svn commit: r1575489 - in /mahout/site/mahout_cms/trunk/content/users/dim-reduction: ./ ssvd.mdtext

[dl...@apache.org]

Author: dlyubimov
Date: Sat Mar  8 05:31:28 2014
New Revision: 1575489

URL: http://svn.apache.org/r1575489
Log:
Trying to use CMS for the first time.

Added:
    mahout/site/mahout_cms/trunk/content/users/dim-reduction/
    mahout/site/mahout_cms/trunk/content/users/dim-reduction/ssvd.mdtext   (with props)

Added: mahout/site/mahout_cms/trunk/content/users/dim-reduction/ssvd.mdtext
URL: http://svn.apache.org/viewvc/mahout/site/mahout_cms/trunk/content/users/dim-reduction/ssvd.mdtext?rev=1575489&view=auto
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+Title:     Stochastic SVD
+Notice:    Licensed to the Apache Software Foundation (ASF) under one
+           or more contributor license agreements.  See the NOTICE file
+           distributed with this work for additional information
+           regarding copyright ownership.  The ASF licenses this file
+           to you under the Apache License, Version 2.0 (the
+           "License"); you may not use this file except in compliance
+           with the License.  You may obtain a copy of the License at
+           .
+             http://www.apache.org/licenses/LICENSE-2.0
+           .
+           Unless required by applicable law or agreed to in writing,
+           software distributed under the License is distributed on an
+           "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+           KIND, either express or implied.  See the License for the
+           specific language governing permissions and limitations
+           under the License.
+
+## Stochastic Singular Value Decomposition ##
+
+Stochastic SVD method in Mahout produces reduced rank Singular Value Decomposition output in its strict mathematical definition: A=USV'
+
+**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][1] "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.
+
+**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.
+
+
+  [1]: http://amath.colorado.edu/faculty/martinss/Pubs/2012_halko_dissertation.pdf
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