Matrix algebra underpins the way many Big Data algorithms and data structures are composed: full-text search can be viewed as doing matrix multiplication of the term-document matrix by the query vector (giving a vector over documents where the components are the relevance score), computing co-occurrences in a collaborative filtering context (people who viewed X also viewed Y, or ratings-based CF like the Netflix Prize contest) is taking the squaring the user-item interaction matrix, calculating users who are k-degrees separated from each other in a social network or web-graph can be found by looking at the k-fold product of the graph adjacency matrix, and the list goes on (and these are all cases where the linear structure of the matrix is preserved!)
Each of these examples deal with cases of matrices which tend to be tremendously large (often millions to tens of millions to hundreds of millions of rows or more, by sometimes a comparable number of columns), but also rather sparse. Sparse matrices are nice in some respects: dense matrices which are 10^7 on a side would have 100 trillion non-zero entries! But the sparsity is often problematic, because any given two rows (or columns) of the matrix may have zero overlap. Additionally, any machine-learning work done on the data which comprises the rows has to deal with what is known as "the curse of dimensionality", and for example, there are too many columns to train most regression or classification problems on them independently.
One of the more useful approaches to dealing with such huge sparse data sets is the concept of dimensionality reduction, where a lower dimensional space of the original column (feature) space of your data is found / constructed, and your rows are mapped into that subspace (or sub-manifold). In this reduced dimensional space, "important" components to distance between points are exaggerated, and unimportant ones washed away, and additionally, sparsity of your rows is traded for drastically reduced dimensional, but dense "signatures". While this loss of sparsity can lead to its own complications, a proper dimensionality reduction can help reveal the most important features of your data, expose correlations among your supposedly independent original variables, and smooth over the zeroes in your correlation matrix.
One of the most straightforward techniques for dimensionality reduction is the matrix decomposition: singular value decomposition, eigen decomposition, non-negative matrix factorization, etc. In their truncated form these decompositions are an excellent first approach toward linearity preserving unsupervised feature selection and dimensional reduction. Of course, sparse matrices which don't fit in RAM need special treatment as far as decomposition is concerned. Parallelizable and/or stream-oriented algorithms are needed.
Singular Value Decomposition
Currently implemented in Mahout (as of 0.3, the first release with MAHOUT-180 applied), are two scalable implementations of SVD, a stream-oriented implementation using the Asymmetric Generalized Hebbian Algorithm outlined in Genevieve Gorrell & Brandyn Webb's paper (Gorrell and Webb 2005); and there is a Lanczos implementation, both single-threaded, and in the o.a.m.math.decomposer.lanczos package (math module), as a hadoop map-reduce (series of) job(s) in o.a.m.math.hadoop.decomposer package (core module). Coming soon: stochastic decomposition.
SSVD method supports dimensionality reduction by means of PCA as an alternative to the Lanczos tutorial, see SSVD documentation for the details.
The Lanczos algorithm is designed for eigen-decomposition, but like any such algorithm, getting singular vectors out of it is immediate (singular vectors of matrix A are just the eigenvectors of A^t * A or A * A^t). Lanczos works by taking a starting seed vector v (with cardinality equal to the number of columns of the matrix A), and repeatedly multiplying A by the result: v' = A.times(v) (and then subtracting off what is proportional to previous v''s, and building up an auxiliary matrix of projections). In the case where A is not square (in general: not symmetric), then you actually want to repeatedly multiply A*A^t by v: v' = (A * A^t).times(v), or equivalently, in Mahout, A.timesSquared(v) (timesSquared is merely an optimization: by changing the order of summation in A*A^t.times(v), you can do the same computation as one pass over the rows of A instead of two).
After k iterations of v_i = A.timesSquared(v_(i-1)), a k- by -k tridiagonal matrix has been created (the auxiliary matrix mentioned above), out of which a good (often extremely good) approximation to k of the singular values (and with the basis spanned by the v_i, the k singular vectors may also be extracted) of A may be efficiently extracted. Which k? It's actually a spread across the entire spectrum: the first few will most certainly be the largest singular values, and the bottom few will be the smallest, but you have no guarantee that just because you have the n'th largest singular value of A, that you also have the (n-1)'st as well. A good rule of thumb is to try and extract out the top 3k singular vectors via Lanczos, and then discard the bottom two thirds, if you want primarily the largest singular values (which is the case for using Lanczos for dimensional reduction).
Lanczos is "embarassingly parallelizable": matrix multiplication of a matrix by a vector may be carried out row-at-a-time without communication until at the end, the results of the intermediate matrix-by-vector outputs are accumulated on one final vector. When it's truly A.times(v), the final accumulation doesn't even have collision / synchronization issues (the outputs are individual separate entries on a single vector), and multicore approaches can be very fast, and there should also be tricks to speed things up on Hadoop. In the asymmetric case, where the operation is A.timesSquared(v), the accumulation does require synchronization (the vectors to be summed have nonzero elements all across their range), but delaying writing to disk until Mapper close(), and remembering that having a Combiner be the same as the Reducer, the bottleneck in accumulation is nowhere near a single point.
The Mahout DistributedLanzcosSolver is invoked by the <MAHOUT_HOME>/bin/mahout svd command. This command takes the following arguments (which can be reproduced by just entering the command with no arguments):
The short form invocation may be used to perform the SVD on the input data:
The --input argument is the location on HDFS where a SequenceFile<Writable,VectorWritable> (preferably SequentialAccessSparseVectors instances) lies which you wish to decompose. Each vector of which has --numcols entries. --numRows is the number of input rows and is used to properly size the matrix data structures.
