## Implementing Truncated Matrix Decompositions for Core.Matrix

Eigendecompositions and Singular Value Decompositions appear in a variety of settings in machine learning and data mining. The eigendecomposition looks like so:

$$\mathbf{A}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{-1}$$

$\mathbf{Q}$ contains the eigenvectors of $\mathbf{A}$ and $\mathbf{\Lambda}$ is a diagonal matrix containing the eigenvalues.

The singular value decomposition looks like:

$$\mathbf{A} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^*$$

$\mathbf{U}$ contains the eigenvectors of the covariance matrix $\mathbf{A}\mathbf{A^T}$. $\mathbf{V}$ contains the eigenvectors of the gram matrix $\mathbf{A^T}\mathbf{A}$.

The truncated variants of these decompositions allow us to compute only a few eigenvalues(vectors) or singular values (vectors).

This is important since (i) a lot of times, the smaller eigenvalues are discarded, and (ii) you don’t want to compute the entire decomposition and retain only a few of the rows and columns of the computed matrices each time.

For core.matrix, I implemented these truncated decompositions in Kublai. Details below.

For large matrices , on consumer grade hardware (like your laptop), it is near impossible to compute a full decomposition. If your algorithm performs a decomposition per iteration, then things get worse.

The excellent ARPACK library implements an efficient truncated SVD that is leveraged by several popular numerical libraries like the popular Python library scikit learn and Apache Spark.

ARPACK’s eigendecomposition is not tied to any particular matrix library. You only need to supply a routine that multiplies a vector with the input matrix (which ARPACK uses for its power-iteration step).

Clojure’s budding matrix library core.matrix implements both the eigen and singular value decompositions but doesn’t contain a truncated version.

In this post, I will describe how I used the ARPACK library to implement these truncated decompositions for core.matrix. The resulting implementation is independent of the particular implementation of core.matrix being used.

While the details of the post are tied to core.matrix, the lessons can be transferred to other numerical libraries in other languages.

The excellent netlib library provides a clean java interface to the ARPACK library (which is implemented in FORTRAN).

I will assume that you have ARPACK installed on your machine. If so, netlib will be able to invoke ARPACK.

### Truncated Eigendecomposition:

There are a symmetric eigendecomposition that I have currently implemented (and a non-symmetric version is on the way).

Invoking the truncated eigendecompositon routine is trival:

• Call the DSAUPD routine (for symmetric matrices. Use DNEUPD for the opposite) till the IDO flag is set to 99.

• Once this is done, the eigen vectors and values can be retrieved by a call to DSEUPD. The eigenpairs are returned in ascending order of eigenvalues.

And that’s it. There are a few flags that tell you if there are fatal errors which you need to check that the docs for DSAUPD and DSEUPD contain.

### Truncated Singular Value Decomposition:

The truncated SVD can just invoke the eigendecomposition on the gram and covariance matrices. No ARPACK calls are needed here.

The implementation for both the decompositions is available in this github repository.

### Usage

This module can be used in the following fashion:

For computing symmetric eigendecompositions:

  1 2 3 4 5 6 7 8 9 10 user> (def M (matrix [[1 2 3 4] [2 5 6 7] [3 6 8 9] [4 7 9 10]])) #'user/M user> (use 'kublai.core :reload-all) nil user> (eigs M 2 :symmetric) ;; compute 2 eigenvectors for this matrix {:Q [[-0.22593827269074584 -0.4432218615090191 -0.5727878807113498 -0.6514754961809404] [0.7253136654558885 0.3184697313242928 0.1424607347013554 -0.5934661371986827]], :A [[24.06253512439672 0.0] [0.0 -0.8054849155764637]]} user> 

For computing a truncated SVD:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 user> (def M (matrix [[1 2 3 4] [5 6 7 8] [9 10 11 12] [13 14 15 16]])) #'user/M user> (svd M 2) {:U [[-0.13472212372225584 0.8257420598345273] [-0.3407576960799602 0.4288172018031381] [-0.5467932684376645 0.03189234377176592] [-0.7528288407953688 -0.365032514259624]], :V* [[0.4284123959267892 0.4743725155726848 0.5203326352185806 0.5662927548644766] [0.7186534763126667 0.27380780936493887 -0.17103785758268963 -0.6158835245304229]], :S [[38.62265683187287 0.0] [0.0 2.0713230668787377]]} user> (clojure.pprint/pprint (svd M 2)) {:U [[-0.13472212372225592 -0.825742059834525] [-0.34075769607996026 -0.42881720180314464] [-0.5467932684376648 -0.03189234377175876] [-0.7528288407953688 0.3650325142596215]], :V* [[-0.4284123959267895 -0.4743725155726852 -0.5203326352185804 -0.5662927548644764] [-0.7186534763126535 -0.27380780936497917 0.1710378575827312 0.615883524530409]], :S [[38.62265683187287 0.0] [0.0 2.0713230668787403]]} nil 

### Efficiency

In this example I shall demonstrate how valuable a truncated SVD is. Say we have a very large matrix and we only need 10 singular values/vectors. We will be using the vectorz-clj implementation of core.matrix.

 1 2 3 4 5 6 (let [M1 (reshape (matrix (range 500000)) [10000 50]) M2 (reshape (matrix (range 5000000)) [10000 500]) M3 (reshape (matrix (range 5000000)) [1000 5000])] (time (kublai/svd M1 10)) (time (kublai/svd M2 10)) (time (kublai/svd M3 10))) 

Here, M1 is a 10000 x 50 matrix, M2 is a 10000 x 500 matrix, and M3 is a 1000 x 5000 matrix.

And the results are:

"Elapsed time: 17372.943 msecs"
"Elapsed time: 78085.404 msecs"
"Elapsed time: 41511.266 msecs"

Now say we had to run a full decomposition on these matrices (using the standard stuff that ships with core.matrix), the results look like:

"Elapsed time: 19617.777 msecs"
"Elapsed time: 157861.467 msecs"
"Elapsed time: 97627.002 msecs"

For the larger matrices this is nearly twice as long as the truncated versions.

(Experiments on a Macbook Air with 8 GB of memory and a 1.8 Ghz i5).