GitHub user hl475 opened a pull request:
https://github.com/apache/spark/pull/16556
[SPARK-19184][MLlib] Improve numerical stability for method tallSkinnyQR.
## What changes were proposed in this pull request?
In method tallSkinnyQR, the final Q is calculated by A * inv(R). When the
upper triangular matrix R is ill-conditioned, computing the inverse of R can
result in catastrophic cancellation. Instead, we should consider using a
forward solve for solving Q such that Q * R = A.
## How was this patch tested?
unit tests and the following codes in Spark Shell:
```
import org.apache.spark.mllib.linalg.{Matrices, Vector, Vectors}
import org.apache.spark.mllib.linalg.distributed.RowMatrix
// Create RowMatrix A.
val mat = Seq(Vectors.dense(1,1,1,1), Vectors.dense(0, 1E-5, 1,1),
Vectors.dense(0,0,1E-10,1), Vectors.dense(0,0,0,1E-14))
val denseMat = new RowMatrix(sc.parallelize(mat, 2))
// Apply tallSkinnyQR to A.
val result = denseMat.tallSkinnyQR(true)
// Print the calculated Q and R.
result.Q.rows.collect.foreach(println)
result.R
// Calculate Q*R. Ideally, this should be close to A.
val reconstruct = result.Q.multiply(result.R)
reconstruct.rows.collect.foreach(println)
// Calculate Q'*Q. Ideally, this should be close to the identity matrix.
result.Q.computeGramianMatrix()
System.exit(0)
```
I first create a 4 by 4 RowMatrix A =
(1,1,1,1;0,1E-5,0,0;0,0,1E-10,1;0,0,0,1E-14), and then I apply method
tallSkinnyQR to A to find RowMatrix Q and Matrix R such that A = Q*R. In this
case, A is ill-conditioned and so is R.
You can merge this pull request into a Git repository by running:
$ git pull https://github.com/hl475/spark tallSkinnyQR
Alternatively you can review and apply these changes as the patch at:
https://github.com/apache/spark/pull/16556.patch
To close this pull request, make a commit to your master/trunk branch
with (at least) the following in the commit message:
This closes #16556
----
commit db737275d563b6057a5b4a2f116badc1f362f2f0
Author: Huamin Li <huami...@huamins-mbp.wireless.yale.internal>
Date: 2017-01-12T02:13:07Z
modify tallSkinnyQR so it will use forward solve instead of computing the
inverse of upper triangular matrix R.
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