# Support for dimensional reduction

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 | http://en.wikipedia.org/wiki/Lanczos_algorithm] 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.

## Lanczos

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 AA^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 AA^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).

### Parallelization Stragegy

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.

### Mahout usage

The Mahout DistributedLanzcosSolver is invoked by the