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diff --git a/docs/mllib-linear-algebra.md b/docs/mllib-linear-algebra.md new file mode 100644 index 0000000000..cc203d833d --- /dev/null +++ b/docs/mllib-linear-algebra.md @@ -0,0 +1,61 @@ +--- +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 %} |