[jira] [Created] (SPARK-19184) Improve numerical stability for method tallSkinnyQR.

["Huamin Li (JIRA)" <ji...@apache.org>]

Huamin Li created SPARK-19184:
---------------------------------

             Summary: Improve numerical stability for method tallSkinnyQR.
                 Key: SPARK-19184
                 URL: https://issues.apache.org/jira/browse/SPARK-19184
             Project: Spark
          Issue Type: Improvement
          Components: MLlib
    Affects Versions: 2.2.0
            Reporter: Huamin Li
            Priority: Minor


In method tallSkinnyQR, the final Q is calculated by A * inv(R) ([Github Link|https://github.com/apache/spark/blob/master/mllib/src/main/scala/org/apache/spark/mllib/linalg/distributed/RowMatrix.scala#L562]). 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.

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.

See codes in Spark Shell:
{code:none}
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)
{code}

it will output the following results:
{code:none}
scala> result.Q.rows.collect.foreach(println)
[1.0,0.0,0.0,1.5416524685312E13]
[0.0,0.9999999999999999,0.0,8011776.0]
[0.0,0.0,1.0,0.0]
[0.0,0.0,0.0,1.0]

scala> result.R
1.0  1.0     1.0      1.0
0.0  1.0E-5  1.0      1.0
0.0  0.0     1.0E-10  1.0
0.0  0.0     0.0      1.0E-14

scala> reconstruct.rows.collect.foreach(println)
[1.0,1.0,1.0,1.15416524685312]
[0.0,9.999999999999999E-6,0.9999999999999999,1.00000008011776]
[0.0,0.0,1.0E-10,1.0]
[0.0,0.0,0.0,1.0E-14]

scala> result.Q.computeGramianMatrix()
1.0                 0.0                 0.0  1.5416524685312E13
0.0                 0.9999999999999998  0.0  8011775.999999999
0.0                 0.0                 1.0  0.0
1.5416524685312E13  8011775.999999999   0.0  2.3766923337289844E26
{code}

With forward solve for solving Q such that Q * R = A rather than computing the inverse of R, it will output the following results instead:
{code:none}
scala> result.Q.rows.collect.foreach(println)
[1.0,0.0,0.0,0.0]
[0.0,1.0,0.0,0.0]
[0.0,0.0,1.0,0.0]
[0.0,0.0,0.0,1.0]

scala> result.R
1.0  1.0     1.0      1.0
0.0  1.0E-5  1.0      1.0
0.0  0.0     1.0E-10  1.0
0.0  0.0     0.0      1.0E-14

scala> reconstruct.rows.collect.foreach(println)
[1.0,1.0,1.0,1.0]
[0.0,1.0E-5,1.0,1.0]
[0.0,0.0,1.0E-10,1.0]
[0.0,0.0,0.0,1.0E-14]

scala> result.Q.computeGramianMatrix()
1.0  0.0  0.0  0.0
0.0  1.0  0.0  0.0
0.0  0.0  1.0  0.0
0.0  0.0  0.0  1.0
{code}




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