而最佳k中心聚类结果为17％。 为了进行比较，在相同的FP速率下，用于层次团聚聚类（2）的最佳TP率为19％，而（4）中所述的工程工具（它提供了更多的生物学知识）的TP率为43％。
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 ito sentence k based on the cost of encoding the words in sentenceiusing 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。与k-means聚类（1）等度量空间聚类技术不同，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）
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 in 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 York City, 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).
我们还应用了亲和力传播来探讨根据估计的商业航空公司旅行时间来确定数量有限的加拿大和美国城市，这些城市最容易被其他城市的大部分子集访问的问题。 每个数据点都是一个城市，相似度s（i，k）设置为乘飞机从城市i到城市k所花费的负时间，包括估计的中途停留时间（3）。
由于逆风，在许多情况下，传播时间会根据行进方向而有所不同，因此36％的相似性是不对称的。 此外，对于97％的城市对i和k，存在第三个城市j，从而违反了三角不等式，因为从i到k的行程在城市j中包含了较长的中途停留延迟，因此花费的时间比城市i的总和更长。 从i到j和j到k的旅行持续时间。当“最易到达的城市”的数量限制为7个（通过适当地调整输入偏好）时，确定了图4中的B到E所示的城市。 。 有趣的是，没有选择几个主要城市，这是因为繁重的国际旅行使它们不适合作为容易到达的国内目的地（例如，纽约市，洛杉矶），或者因为可以通过其他目的地（例如，亚特兰大）更有效地访问其社区 ，费城和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）。
Fig. 2. Clustering faces. Exemplars minimizing the standard squared error measure of similarity were identified from 900 normalized face images (3). For a common preference of −600, affinity propagation found 62 clusters, and the average squared error was 108. For comparison, the best of 100 runs of k-centers clustering with different random initializations achieved a worse average squared error of 119. (A) The 15 images with highest squared error under either affinity propagation or k-centers clustering are shown in the top row. The middle and bottom rows show the exemplars assigned by the two methods, and the boxes show which of the two methods performed better for that image, in terms of squared error. Affinity propagation found higher-quality exemplars. (B) T h e average squared error achieved by a single run of affinity propagation and 10,000 runs of k-centers clustering, versus the number of clusters. The colored bands show different percentiles of squared error, and the number of exemplars corresponding to the result from (A) is indicated. © The above procedure was repeated using the sum of absolute errors as the measure of similarity, which is also a popular optimization criterion.
图2。聚集人脸。从900幅标准化人脸图像中识别出最小化相似度标准平方误差的样本（3）。对于−600的共同偏好，亲和传播发现62个簇，平均平方误差为108。作为比较，在100次k-中心聚类中，使用不同随机初始化的最佳聚类结果的平均平方误差为119。（A）
最上面一行显示了在亲和传播或k-中心聚类下误差最大的15幅图像。中间和底部的行显示了这两种方法指定的示例，方框显示了从平方误差的角度看，这两种方法中哪一种对该图像的性能更好。亲和传播发现了更高质量的样本。（B）
单次亲和力传播和10000次k-中心聚类所获得的平均平方误差与聚类数量的关系。彩色带显示不同百分位的平方误差，并指出与（A）结果相对应的样本数。（C）
以绝对误差之和作为相似性度量，重复上述过程，这也是一种流行的优化准则。
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© = −∑i=1 N s(i,ci). 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 constraints on the labels and the energy function (2).
