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From conflue...@apache.org
Subject [CONF] Apache Mahout > Spectral Clustering
Date Fri, 29 Jul 2011 11:37:01 GMT
Space: Apache Mahout (https://cwiki.apache.org/confluence/display/MAHOUT)
Page: Spectral Clustering (https://cwiki.apache.org/confluence/display/MAHOUT/Spectral+Clustering)

Change Comment:
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added a few supporting links for b/g reading

Edited by Dan Brickley:
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Spectral clustering, a more powerful and specialized algorithm (compared to K-means), derives
its name from spectral analysis of a graph, which is how the data are represented. Each object
to be clustered can initially be represented as an _n_\-dimensional numeric vector, but the
difference with this algorithm is that there must also be some method for performing a comparison
between each object and expressing this comparison as a scalar.

This _n_ by _n_ comparison of all objects with all others forms the _affinity_ matrix, which
can be intuitively thought of as a rough representation of an underlying undirected, weighted,
and fully-connected graph whose edges express the relative relationships, or affinities, between
each pair of objects in the original data. This affinity matrix forms the basis from which
the two spectral clustering algorithms operate.

The equation by which the affinities are calculated can vary depending on the user's circumstances;
typically, the equation takes the form of:

exp( _d{_}{^}2^ / _c_ )

where _d_ is the Euclidean distance between a pair of points, and _c_ is a scaling factor.
_c_ is often calculated relative to a _k_\-neighborhood of closest points to the current point;
all other affinities are set to 0 outside of the neighborhood. Again, this formula can vary
depending on the situation (e.g. a fully-connected graph would ignore the _k_\-neighborhood
and calculate affinities for all pairs of points).

[Full overview on spectral clustering|http://spectrallyclustered.wordpress.com/2010/05/27/intro-and-spectral-clustering-101/]

h2. K-Means Spectral Clustering

h3. Overview

This consists of a few basic steps of generalized spectral clustering, followed by standard
k-means clustering over the intermediate results. Again, this process begins with an affinity
matrix *A* - whether or not it is fully-connected depends on the user's need.

*A* is then transformed into a pseudo-Laplacian matrix via a multiplication with a diagonal
matrix whose entries consist of the sums of the rows of *A*. The sums are modified to be the
inverse square root of their original values. The final operation looks something like:

L = D^{-1/2} A D^{-1/2}

*L* has some properties that are of interest to us; most importantly, while it is symmetric
like *A*, it has a more stable eigen-decomposition. *L* is decomposed into its constituent
eigenvectors and corresponding eigenvalues (though the latter will not be needed for future
calculations); the matrix of eigenvectors, *U*, is what we are now interested in.

Assuming *U* is a column matrix (the eigenvectors comprise the columns), then we will now
use the _rows_ of *U* as proxy data for the original data points. We will run each row through
standard K-means clustering, and the label that each proxy point receives will be transparently
assigned to the corresponding original data point, resulting in the final clustering assignments.

[Full overview on k-means spectral clustering|http://spectrallyclustered.wordpress.com/2010/06/05/sprint-1-k-means-spectral-clustering/]

h3. Implementation

The Mahout implementation consists of a single driver - SpectralKMeansDriver - calling upon
several common utilities. The driver performs the operations in sequential fashion: reading
in and constructing the affinity matrix, building the diagonal matrix, building the pseudo-Laplacian
and decomposing it, and clustering the components.

The affinity matrix input is the most important part of this algorithm. It consists of text
files which follow a specific format: that of a weighted, undirected graph. In order to represent
a graph in text files, each line of a text file represents a single directional edge between
two nodes. There are three comma-separated values on the line. The first number indicates
the source node, the second is the destination node, and the third is the weight. For example:

0, 1, 2.5

would indicate the directional edge from node 0 to node 1 has a weight of 2.5. *Please note:
as of 8/16/2010, Eigencuts assumes the affinity matrix is symmetric, hence there should be
a corresponding line in the text file of: 1, 0, 2.5.* Also, each node should be an integer
value.

M/R jobs written for SpectralKMeans:
* AffinityMatrixInputJob (reads the raw input into a DistributedRowMatrix)
* MatrixDiagonalizeJob (constructs the diagonal matrix)
* UnitVectorizerJob (converts the eigenvector matrix *U* to unit rows)
* VectorMatrixMultiplicationJob (multiplies *D* with *A*)

M/R jobs already in Mahout that were used:
* DistributedRowMatrix.transpose()
* DistributedLanczosSolver
* EigenVerfierJob
* KMeansDriver

h2. Eigencuts Spectral Clustering

h3. Overview

Intuitively, Eigencuts can be thought of as part 2 of the K-means algorithm, in that it performs
the same initial steps up until the k-means clustering. The algorithm uses the same affinity
matrix *A*, constructs the same diagonal matrix *D*, performs the same multiplication to create
the pseudo-Laplacian *L*, and conducts an eigen-decomposition on *L* to obtain the eigenvectors
and eigenvalues. But here is where the two algorithms differentiate.

For each eigenvector, we wish to determine how stable its flow of probability is within the
underlying graph of the original data. Intuitively, this is intrinsically related to the min-cut,
max-flow problem of finding bottlenecks: if we perturb the flow rate on a specific edge, and
overall the flow is stable, then we can conclude that this edge was not a bottleneck. If,
however, perturbing an edge significantly alters the overall flow, we know this edge's eigenflow
is very unstable and is a bottleneck.

