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layout: default
title: Spectral Clustering
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# *StreamingKMeans* algorithm
The *StreamingKMeans* algorithm is a variant of Algorithm 1 from [Shindler et al][1] and consists of two steps:
1. Streaming step
2. BallKMeans step.
The streaming step is a randomized algorithm that makes one pass through the data and
produces as many centroids as it determines is optimal. This step can be viewed as
a preparatory dimensionality reduction. If the size of the data stream is *n* and the
expected number of clusters is *k*, the streaming step will produce roughly *k\*log(n)*
clusters that will be passed on to the BallKMeans step which will further reduce the
number of clusters down to *k*. BallKMeans is a randomized Lloyd-type algorithm that
has been studied in detail, see [Ostrovsky et al][2].
## Streaming step
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### Overview
The streaming step is a derivative of the streaming
portion of Algorithm 1 in [Shindler et al][1]. The main difference between the two is that
Algorithm 1 of [Shindler et al][1] assumes
the knowledge of the size of the data stream and uses it to set a key parameter
for the algorithm. More precisely, the initial *distanceCutoff* (defined below), which is
denoted by *f* in [Shindler et al][1], is set to *1/(k(1+log(n))*. The *distanceCutoff* influences the number of clusters that the algorithm
will produce.
In contrast, Mahout implementation does not require the knowledge of the size of the
data stream. Instead, it dynamically re-evaluates the parameters that depend on the size
of the data stream at runtime as more and more data is processed. In particular,
the parameter *numClusters* (defined below) changes its value as the data is processed.
###Parameters
- **numClusters** (int): Conceptually, *numClusters* represents the algorithm's guess at the optimal
number of clusters it is shooting for. In particular, *numClusters* will increase at run
time as more and more data is processed. Note that â€˘numClustersâ€˘ is not the number of clusters that the algorithm will produce. Also, *numClusters* should not be set to the final number of clusters that we expect to receive as the output of *StreamingKMeans*.
- **distanceCutoff** (double): a parameter representing the value of the distance between a point and
its closest centroid after which
the new point will definitely be assigned to a new cluster. *distanceCutoff* can be thought
of as an estimate of the variable *f* from Shindler et al. The default initial value for
*distanceCutoff* is *1.0/numClusters* and *distanceCutoff* grows as a geometric progression with
common ratio *beta* (see below).
- **beta** (double): a constant parameter that controls the growth of *distanceCutoff*. If the initial setting of *distanceCutoff* is *d0*, *distanceCutoff* will grow as the geometric progression with initial term *d0* and common ratio *beta*. The default value for *beta* is 1.3.
- **clusterLogFactor** (double): a constant parameter such that *clusterLogFactor* *log(numProcessedPoints)* is the runtime estimate of the number of clusters to be produced by the streaming step. If the final number of clusters (that we expect *StreamingKMeans* to output) is *k*, *clusterLogFactor* can be set to *k*.
- **clusterOvershoot** (double): a constant multiplicative slack factor that slows down the collapsing of clusters. The default value is 2.
###Algorithm
The algorithm processes the data one-by-one and makes only one pass through the data.
The first point from the data stream will form the centroid of the first cluster (this designation may change as more points are processed). Suppose there are *r* clusters at one point and a new point *p* is being processed. The new point can either be added to one of the existing *r* clusters or become a new cluster. To decide:
- let *c* be the closest cluster to point *p*
- let *d* be the distance between *c* and *p*
- if *d > distanceCutoff*, create a new cluster from *p* (*p* is too far away from the clusters to be part of any one of them)
- else (*d <= distanceCutoff*), create a new cluster with probability *d / distanceCutoff* (the probability of creating a new cluster increases as *d* increases).
There will be either *r* or *r+1* clusters after processing a new point.
As the number of clusters increases, it will go over the *clusterOvershoot \* numClusters* limit (*numClusters* represents a recommendation for the number of clusters that the streaming step should aim for and *clusterOvershoot* is the slack). To decrease the number of clusters the existing clusters
are treated as data points and are re-clustered (collapsed). This tends to make the number of clusters go down. If the number of clusters is still too high, *distanceCutoff* is increased.
## BallKMeans step
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### Overview
The algorithm is a Lloyd-type algorithm that takes a set of weighted vectors and returns k centroids, see [Ostrovsky et al][2] for details. The algorithm has two stages:
1. Seeding
2. Ball k-means
The seeding stage is an initial guess of where the centroids should be. The initial guess is improved using the ball k-means stage.
### Parameters
* **numClusters** (int): the number k of centroids to return. The algorithm will return exactly this number of centroids.
* **maxNumIterations** (int): After seeding, the iterative clustering procedure will be run at most *maxNumIterations* times. 1 or 2 iterations are recommended. Increasing beyond this will increase the accuracy of the result at the expense of runtime. Each successive iteration yields diminishing returns in lowering the cost.
* **trimFraction** (double): Outliers are ignored when computing the center of mass for a cluster. For any datapoint *x*, let *c* be the nearest centroid. Let *d* be the minimum distance from *c* to another centroid. If the distance from *x* to *c* is greater than *trimFraction \* d*, then *x* is considered an outlier during that iteration of ball k-means. The default is 9/10. In [Ostrovsky et al][2], the authors use *trimFraction* = 1/3, but this does not mean that 1/3 is optimal in practice.
* **kMeansPlusPlusInit** (boolean): If true, the seeding method is k-means++. If false, the seeding method is to select points uniformly at random. The default is true.
* **correctWeights** (boolean): If *correctWeights* is true, outliers will be considered when calculating the weight of centroids. The default is true. Note that outliers are not considered when calculating the position of centroids.
* **testProbability** (double): If *testProbability* is *p* (0 < *p* < 1), the data (of size n) is partitioned into a test set (of size *p\*n*) and a training set (of size *(1-p)\*n*). If 0, no test set is created (the entire data set is used for both training and testing). The default is 0.1 if *numRuns* > 1. If *numRuns* = 1, then no test set should be created (since it is only used to compare the cost between different runs).
* **numRuns** (int): This is the number of runs to perform. The solution of lowest cost is returned. The default is 1 run.
###Algorithm
The algorithm can be instructed to take multiple independent runs (using the *numRuns* parameter) and the algorithm will select the best solution (i.e., the one with the lowest cost). In practice, one run is sufficient to find a good solution.
Each run operates as follows: a seeding procedure is used to select k centroids, and then ball k-means is run iteratively to refine the solution.
The seeding procedure can be set to either 'uniformly at random' or 'k-means++' using *kMeansPlusPlusInit* boolean variable. Seeding with k-means++ involves more computation but offers better results in practice.
Each iteration of ball k-means runs as follows:
1. Clusters are formed by assigning each datapoint to the nearest centroid
2. The centers of mass of the trimmed clusters (see *trimFraction* parameter above) become the new centroids
The data may be partitioned into a test set and a training set (see *testProbability*). The seeding procedure and ball k-means run on the training set. The cost is computed on the test set.
##Usage of *StreamingKMeans*
bin/mahout streamingkmeans
-i
-o