view release on metacpan or  search on metacpan

lib/Algorithm/LinearManifoldDataClusterer.pm  view on Meta::CPAN

As to why we need machine learning and data clustering on manifolds, there exist many
important applications in which the measured data resides on a nonlinear manifold.
For example, when you record images of a human face from different angles, all the
image pixels taken together fall on a low-dimensional surface in a high-dimensional
measurement space. The same is believed to be true for the satellite images of a land
mass that are recorded with the sun at different angles with respect to the direction
of the camera.

Reducing the dimensionality of the sort of data mentioned above is critical to the
proper functioning of downstream classification algorithms, and the most popular
traditional method for dimensionality reduction is the Principal Components Analysis
(PCA) algorithm.  However, using PCA is tantamount to passing a linear least-squares
hyperplane through the surface on which the data actually resides.  As to why that
might be a bad thing to do, just imagine the consequences of assuming that your data
falls on a straight line when, in reality, it falls on a strongly curving arc.  This
is exactly what happens with PCA --- it gives you a linear manifold approximation to
your data that may actually reside on a curved surface.

That brings us to the purpose of this module, which is to cluster data that resides
on a nonlinear manifold.  Since a nonlinear manifold is locally linear, we can think
of each data cluster on a nonlinear manifold as falling on a locally linear portion
of the manifold, meaning on a hyperplane.  The logic of the module is based on
finding a set of hyperplanes that best describes the data, with each hyperplane
derived from a local data cluster.  This is like constructing a piecewise linear
approximation to data that falls on a curve as opposed to constructing a single
straight line approximation to all of the data.  So whereas the frequently used PCA
algorithm gives you a single hyperplane approximation to all your data, what this
module returns is a set of hyperplane approximations, with each hyperplane derived by
applying the PCA algorithm locally to a data cluster.

That brings us to the problem of how to actually discover the best set of hyperplane
approximations to the data.  What is probably the most popular algorithm today for
that purpose is based on the following key idea: Given a set of subspaces to which a
data element can be assigned, you assign it to that subspace for which the
B<reconstruction error> is the least.  But what do we mean by a B<subspace> and what
is B<reconstruction error>?

To understand the notions of B<subspace> and B<reconstruction-error>, let's revisit
the traditional approach of dimensionality reduction by the PCA algorithm.  The PCA
algorithm consists of: (1) Subtracting from each data element the global mean of the
data; (2) Calculating the covariance matrix of the data; (3) Carrying out an
eigendecomposition of the covariance matrix and ordering the eigenvectors according
to decreasing values of the corresponding eigenvalues; (4) Forming a B<subspace> by
discarding the trailing eigenvectors whose corresponding eigenvalues are relatively
small; and, finally, (5) projecting all the data elements into the subspace so
formed. The error incurred in representing a data element by its projection into the
subspace is known as the B<reconstruction error>.  This error is the projection of
the data element into the space spanned by the discarded trailing eigenvectors.

I<In linear-manifold based machine learning, instead of constructing a single
subspace in the manner described above, we construct a set of subspaces, one for each
data cluster on the nonlinear manifold.  After the subspaces have been constructed, a
data element is assigned to that subspace for which the reconstruction error is the
least.> On the face of it, this sounds like a chicken-and-egg sort of a problem.  You
need to have already clustered the data in order to construct the subspaces at
different places on the manifold so that you can figure out which cluster to place a
data element in.

Such problems, when they do possess a solution, are best tackled through iterative
algorithms in which you start with a guess for the final solution, you rearrange the
measured data on the basis of the guess, and you then use the new arrangement of the
data to refine the guess.  Subsequently, you iterate through the second and the third
steps until you do not see any discernible changes in the new arrangements of the
data.  This forms the basis of the clustering algorithm that is described under
B<Phase 1> in the section that follows.  This algorithm was first proposed in the
article "Dimension Reduction by Local Principal Component Analysis" by Kambhatla and
Leen that appeared in the journal Neural Computation in 1997.

Unfortunately, experiments show that the algorithm as proposed by Kambhatla and Leen
is much too sensitive to how the clusters are seeded initially.  To get around this
limitation of the basic clustering-by-minimization-of-reconstruction-error, this
module implements a two phased approach.  In B<Phase 1>, we introduce a multiplier
effect in our search for clusters by looking for C<M*K> clusters instead of the main
C<K> clusters.  In this manner, we increase the odds that each original cluster will
be visited by one or more of the C<M*K> randomly selected seeds at the beginning,
where C<M> is the integer value given to the constructor parameter
C<cluster_search_multiplier>.  Subsequently, we merge the clusters that belong
together in order to form the final C<K> clusters.  That work is done in B<Phase 2>
of the algorithm.

For the cluster merging operation in Phase 2, we model the C<M*K> clusters as the
nodes of an attributed graph in which the weight given to an edge connecting a pair
of nodes is a measure of the similarity between the two clusters corresponding to the
two nodes.  Subsequently, we use spectral clustering to merge the most similar nodes
in our quest to partition the data into C<K> clusters.  For that purpose, we use the
Shi-Malik normalized cuts algorithm.  The pairwise node similarity required by this
algorithm is measured by the C<pairwise_cluster_similarity()> method of the
C<LinearManifoldDataClusterer> class.  The smaller the overall reconstruction error
when all of the data elements in one cluster are projected into the other's subspace
and vice versa, the greater the similarity between two clusters.  Additionally, the
smaller the distance between the mean vectors of the clusters, the greater the
similarity between two clusters.  The overall similarity between a pair of clusters
is a combination of these two similarity measures.

