spark-instrumented-optimizer/docs/mllib-linear-algebra.md

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global	MLlib - Linear Algebra

• Table of contents {:toc}

Singular Value Decomposition

Singular Value Decomposition for Tall and Skinny matrices. Given an $m \times n$ matrix $A$, we can compute matrices $U,S,V$ such that

\[ A = U \cdot S \cdot V^T \]

There is no restriction on m, but we require n^2 doubles to fit in memory locally on one machine. Further, n should be less than m.

The decomposition is computed by first computing $A^TA = V S^2 V^T$, computing SVD locally on that (since $n \times n$ is small), from which we recover $S$ and $V$. Then we compute U via easy matrix multiplication as $U = A \cdot V \cdot S^{-1}$.

Only singular vectors associated with largest k singular values are recovered. If there are k such values, then the dimensions of the return will be:

• $S$ is $k \times k$ and diagonal, holding the singular values on diagonal.
• $U$ is $m \times k$ and satisfies $U^T U = \mathop{eye}(k)$.
• $V$ is $n \times k$ and satisfies $V^T V = \mathop{eye}(k)$.

All input and output is expected in sparse matrix format, 0-indexed as tuples of the form ((i,j),value) all in SparseMatrix RDDs. Below is example usage.

{% highlight scala %}

import org.apache.spark.SparkContext import org.apache.spark.mllib.linalg.SVD import org.apache.spark.mllib.linalg.SparseMatrix import org.apache.spark.mllib.linalg.MatrixEntry

// Load and parse the data file val data = sc.textFile("mllib/data/als/test.data").map { line => val parts = line.split(',') MatrixEntry(parts(0).toInt, parts(1).toInt, parts(2).toDouble) } val m = 4 val n = 4 val k = 1

// recover largest singular vector val decomposed = SVD.sparseSVD(SparseMatrix(data, m, n), k) val = decomposed.S.data

println("singular values = " + s.toArray.mkString) {% endhighlight %}