# RFC on Sparse matrices in R

Roger Koenker and Pin Ng have provided a sparse matrix implementation for R in the SparseM package, which is based on Fortran code in sparskit and a modified version of the sparse Cholesky factorization written by Esmond Ng and Barry Peyton. The modified version is distributed as part of PCx by Czyzyk, Mehrotra, Wagner, and Wright and is copywrite by the University of Chicago.

Recently I become very interested in certain sparse matrix calculations myself and have looked at some of the available Open Source software for the sparse Cholesky decomposition. While I certainly appreciate the work that Roger and Pin have done I will propose a slightly different implementation.

## Representations of sparse matrices

Conceptually, the simplest representation of a sparse matrix is as a triplet of an integer vector i giving the row numbers, an integer vector j giving the column numbers, and a numeric vector x giving the non-zero values in the matrix. An S4 class definition might be

```# Sparse general matrix in triplet format
setClass("tripletMatrix",
representation(i = "integer", j = "integer", x = "numeric",
Dim = "integer"))
```

The triplet representation is row-oriented if elements in the same row were adjacent and column-oriented if elements in the same column were adjacent. The compressed sparse row (csr) (or compressed sparse column - csc) representation is similar to row-oriented triplet (column-oriented triplet) except that i (j) just stores the index of the first element in the row (column). (There are a couple of other details but that is the gist of it.) These compressed representations remove the redundant row (column) indices and provide faster access to a given location in the matrix because you only need to check one row (column).

The preferred representation of sparse matrices in the SparseM package is csr. Matlab uses csc. We hope that Octave will also use this representation. There are certain advantages to csc in systems like R and Matlab where dense matrices are stored in column-major order. For example, Sivan Toledo's TAUCS library and Tim Davis's UMFPACK library are both based on csc and can both use level-3 BLAS in certain sparse matrix computations.

I feel that compatibility with Matlab (and, we hope, Octave), the ability to use level-3 BLAS, and the availability of the csc-based TAUCS, UMFPACK, and AMD libraries favors csc as the preferred sparse matrix representation.

## Applications of sparse matrices

I imagine that the main applications of sparse matrices in R will be for parameter estimation in very large linear models and for large sparse contingency tables.

As Roger and Pin have pointed out, the key to estimating parameters in large linear models quickly and with minimal storage requirements will be in providing a way for `model.matrix` to generate a sparse model matrix `X` or a sparse symmetric representation of `X'X` and `X'y`.

Assuming that we have a sparse representation of the model matrix as `mm` and a sparse or dense representation of the response as `y`, the coefficients can be estimated as

```solve(crossprod(mm), crossprod(mm, y))
```

I think that the multifrontal sparse Cholesky in the TAUCS library is one of the best currently available ways to do this and have implemented a solution based on that in the `taucs` package for R. I use the approximate minimal degree ordering determined by Tim Davis's AMD library to reduce fill-in.

For statistical analysis of a linear model we probably also want at least the standard errors of the coefficient estimates which means we want an inverse of the Cholesky factor. TAUCS has an inverse factorization routine `taucs_ccs_factor_xxt` that can provide a sparse representation of the inverse. I think that we want to use that for a linear model analysis. We can use the multifrontal solver if we only want coefficients.

When working with linear models there will be a tradeoff between the speed boost available by reordering rows and columns and the statistical information available in the original ordering of the rows and columns. For example, the simplest way to determine the sequential sums of squares of the terms in the model is to maintain the column ordering in `X` but that could result in dramatic amounts of fill-in for the sparse Cholesky and especially for the inverse factorization. I think it is best to compromise and obtain the inverse factorization of the reordered matrix. This can provide standard errors and correlations of coefficients but not the sequential sums of squares. (At least I don't know how to get them from the reordered matrix.)

Sparse contingency tables can be easily constructed and manipulated. I understand that Kurt Hornik would like to use them but I'm not sure exactly what operations he needs.

I have a hybrid application involving large linear mixed models with partially crossed grouping factors. For these I need to manipulate both sparse contingency tables and some associated sparse positive definite matrices.

## Utilities for sparse matrices

TAUCS has a convenient C struct for the csc representation of a matrix. I have written functions to transfer from an S4 object to the TAUCS struct and back. TAUCS also has routines for multiplication by dense matrices and for symmetric permutation of rows and columns (needed in the Cholesky factorization routines).

UMFPACK is a set of routines for solving unsymmetric sparse linear systems with the Unsymmetric MultiFrontal method. It has a couple of very convenient routines for switching between csc and a triplet representation. The triplet to csc converter is quite general in that it allows redundant triplet representations (more than one entry for the same position - multiple entries have their values summed) and arbitrary ordering. This allows convenient creation of sparse contingency tables (build up the triplet representation then compress it). As described in the UMFPACK documentation, it also allows simple ways to write operations like transposition of matrices (convert csc to triplet, interchange i and j, convert back to csc).

As a side note, it appears that the UMFPACK/AMD form of the csc representation is more strict than the TAUCS representation. If one applies `taucs_ccs_permute_symmetrically` to a csc matrix (in TAUCS these are called ccs) the result does not have the rows in increasing order within each column. AMD doesn't like this and I find it confusing when trying to examine the matrix. Again the csc to triplet to csc conversion can be used to remove this problem.

• Modify `model.matrix` to produce a sparse representation of `X` or of `X'X` and `X'y`. It would be convenient to get a sparse representation of the model matrix but the big payoff would be in getting the crossproduct matrix. However, it is difficult to decide where the non-zero elements in the crossproduct matrix are when you are sequentially examining the rows of `X`. It may be possible to perform the operation on chunks of rows and use the csc to triplet to csc transformation if new non-zero elements are found while processing a chunk.