Mahout has a distributed implementation of QR decomposition for tall thin matrices1.

For the classic QR decomposition of the form `\(\mathbf{A}=\mathbf{QR},\mathbf{A}\in\mathbb{R}^{m\times n}\)`

a distributed version is fairly easily achieved if `\(\mathbf{A}\)`

is tall and thin such that `\(\mathbf{A}^{\top}\mathbf{A}\)`

fits in memory, i.e. *m* is large but *n* < ~5000 Under such circumstances, only `\(\mathbf{A}\)`

and `\(\mathbf{Q}\)`

are distributed matrices and `\(\mathbf{A^{\top}A}\)`

and `\(\mathbf{R}\)`

are in-core products. We just compute the in-core version of the Cholesky decomposition in the form of `\(\mathbf{LL}^{\top}= \mathbf{A}^{\top}\mathbf{A}\)`

. After that we take `\(\mathbf{R}= \mathbf{L}^{\top}\)`

and `\(\mathbf{Q}=\mathbf{A}\left(\mathbf{L}^{\top}\right)^{-1}\)`

. The latter is easily achieved by multiplying each vertical block of `\(\mathbf{A}\)`

by `\(\left(\mathbf{L}^{\top}\right)^{-1}\)`

. (There is no actual matrix inversion happening).

Mahout `dqrThin(...)`

is implemented in the mahout `math-scala`

algebraic optimizer which translates Mahoutâ€™s R-like linear algebra operators into a physical plan for both Spark and H2O distributed engines.

```
def dqrThin[K: ClassTag](A: DrmLike[K], checkRankDeficiency: Boolean = true): (DrmLike[K], Matrix) = {
if (drmA.ncol > 5000)
log.warn("A is too fat. A'A must fit in memory and easily broadcasted.")
implicit val ctx = drmA.context
val AtA = (drmA.t %*% drmA).checkpoint()
val inCoreAtA = AtA.collect
val ch = chol(inCoreAtA)
val inCoreR = (ch.getL cloned) t
if (checkRankDeficiency && !ch.isPositiveDefinite)
throw new IllegalArgumentException("R is rank-deficient.")
val bcastAtA = sc.broadcast(inCoreAtA)
val Q = A.mapBlock() {
case (keys, block) => keys -> chol(bcastAtA).solveRight(block)
}
Q -> inCoreR
}
```

The scala `dqrThin(...)`

method can easily be called in any Spark or H2O application built with the `math-scala`

library and the corresponding `Spark`

or `H2O`

engine module as follows:

```
import org.apache.mahout.math._
import decompositions._
import drm._
val(drmQ, inCoreR) = dqrThin(drma)
```