25ad8f9301
While play-testing the Scala and Java code examples in the MLlib docs, I noticed a number of small compile errors, and some typos. This led to finding and fixing a few similar items in other docs. Then in the course of building the site docs to check the result, I found a few small suggestions for the build instructions. I also found a few more formatting and markdown issues uncovered when I accidentally used maruku instead of kramdown. Author: Sean Owen <sowen@cloudera.com> Closes #653 from srowen/SPARK-1727 and squashes the following commits: 6e7c38a [Sean Owen] Final doc updates - one more compile error, and use of mean instead of sum and count 8f5e847 [Sean Owen] Fix markdown syntax issues that maruku flags, even though we use kramdown (but only those that do not affect kramdown's output) 99966a9 [Sean Owen] Update issue tracker URL in docs 23c9ac3 [Sean Owen] Add Scala Naive Bayes example, to use existing example data file (whose format needed a tweak) 8c81982 [Sean Owen] Fix small compile errors and typos across MLlib docs
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| global | <a href="mllib-guide.html">MLlib</a> - Dimensionality Reduction |
- Table of contents {:toc}
Dimensionality reduction is the process of reducing the number of variables under consideration. It is used to extract latent features from raw and noisy features, or compress data while maintaining the structure. In this release, we provide preliminary support for dimensionality reduction on tall-and-skinny matrices.
Singular value decomposition (SVD)
Singular value decomposition (SVD)
factorizes a matrix into three matrices: U, \Sigma, and V such that
\[ A = U \Sigma V^T, \]
where
Uis an orthonormal matrix, whose columns are called left singular vectors,\Sigmais a diagonal matrix with non-negative diagonals in descending order, whose diagonals are called singular values,Vis an orthonormal matrix, whose columns are called right singular vectors.
For large matrices, usually we don't need the complete factorization but only the top singular values and its associated singular vectors. This can save storage, and more importantly, de-noise and recover the low-rank structure of the matrix.
If we keep the top k singular values, then the dimensions of the return will be:
$U$:$m \times k$,$\Sigma$:$k \times k$,$V$:$n \times k$.
In this release, we provide SVD computation to row-oriented matrices that have only a few columns,
say, less than 1000, but many rows, which we call tall-and-skinny.
val mat: RowMatrix = ...
// Compute the top 20 singular values and corresponding singular vectors. val svd: SingularValueDecomposition[RowMatrix, Matrix] = mat.computeSVD(20, computeU = true) val U: RowMatrix = svd.U // The U factor is a RowMatrix. val s: Vector = svd.s // The singular values are stored in a local dense vector. val V: Matrix = svd.V // The V factor is a local dense matrix. {% endhighlight %}
Principal component analysis (PCA)
Principal component analysis (PCA) is a statistical method to find a rotation such that the first coordinate has the largest variance possible, and each succeeding coordinate in turn has the largest variance possible. The columns of the rotation matrix are called principal components. PCA is used widely in dimensionality reduction.
In this release, we implement PCA for tall-and-skinny matrices stored in row-oriented format.
The following code demonstrates how to compute principal components on a tall-and-skinny RowMatrix
and use them to project the vectors into a low-dimensional space.
The number of columns should be small, e.g, less than 1000.
{% highlight scala %} import org.apache.spark.mllib.linalg.Matrix import org.apache.spark.mllib.linalg.distributed.RowMatrix
val mat: RowMatrix = ...
// Compute the top 10 principal components. val pc: Matrix = mat.computePrincipalComponents(10) // Principal components are stored in a local dense matrix.
// Project the rows to the linear space spanned by the top 10 principal components. val projected: RowMatrix = mat.multiply(pc) {% endhighlight %}