For additional information regarding the theoretical underpinnings of the algorithm
implemented in this module, visit
L<https://engineering.purdue.edu/kak/Tutorials/ClusteringDataOnManifolds.pdf>


=head1 SUMMARY OF THE ALGORITHM

We now present a summary of the two phases of the algorithm implemented in this
module.  Note particularly the important role played by the constructor parameter
C<cluster_search_multiplier>.  It is only when the integer value given to this
parameter is greater than 1 that Phase 2 of the algorithm kicks in.

=over 4

=item B<Phase 1:>

Through iterative minimization of the total reconstruction error, this phase of the
algorithm returns C<M*K> clusters where C<K> is the actual number of clusters you
expect to find in your data and where C<M> is the integer value given to the
constructor parameter C<cluster_search_multiplier>.  As previously mentioned, on
account of the sensitivity of the reconstruction-error based clustering to how the
clusters are initially seeded, our goal is to look for C<M*K> clusters with the idea
of increasing the odds that each of the C<K> clusters will see at least one seed at
the beginning of the algorithm.

=over 4

=item Step 1:

lib/Algorithm/LinearManifoldDataClusterer.pm  view on Meta::CPAN

Phase 1 of the algorithm to construct C<M*K> clusters as opposed to just C<K>
clusters. Subsequently, in Phase 2, the module uses inter-cluster similarity based
graph partitioning to merge the C<M*K> clusters into C<K> clusters.

=item C<max_iterations>:

This hard limits the number of iterations in Phase 1 of the algorithm.  Ordinarily,
the iterations stop automatically when the change in the total reconstruction error
from one iteration to the next is less than the value specified by the parameter
C<delta_reconstruction_error>.

=item C<delta_reconstruction_error>:

It is this parameter that determines when the iterations will actually terminate in
Phase 1 of the algorithm.  When the difference in the total reconstruction error from
one iteration to the next is less than the value given to this parameter, the
iterations come to an end. B<IMPORTANT: I have noticed that the larger the number of
data samples that need to be clustered, the larger must be the value give to this
parameter.  That makes intuitive sense since the total reconstruction error is the
sum of all such errors for all of the data elements.> Unfortunately, the best value
for this parameter does NOT appear to depend linearly on the total number of data
records to be clustered.

=item C<terminal_output>:

When this parameter is set, you will see in your terminal window the different
clusters as lists of the symbolic tags associated with the data records.  You will
also see in your terminal window the output produced by the different steps of the
graph partitioning algorithm as smaller clusters are merged to form the final C<K>
clusters --- assuming that you set the parameter C<cluster_search_multiplier> to an
integer value that is greater than 1.

=item C<visualize_each_iteration>:

As its name implies, when this option is set to 1, you'll see 3D plots of the
clustering results for each iteration --- but only if your data is 3-dimensional.

=item C<show_hidden_in_3D_plots>:

This parameter is important for controlling the visualization of the clusters on the
surface of a sphere.  If the clusters are too spread out, seeing all of the clusters
all at once can be visually confusing.  When you set this parameter, the clusters on
the back side of the sphere will not be visible.  Note that no matter how you set
this parameter, you can interact with the 3D plot of the data and rotate it with your
mouse pointer to see all of the data that is output by the clustering code.

=item C<make_png_for_each_iteration>:

If you set this option to 1, the module will output a Gnuplot in the form of a PNG
image for each iteration in Phase 1 of the algorithm.  In Phase 2, the module will
output the clustering result produced by the graph partitioning algorithm.

=back

=over

=item B<get_data_from_csv()>:

    $clusterer->get_data_from_csv();

As you can guess from its name, the method extracts the data from the CSV file named
in the constructor.

=item B<linear_manifold_clusterer()>:

    $clusterer->linear_manifold_clusterer();   

    or 

    my $clusters = $clusterer->linear_manifold_clusterer();

This is the main call to the linear-manifold based clusterer.  The first call works
by side-effect, meaning that you will see the clusters in your terminal window and
they would be written out to disk files (depending on the constructor options you
have set).  The second call also returns the clusters as a reference to an array of
anonymous arrays, each holding the symbolic tags for a cluster.

=item B<display_reconstruction_errors_as_a_function_of_iterations()>:

    $clusterer->display_reconstruction_errors_as_a_function_of_iterations();

This method would normally be called after the clustering is completed to see how the
reconstruction errors decreased with the iterations in Phase 1 of the overall
algorithm.

=item B<write_clusters_to_files()>:

    $clusterer->write_clusters_to_files($clusters);

As its name implies, when you call this methods, the final clusters would be written
out to disk files.  The files have names like:

     cluster0.txt 
     cluster1.txt 
     cluster2.txt
     ...
     ...

Before the clusters are written to these files, the module destroys all files with
such names in the directory in which you call the module.

=item B<visualize_clusters_on_sphere()>:

    $clusterer->visualize_clusters_on_sphere("final clustering", $clusters);

or

    $clusterer->visualize_clusters_on_sphere("final_clustering", $clusters, "png");

If your data is 3-dimensional and it resides on the surface of a sphere (or in the
vicinity of such a surface), you may be able to use these methods for the
visualization of the clusters produced by the algorithm.  The first invocation
produces a Gnuplot in a terminal window that you can rotate with your mouse pointer.
The second invocation produces a `.png' image of the plot.

=item B<auto_retry_clusterer()>:

    $clusterer->auto_retry_clusterer();

or