After execution, the --output directory will have a file named "rawEigenvectors" containing the raw eigenvectors. As the DistributedLanczosSolver sometimes produces "extra" eigenvectors, whose eigenvalues aren't valid, and also scales all of the eigenvalues down by the max eignenvalue (to avoid floating point overflow), there is an additional step which spits out the nice correctly scaled (and non-spurious) eigenvector/value pairs. This is done by the "cleansvd" shell script step (c.f. EigenVerificationJob).
If you have run he short form svd invocation above and require this "cleaning" of the eigen/singular output you can run "cleansvd" as a separate command:
The --corpusInput is the input path from the previous step, --eigenInput is the output from the previous step (<output>/rawEigenvectors), and --output is the desired output path (same as svd argument). The two "cleaning" params are --maxError - the maximum allowed 1-cosAngle(v, A.timesSquared(v)), and --minEigenvalue. Eigenvectors which have too large error, or too small eigenvalue are discarded. Optional argument: --inMemory, if you have enough memory on your local machine (not on the hadoop cluster nodes!) to load all eigenvectors into memory at once (at least 8 bytes/double * rank * numCols), then you will see some speedups on this cleaning process.
After execution, the --output directory will have a file named "cleanEigenvectors" containing the clean eigenvectors.
These two steps can also be invoked together by the svd command by using the long form svd invocation:
After execution, the --output directory will contain two files: the "rawEigenvectors" and the "cleanEigenvectors".
TODO: also allow exclusion based on improper orthogonality (currently computed, but not checked against constraints).
Example: SVD of ASF Mail Archives on Amazon Elastic MapReduce
This section walks you through a complete example of running the Mahout SVD job on Amazon Elastic MapReduce cluster and then preparing the output to be used for clustering. This example was developed as part of the effort to benchmark Mahout's clustering algorithms using a large document set (see MAHOUT-588). Specifically, we use the ASF mail archives located at http://aws.amazon.com/datasets/7791434387204566. You will need to likely run seq2sparse on these first. See $MAHOUT_HOME/examples/bin/build-asf-email.sh (on trunk) for examples of processing this data.
At a high level, the steps we're going to perform are:
bin/mahout svd (original -> svdOut)
bin/mahout cleansvd ...
bin/mahout transpose svdOut -> svdT
bin/mahout transpose original -> originalT
bin/mahout matrixmult originalT svdT -> newMatrix
bin/mahout kmeans newMatrix
Note: Some of this work is due in part to credits donated by the Amazon Elastic MapReduce team.
1. Launch EMR Cluster
For a detailed explanation of the steps involved in launching an Amazon Elastic MapReduce cluster for running Mahout jobs, please read the "Building Vectors for Large Document Sets" section of Mahout on Elastic MapReduce.
In the remaining steps below, remember to replace JOB_ID with the Job ID of your EMR cluster.
2. Load Mahout 0.5+ JAR into S3
These steps were created with the mahout-0.5-SNAPSHOT because they rely on the patch for MAHOUT-639
3. Copy TFIDF Vectors into HDFS
Before running your SVD job on the vectors, you need to copy them from S3 to your EMR cluster's HDFS.
4. Run the SVD Job
Now you're ready to run the SVD job on the vectors stored in HDFS:
This will run 100 iterations of the LanczosSolver SVD job to produce 87 eigenvectors in:
Only 87 eigenvectors were produced because of the cleanup step, which removes any duplicate eigenvectors caused by convergence issues and numeric overflow and any that don't appear to be "eigen" enough (ie, they don't satisfy the eigenvector criterion with high enough fidelity). - Jake Mannix
5. Transform your TFIDF Vectors into Mahout Matrix
The tfidf vectors created by the seq2sparse job are SequenceFile<Text,VectorWritable>. The Mahout RowId job transforms these vectors into a matrix form that is a SequenceFile<IntWritable,VectorWritable> and a SequenceFile<IntWritable,Text> (where the original one is the join of these new ones, on the new int key).
This is not a distributed job and will only run on the master server in your EMR cluster. The job produces the following output:
where docIndex is the SequenceFile<IntWritable,Text> and matrix is SequenceFile<IntWritable,VectorWritable>.
6. Transpose the Matrix
Our ultimate goal is to multiply the TFIDF vector matrix times our SVD eigenvectors. For the mathematically inclined, from the rowid job, we now have an m x n matrix T (m=6076937, n=20444). The SVD eigenvector matrix E is p x n (p=87, n=20444). So to multiply these two matrices, I need to transpose E so that the number of columns in T equals the number of rows in E (i.e. E^T is n x p) the result of the matrixmult would give me an m x p matrix (m=6076937, p=87).
However, in practice, computing the matrix product of two matrices as a map-reduce job is efficiently done as a map-side join on two row-based matrices with the same number of rows, and the columns are the ones which are different. In particular, if you take a matrix X which is represented as a set of numRowsX rows, each of which has numColsX, and another matrix with numRowsY == numRowsX, each of which has numColsY (!= numColsX), then by summing the outer-products of each of the numRowsX pairs of vectors, you get a matrix of with numRowsZ == numColsX, and numColsZ == numColsY (if you instead take the reverse outer product of the vector pairs, you can end up with the transpose of this final result, with numRowsZ == numColsY, and numColsZ == numColsX). - Jake Mannix
Thus, we need to transpose the matrix using Mahout's Transpose Job:
This job requires the patch to MAHOUT-639
The job creates the following output:
7. Transpose Eigenvectors
If you followed Jake's explanation in step 6 above, then you know that we also need to transpose the eigenvectors:
Note: You need to use the same number of reducers that was used for transposing the matrix you are multiplying the vectors with.
The job creates the following output:
8. Matrix Multiplication
Lastly, we need to multiply the transposed vectors using Mahout's matrixmult job:
This job produces output such as: