Stochastic Singular Value Decomposition

Stochastic SVD method in Mahout produces reduced rank Singular Value Decomposition output in its strict mathematical definition: ` (\mathbf{A\approx U}\boldsymbol{\Sigma}\mathbf{V}^{\top})`.

##The benefits over other methods are:

reduced flops required compared to Krylov subspace methods
In map-reduce world, a fixed number of MR iterations required regardless of rank requested
Tweak precision/speed balance with options.
A is a Distributed Row Matrix where rows may be identified by any Writable (such as a document path). As such, it would work directly on the output of seq2sparse.
As of 0.7 trunk, includes PCA and dimensionality reduction workflow (EXPERIMENTAL! Feedback on performance/other PCA related issues/ blogs is greatly appreciated.)

Map-Reduce characteristics:

SSVD uses at most 3 MR sequential steps (map-only + map-reduce + 2 optional parallel map-reduce jobs) to produce reduced rank approximation of U, V and S matrices. Additionally, two more map-reduce steps are added for each power iteration step if requested.

##Potential drawbacks:

potentially less precise (but adding even one power iteration seems to fix that quite a bit).

##Documentation

Overview and Usage

Note: Please use 0.6 or later! for PCA workflow, please use 0.7 or later.

##Publications

Nathan Halko’s dissertation “Randomized methods for computing low-rank approximations of matrices” contains comprehensive definition of parallelization strategy taken in Mahout SSVD implementation and also some precision/scalability benchmarks, esp. w.r.t. Mahout Lanczos implementation on a typical corpus data set.

Halko, Martinsson, Tropp paper discusses family of random projection-based algorithms and contains theoretical error estimates.

R simulation

Non-parallel SSVD simulation in R with power iterations and PCA options. Note that this implementation is not most optimal for sequential flow solver, but it is for demonstration purposes only.

However, try this R code to simulate a meaningful input:

tests.R

n<-1000
m<-2000
k<-10
 
qi<-1
 
#simulated input
svalsim<-diag(k:1)
 
usim<- qr.Q(qr(matrix(rnorm(m*k, mean=3), nrow=m,ncol=k)))
vsim<- qr.Q(qr( matrix(rnorm(n*k,mean=5), nrow=n,ncol=k)))
 
 
x<- usim %*% svalsim %*% t(vsim)

and try to compare ssvd.svd(x) and stock svd(x) performance for the same rank k, notice the difference in the running time. Also play with power iterations (qIter) and compare accuracies of standard svd and SSVD.

Note: numerical stability of R algorithms may differ from that of Mahout’s distributed version. We haven’t studied accuracy of the R simulation. For study of accuracy of Mahout’s version, please refer to Nathan’s dissertation as referenced above.

Modified SSVD Algorithm.

Given an \(m\times n\) matrix \(\mathbf{A}\), a target rank \(k\in\mathbb{N}_{1}\) , an oversampling parameter \(p\in\mathbb{N}_{1}\), and the number of additional power iterations \(q\in\mathbb{N}_{0}\), this procedure computes an \(m\times\left(k+p\right)\) SVD \(\mathbf{A\approx U}\boldsymbol{\Sigma}\mathbf{V}^{\top}\):

Create seed for random \(n\times\left(k+p\right)\) matrix \(\boldsymbol{\Omega}\). The seed defines matrix \(\mathbf{\Omega}\) using Gaussian unit vectors per one of suggestions in Halko, Martinsson, Tropp.
\(\mathbf{Y=A\boldsymbol{\Omega}},\,\mathbf{Y}\in\mathbb{R}^{m\times\left(k+p\right)}\)
Column-orthonormalize \(\mathbf{Y}\rightarrow\mathbf{Q}\) by computing thin decomposition \(\mathbf{Y}=\mathbf{Q}\mathbf{R}\). Also, \(\mathbf{Q}\in\mathbb{R}^{m\times\left(k+p\right)},\,\mathbf{R}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\). I denote this as \(\mathbf{Q}=\mbox{qr}\left(\mathbf{Y}\right).\mathbf{Q}\)
\(\mathbf{B}_{0}=\mathbf{Q}^{\top}\mathbf{A}:\,\,\mathbf{B}\in\mathbb{R}^{\left(k+p\right)\times n}\).
If \(q>0\) repeat: for \(i=1..q\): \(\mathbf{B}_{i}^{\top}=\mathbf{A}^{\top}\mbox{qr}\left(\mathbf{A}\mathbf{B}_{i-1}^{\top}\right).\mathbf{Q}\) (power iterations step).
Compute Eigensolution of a small Hermitian \(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}=\mathbf{\hat{U}}\boldsymbol{\Lambda}\mathbf{\hat{U}}^{\top}\), \(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\).
Singular values \(\mathbf{\boldsymbol{\Sigma}}=\boldsymbol{\Lambda}^{0.5}\), or, in other words, \(s_{i}=\sqrt{\sigma_{i}}\).
If needed, compute \(\mathbf{U}=\mathbf{Q}\hat{\mathbf{U}}\).
If needed, compute \(\mathbf{V}=\mathbf{B}_{q}^{\top}\hat{\mathbf{U}}\boldsymbol{\Sigma}^{-1}\). Another way is \(\mathbf{V}=\mathbf{A}^{\top}\mathbf{U}\boldsymbol{\Sigma}^{-1}\).