可以将亲和传播视为一种搜索能量函数（7）的最小值的方法，该函数取决于与N个数据点相对应的一组N个隐藏标签c1，…，cN。 每个标签都指示该点所属的样本，因此s（i，ci）是数据点i与其样本的相似性。 ci = i是一种特殊情况，指示点i本身就是一个示例，因此s（i，ci）是点i的输入偏好。 并非标签的所有配置都有效； 当对于每个点i，如果其他某个点i’选择i作为其示例（即ci’= i），则配置c就是有效的，那么我必须是一个示例（即ci = i）。 有效配置的能量为E（c）= -∑i = 1 N s（i，ci）。 精确地最小化能量在计算上是棘手的，因为这种最小化问题的特例是NP-hard k中值问题（8）。 但是，用于亲和力传播的更新规则对应于定点递归，用于最小化Bethe自由能（9）近似。 亲和力传播最容易在因子图（10）中作为max-sum算法的实例得出，该因子图描述了标签和能量函数（2）的约束。
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 a n d 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.
相对于相关技术，亲和传播具有多个优势。 诸如k中心聚类（1），k均值聚类（1）和期望最大化（EM）算法（11）之类的方法在每个步骤都存储了相对较小的估计聚类中心集。 这些技术通过从大量簇开始然后修剪的方法得到了改进（12），但是它们仍然依赖于随机采样并做出无法从中恢复的硬修剪决策。 相反，通过同时将所有数据点视为候选中心并逐渐识别群集，亲和力传播能够避免由于不幸的初始化和艰难的决策而导致的许多不良解决方案。 马尔可夫链蒙特卡罗技术（13）随机搜索良好的解决方案，但没有共享亲和力传播的优点，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.
层次性聚类聚类（14）和光谱聚类（15）解决了递归比较点对对以找到数据分区的完全不同的问题。 这些技术不需要群集内的所有点都类似于单个中心，因此不适合许多任务。 特别地，可以通过不幸的成对分组序列将不应该在同一聚类中的两个点分组在一起。
Fig. 3. Detecting genes. Affinity propagation was used to detect putative exons (data points) comprising genes from mouse chromosome 1. Here, squared error is not appropriate as a measure of similarity, but instead similarity values were derived from a cost function measuring proximity o f t h e p u t a t i v e e x o n s i n t h e g e n o m e a n d c o expression of the putative exons across 12 tissue samples (3). (A) A small portion of the data and the emergence of clusters during each iteration of affinity propagation are shown. In each picture, the 100 boxes outlined in black correspond to 100 data points (from a total of 75,066 putative exons), and the 12 colored blocks in each box indicate the transcription levels of the corresponding DNA segment in 12 tissue samples. The box on the far left corresponds to an artificial data point with infinite preference that is used to account for nonexon regions (e.g., introns). Lines connecting data points indicate potential assignments, where gray lines indicate assignments that currently have weak evidence and solid lines indicate assignments that currently have strong evidence. (B) Performance on minimizing the reconstruction error of genes, for different numbers of detected clusters. For each number of clusters, affinity propagation took 6 min, whereas 10,000 runs of k-centers clustering took 208 hours on the same computer. In each case, affinity propagation achieved a significantly lower reconstruction error than k-centers clustering. © A plot of true-positive rate versus false-positive rate for detecting exons [using labels from RefSeq (5)] shows that affinity propagation also performs better at detecting biologically verified exons than k-centers clustering.
图3.检测基因。 使用亲和力传播来检测包含来自小鼠染色体1的基因的推定外显子（数据点）。此处，平方误差不适合作为相似性的度量，而是相似性值是从测量推定外显子的接近度的成本函数推导得出的， 跨12个组织样本的外显子（3）。 （A）显示了数据的一小部分和亲和力传播的每次迭代过程中簇的出现。 在每张图片中，黑色轮廓的100个框对应于100个数据点（来自总共75,066个假定的外显子），每个框中的12个彩色块表示12个组织样本中相应DNA片段的转录水平。 最左侧的框对应于具有无限优先级的人工数据点，该人工数据点用于说明非外显子区域（例如内含子）。 连接数据点的线表示潜在的分配，其中灰线表示当前证据不足的分配，实线表示当前证据较强的分配。 （B）对于不同数目的检测到的簇，使基因的重建误差最小化的性能。 对于每个数量的群集，亲和力传播花费6分钟，而在同一台计算机上进行10,000次k-中心群集花费了208小时。 在每种情况下，亲和力传播实现的重建误差均远低于k中心聚类。 （C）检测外显子的真阳性率与假阳性率的关系图（使用来自RefSeq（5）的标记）显示，亲和力传播在检测经过生物学验证的外显子方面也比k中心聚类更好。
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中值问题可以放宽以形成具有恒定因子近似的线性程序。 在那里，假定输入为度量，即非负，对称并满足三角不等式。 相反，亲和力传播可以将一般的非度量相似性作为输入。
亲和力传播还提供了一种在实践中效果很好的概念上新的方法。 线性编程的松弛难以解决，需要应用复杂的软件包（例如CPLEX），而亲缘传播则利用直观的消息更新功能，可以在几行代码中实现（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.
