62 lines
1.8 KiB
Markdown
62 lines
1.8 KiB
Markdown
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---
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layout: global
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title: MLlib - Linear Algebra
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---
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* Table of contents
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{:toc}
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# Singular Value Decomposition
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Singular Value `Decomposition` for Tall and Skinny matrices.
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Given an `$m \times n$` matrix `$A$`, we can compute matrices `$U,S,V$` such that
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`\[
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A = U \cdot S \cdot V^T
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\]`
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There is no restriction on m, but we require n^2 doubles to
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fit in memory locally on one machine.
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Further, n should be less than m.
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The decomposition is computed by first computing `$A^TA = V S^2 V^T$`,
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computing SVD locally on that (since `$n \times n$` is small),
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from which we recover `$S$` and `$V$`.
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Then we compute U via easy matrix multiplication
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as `$U = A \cdot V \cdot S^{-1}$`.
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Only singular vectors associated with largest k singular values
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are recovered. If there are k
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such values, then the dimensions of the return will be:
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* `$S$` is `$k \times k$` and diagonal, holding the singular values on diagonal.
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* `$U$` is `$m \times k$` and satisfies `$U^T U = \mathop{eye}(k)$`.
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* `$V$` is `$n \times k$` and satisfies `$V^T V = \mathop{eye}(k)$`.
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All input and output is expected in sparse matrix format, 0-indexed
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as tuples of the form ((i,j),value) all in
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SparseMatrix RDDs. Below is example usage.
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{% highlight scala %}
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import org.apache.spark.SparkContext
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import org.apache.spark.mllib.linalg.SVD
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import org.apache.spark.mllib.linalg.SparseMatrix
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import org.apache.spark.mllib.linalg.MatrixEntry
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// Load and parse the data file
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val data = sc.textFile("mllib/data/als/test.data").map { line =>
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val parts = line.split(',')
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MatrixEntry(parts(0).toInt, parts(1).toInt, parts(2).toDouble)
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}
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val m = 4
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val n = 4
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val k = 1
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// recover largest singular vector
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val decomposed = SVD.sparseSVD(SparseMatrix(data, m, n), k)
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val = decomposed.S.data
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println("singular values = " + s.toArray.mkString)
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{% endhighlight %}
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