add computational cost explanation

Author: rezazadeh

docs/mllib-dimensionality-reduction.md

@@ -39,6 +39,24 @@ If we keep the top $k$ singular values, then the dimensions of the resulting low
 * `$\Sigma$`: `$k \times k$`,
 * `$V$`: `$n \times k$`.
 
+### Performance
+We assume $n$ is smaller than $m$. The singular values and the right singular vectors are derived
+from the eigenvalues and the eigenvectors of the Gramian matrix $A^T A$. The matrix
+storing the right singular vectors $U$, is computed via matrix multiplication as
+$U = A (V S^{-1})$, if requested by user via the computeU parameter. 
+The actual method to use is determined automatically based on the computational cost:
+
+* If n is small ($n < 100$) or $k$ is large compared with $n$ ($k > n / 2$), we compute the Gramian matrix
+first and then compute its top eigenvalues and eigenvectors locally on the driver.
+This requires a single pass with $O(n^2)$ storage on each executor and on the driver, and
+$O(n^2 k)$ time on the driver.
+* Otherwise, we compute $(A^T A) v$ in a distributive way and send it to
+<a href="http://www.caam.rice.edu/software/ARPACK/">ARPACK</a> to
+compute $(A^T^ A)$'s top eigenvalues and eigenvectors on the driver node. This requires $O(k)$
+passes, $O(n)$ storage on each executor, and $O(n k)$ storage on the driver.
+
+## SVD Example
+ 
 MLlib provides SVD functionality to row-oriented matrices, provided in the
 <a href="mllib-basics.html#rowmatrix">RowMatrix</a> class.