Author: squinn
Date: Fri May 2 20:21:23 2014
New Revision: 1592026
URL: http://svn.apache.org/r1592026
Log:
Fixed the latex syntax to match mathjax.
Modified:
mahout/site/mahout_cms/trunk/content/users/clustering/spectral-clustering.mdtext
Modified:
mahout/site/mahout_cms/trunk/content/users/clustering/spectral-clustering.mdtext
URL:
http://svn.apache.org/viewvc/mahout/site/mahout_cms/trunk/content/users/clustering/spectral-clustering.mdtext?rev=1592026&r1=1592025&r2=1592026&view=diff
==============================================================================
---
mahout/site/mahout_cms/trunk/content/users/clustering/spectral-clustering.mdtext
(original)
+++
mahout/site/mahout_cms/trunk/content/users/clustering/spectral-clustering.mdtext
Fri May 2 20:21:23 2014
@@ -6,13 +6,13 @@ Spectral clustering, as its name implies
At its simplest, spectral clustering relies on the following four steps:
- 1. Computing a similarity (or _affinity_) matrix $\mathbf{A}$ from the data.
This involves determining a pairwise distance function $f$ that takes a pair of
data points and returns a scalar.
+ 1. Computing a similarity (or _affinity_) matrix \(\mathbf{A}\) from the
data. This involves determining a pairwise distance function \(f\) that takes a
pair of data points and returns a scalar.
- 2. Computing a graph Laplacian $\mathbf{L}$ from the affinity matrix. There
are several types of graph Laplacians; which is used will often depends on the
situation.
+ 2. Computing a graph Laplacian \(\mathbf{L}\) from the affinity matrix. There
are several types of graph Laplacians; which is used will often depends on the
situation.
- 3. Computing the eigenvectors and eigenvalues of $\mathbf{L}$. The degree of
this decomposition is often modulated by $k$, or the number of clusters. Put
another way, $k$ eigenvectors and eigenvalues are computed.
+ 3. Computing the eigenvectors and eigenvalues of \(\mathbf{L}\). The degree
of this decomposition is often modulated by \(k\), or the number of clusters.
Put another way, \(k\) eigenvectors and eigenvalues are computed.
- 4. The $k$ eigenvectors are used as "proxy" data for the original dataset,
and fed into k-means clustering. The resulting cluster assignments are
transparently passed back to the original data.
+ 4. The \(k\) eigenvectors are used as "proxy" data for the original dataset,
and fed into k-means clustering. The resulting cluster assignments are
transparently passed back to the original data.
For more theoretical background on spectral clustering, such as how affinity
matrices are computed, the different types of graph Laplacians, and whether the
top or bottom eigenvectors and eigenvalues are computed, please read [Ulrike
von Luxburg's article in _Statistics and Computing_ from December
2007](http://link.springer.com/article/10.1007/s11222-007-9033-z). It provides
an excellent description of the linear algebra operations behind spectral
clustering, and imbues a thorough understanding of the types of situations in
which it can be used.
@@ -24,11 +24,11 @@ As of Mahout 0.3, spectral clustering ha
## Input
-The input format for the algorithm currently takes the form of a Hadoop-backed
affinity matrix, in text form. Each line of the text file specifies a single
element of the affinity matrix: the row index $i$, the column index $j$, and
the value:
+The input format for the algorithm currently takes the form of a Hadoop-backed
affinity matrix, in text form. Each line of the text file specifies a single
element of the affinity matrix: the row index \(i\), the column index \(j\),
and the value:
`i, j, value`
-The affinity matrix is symmetric, and any unspecified $i, j$ pairs are assumed
to be 0 for sparsity. The row and column indices are 0-indexed. Thus, only the
non-zero entries of either the upper or lower triangular need be specified.
+The affinity matrix is symmetric, and any unspecified \(i, j\) pairs are
assumed to be 0 for sparsity. The row and column indices are 0-indexed. Thus,
only the non-zero entries of either the upper or lower triangular need be
specified.
**([MAHOUT-1539](https://issues.apache.org/jira/browse/MAHOUT-1539) will allow
for the creation of the affinity matrix to occur as part of the core spectral
clustering algorithm, as opposed to the current requirement that the user
create this matrix themselves and provide it, rather than the original data, to
the algorithm)**
