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+---
+layout: global
+title: 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 %}