We have an [explicit form|http://spectrallyclustered.files.wordpress.com/2010/07/sensitivityequation.png]
of this "sensitivity" calculation ([full post here, under "computing sensitivities"|http://spectrallyclustered.wordpress.com/2010/07/15/sprint-3-three-last-mr-tasks/]).
The next step is called "non-maximal suppression", which effectively means we will ignore
any of the calculated sensitivities for which there exists another more negative sensitivity
in the same neighborhood as the current one, effectively "suppressing" it.

This non-maximal suppression then plays a role in the final affinity-cutting step, where we
"cut" the affinity (set to 0) between any two nodes (effectively destroying the edge between
them) for which the sensitivity calculated at that edge passes some threshold, _and_ for which
the sensitivity was _not_ suppressed in the previous step.

Once the cutting has been completed, the process loops upon itself (starting with the recalculation
of *D* using the modified *A*) until no new cuts in *A* are made in the final step.

[Full overview on Eigencuts spectral clustering|http://spectrallyclustered.wordpress.com/2010/07/06/sprint-3-introduction-to-eigencuts/]

h3. Implementation

Since the first half of Eigencuts uses the same calculations as Spectral K-means, it uses
the same common M/R tasks, both those specific to spectral clustering, as well as those general
to Mahout. Unlike SpectralKMeans, however, there are no DistributedRowMatrix-specific operations
performed, and hence there is no need for the data type at all; Mahout Vectors are used heavily
instead.

Once the initial affinity matrix is constructed, there is a loop within the EigencutsDriver
over the calculation of *D*, the creation of *L* and its eigen-decomposition, the calculation
of the sensitivities, and the actual affinity cuts, such that the loop terminates only when
no cuts are made to the affinity matrix *A*. The final matrix *A* will then be representative
of a graph structure whose data points exist in intra-connected clusters.

M/R tasks specific to Eigencuts:
* EigencutsSensitivityJob (calculates the perturbation effects on edge weights)
* EigencutsAffinityCutsJob (sets edge weights to 0)

M/R tasks within spectral clustering:
* AffinityMatrixInputJob (reads the raw input into a DistributedRowMatrix)
* MatrixDiagonalizeJob (constructs the diagonal matrix)
* VectorMatrixMultiplicationJob (multiplies *D* with *A*)

M/R tasks general to Mahout:
* DistributedLanczosSolver
* EigenVerifierJob

h2. Quickstart

As noted before, the data for both these algorithms - the affinity matrix - is required to
be symmetric. As of the first release, there is no built-in mechanism in Mahout for generating
the affinities from raw data, as the formula this follows varies depending on the user's need,
so it is left as an exercise to the user to generate the affinities prior to using these algorithms.

The affinity input should follow the standard format for textual representation of a graph,
namely:

node_i, node_j, value_ij

For example, the following 3x3 affinity matrix:

|0.0|0.8|0.5|
|0.8|0.0|0.9|
|0.5|0.9|0.0|

would have the following textual input format:

0, 0, 0
0, 1, 0.8
0, 2, 0.5
1, 0, 0.8
1, 1, 0
1, 2, 0.9
2, 0, 0.5
2, 1, 0.9
2, 2, 0

Then simply invoke Spectral K-means or Eigencuts, using the input directory and setting the
other parameters as necessary (e.g. "k" for K-means, "beta" for Eigencuts, etc).

h2. Examples

*NOTE: Am still waiting for Carnegie Mellon/Univ. of Pittsburgh approval for official data
set*

For these algorithms, it is useful to have a viable example, so I have created a small but
effective synthetic data set to show how these algorithms operate. The raw data was generated
synthetically and [can be viewed here|http://dl.dropbox.com/u/1377610/rawdata.csv]. It consists
of 450 two-dimensional points drawn from 3 separate gaussian distributions their own means
and standard deviations ([here is an image of the plotted points|http://spectrallyclustered.files.wordpress.com/2010/07/clusters.png].
This same data in affinity matrix form looks like this: [view the affinity data set|http://dl.dropbox.com/u/1377610/affinity.csv]

In order to run the program, then, for spectral k-means:

bin/mahout spectralkmeans -i /path/to/directory/with/affinity/matrix -o /output/path -k 3
-d 450

and for eigencuts:

bin/mahout eigencuts -i /path/to/directory/with/affinity/matrix -o /output/path -b 2 -d 450

In both cases, the "-d" flag refers to the dimensionality of the affinity matrix; since there
are 450 points, this value will be 450. In spectral k-means, the "-k" is the number of clusters
to estimate. In Eigencuts, the "-b" refers to an initial estimate on the half-life of the
flow of probability in the graph; higher estimates correspond to fewer clusters.

Spectral k-means will eventually yield the clustering results, and there should only be a
single mistake: point 54. This is because that particular point is in between the two left-hand
clusters in the image, and so has high affinities with both clusters.

[Full overview of example here|http://spectrallyclustered.wordpress.com/2010/07/14/sprint-3-quick-update/]

h3. Resources

* http://spectrallyclustered.wordpress.com/2010/05/27/intro-and-spectral-clustering-101/
* http://www.stanford.edu/class/ee378B/papers/luxburg-spectral.pdf
* http://en.wikipedia.org/wiki/Laplacian_matrix
* http://en.wikipedia.org/wiki/Cluster_analysis#Spectral_clustering

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