从本质上讲，亲和力传播与最近用于在完全不同的学科中获得破记录的结果的技术有关（16）。 在“循环图”中递归传播消息的方法（17）已被用于在纠错解码（18，19）中接近香农极限，解决了大小可乘数级的随机可满足性问题（20）， 解决NP硬二维相位展开问题的实例（21），并从成对的立体图像中有效地估计深度（22）。 然而，据我们所知，亲和力传播是利用此思想解决群集数据的古老基础问题的第一种方法。 由于它的简单性，普遍适用性和性能，我们相信亲和力传播将在科学和工程中被证明具有广泛的价值。
Fig. 4. Identifying key sentences and air-travel routing. Affinity propagation can be used to explore the identification of exemplars on the basis of nonstandard optimization criteria. (A) Similarities between pairs of sentences in a draft of this manuscript were constructed by matching words. Four exemplar sentences were identified by affinity propagation and are shown. (B) Affinity propagation was applied to similarities derived from air-travel efficiency (measured by estimated travel time) between the 456 busiest commercial airports in Canada and the United States—the travel times for both direct flights (shown in blue) and indirect flights (not shown), including the mean transfer time of up to a maximum of one stopover, were used as negative similarities (3). © Seven exemplars identified by affinity propagation are color-coded, and the assignments of other cities to these exemplars is shown. Cities located quite near to exemplar cities may be members of other more distant exemplars due to the lack of direct flights between them (e.g., Atlantic City is 100 km from Philadelphia, but is closer in flight time to Atlanta). (D) T h e i n s e t shows that the Canada-USA border roughly divides the Toronto and Philadelphia clusters, due to a larger availability of domestic flights compared to international flights. However, this is not the case on the west coast as shown in (E), because extraordinarily frequent airline service between Vancouver and Seattle connects Canadian cities in the northwest to Seattle.
图4.识别关键句子和空中旅行路线。 亲和力传播可用于基于非标准优化标准探索示例的识别。 （A）通过匹配单词来构造本手稿草稿中成对句子之间的相似性。 通过亲和力传播识别并显示了四个示例性句子。 （B）将亲和力传播应用于源自加拿大和美国456个最繁忙的商业机场之间的空中旅行效率（通过估计的旅行时间衡量）的相似性-直飞（蓝色显示）和间接飞行的旅行时间 （未显示）（包括最多最多一个中途停留的平均转移时间）被用作负相似度（3）。 （C）通过亲和力传播识别的七个样本进行了颜色编码，并显示了其他城市对这些样本的分配。 由于城市之间缺乏直飞航班，因此距离示例城市很近的城市可能是其他较远的示例的成员（例如，大西洋城距离费城100公里，但到亚特兰大的飞行时间更近）。 （D）情况表明，由于与国际航班相比，国内航班的可用性更大，因此加拿大-美国边界将多伦多和费城的集群大致划分。 但是，（E）中所示的西海岸情况并非如此，因为温哥华和西雅图之间非常频繁的航空服务将西北部的加拿大城市连接到西雅图。