SparseMatrix
From Eigen
Contents
Preliminaries
- CCS for Compressed Column Storage is the most common sparse storage scheme (see http://netlib.org/linalg/html_templates/node90.html)
- CRS: Compressed Row Storage, like CCS but row major
- nnz: number of nonzero entries
What's the plan ?
Ideally:
- define our own classes for storage and basic algebra (sum, product, triangular solver)
- a good support of CCS/CRS sparse matrix is certainly the priority as it is the most common storage format
- we could probably adapt the linear solver algorithms of GMM++ (or ITL) to directly use our matrix classes
- see the licensing issues
- provide the ability to use more optimized backends like TAUCS or SuperLU as well as backends for eigen value solvers via a unified API
Current state
- we already have some proof of concept code for CCS matrix (class SparseMatrix) and std::map based matrix (HashMatrix).
- unlike GMM++, our CCS matrix can be filled dynamically in a coherent order, i.e. with increasing i+j*rows (no random write)
- internally, a HashMatrix is simply a std::vector< std::map<int,Scalar> >
- each expression defines a InnerIterator which allows to efficiently traverse the inner coefficients of a sparse (or dense) matrix. Of course, these InnerIterator can be nested exactly like expressions are nested such that our sparse matrices already support expression templates !
- InnerIterator are implemented for:
- a default implementation for MatrixBase
- CwiseUnaryOp
- CwiseBinaryOp (with some shortcomings, see below)
- SparseMatrix
- HashMatrix
- efficient code for the product of two CCS to a CCS.
Issues
The major issue with sparse matrices is that they can only be efficiently traversed in a specific order, i.e., per column for a CCS matrix and per row for a CRS matrix. The nighmare starts when you have to deal with expressions mixing CCS matrices (column major) and CRS matrices (row major).
Binary operators (aka. operator+)
Here the problem is that we cannot mix a CCS with a CRS. In such a case one of the argument have to be evaluated to the other format (let's pick the default format). This requires some modifications in Eigen/Core:
- probably needs to add a SparseBit flag
- needs a more advanced mechanism to determine the storage order of a CwiseBinaryOP<> expression
- needs to modify nesting in ei_traits<CwiseBinaryOP<>> such that if an argument is sparse and has a different storage order than the expression itself, then it has to be evaluated to a sparse temporary.
- needs to modify ei_eval such that a sparse xpr gets evaluated to a sparse matrix (simply add an optional template parameter equal to Xpr::Flags&SparseBit and do the specialization in Eigen/Sparse)
Transpose
Basically, here the challenge is to copy a CRS matrix to a CCS one. I can imagine three strategies:
- copy to a temporary column major HashMatrix and then compress it to a CCS (slow)
- perform a column major traversal of the CRS matrix:
- create rows InnerIterators iters;
- for j=0..cols do
- for i=0..rows do
- if iters[i].index()==j then
- res(i,j) = iters[i].value; iters[i]++
- if iters[i].index()==j then
- for i=0..rows do
- of course creating rows iterators might be too expensive so we need a way to provide lightweight iterators where the reference to the expression, the current outer index etc. are not stored. Maybe we could slightly update the InnerIterator such that they can loop over an arbitrary number of outer columns/rows.
- the third options is like the first one but with a much more efficient data structure. Indeed, during a row major processing the coefficients of the column major destination matrix will be set in coherent order per column. So we can use a special temporary sparse matrix with a column/row vector of linked fixed size vectors. This seems to be the best option. If it is a temporary there is no need to convert it to a compact CCS matrix.
Matrix product (again !)
Here we have to investigate all the possible combinations for the 2 operands and the results (2*2*2=8 possibilities):
lhs | rhs | res | comments |
col | col | col | Loop over rhs in a column major order (j, k), accumulate res.co(j) += rhs(k,j) * lhs.col(k). The accumulation needs a single column temporary which is initialized for each j. If the result of the product is dense (nnz > 4%), then the best is to allocate a dense vector, otherwise a sorted linked list peforms best (the accumulation of each column of lhs is coherent, so we can exploit this coherence to optimize search and insertion in the linked list) |
col | col | row | weird case, let's evaluate to a col major temporary and then transpose |
row | col | * | Loop over the result coefficient in the preferred order and perform coherent dot products |
row | row | * | like col col * |
col | row | row | tedious |
col | row | col | tedious |
Review of some existing libs
GMM++
- native support of basics (add, mul, etc)
- native support of various iterative linear solvers including various pre-conditioners for selfadjoint and non-selfadjoint matrix (most of them come from ITL)
- can use superLU for direct solvers
- provides various sparse matrix formats:
- dense array (col or row) of std::map<int,scalar>
- pro: easy to implement, relatively fast random access (read/write)
- cons: no ideal to use as input of the algorithms, one dimension is dense
- Compressed Col/Row Storage (CCS/CRS) (see http://netlib.org/linalg/html_templates/node91.html)
- pro: compact storage (overhead = (rows/cols + nnz) * sizeof(int) where nnz = number of non zeros), fast to loop over the non-zeros, compatible with other very optimized C library (TAUCS, SuperLU)
- cons: random access (GMM++ does not provide write access to such matrices), one dimension is dense
- block matrix (not documented)
- dense array (col or row) of std::map<int,scalar>
MTL4 / ITL
- native support of basics (in MTL4)
- various iterative linear solvers (in ITL)
- provides CCS/CRS matrix format (read only)
- writes/update of a sparse matrix are done via a generic facade/proxy:
- this object allocate a fixed number of coefficients per column (e.g. 5) and manage overflow using std::maps
- when this object is deleted (after the write operations) this dynamic representation is packed to the underlying matrix
- for dense matrix this facade object degenerate to the matrix itself allowing a uniform interface to fill/update both dense and sparse matrix
boost::ublas
- provides only basic linear algebra
- provides tons of sparse matrix/vector formats:
- CCS/CRS
- std::vector< std::map<int,value> >
- std::map< int, std::map<int,value> >
- coordinate matrix: similar to CCS/CRS but with unsorted entries and the same coeff can appear multiple time (its real value is the sum of its occurances)
- and some other variants
- sparselib++ / IML++: basics, various iterative solvers (in IML++)
- superLU: direct solver
- UMFPACK/CHOLMOD: direct solvers (LU/cholesky), LGPL, depend on lapack
- IETL: eigen problems
- TAUCS: direct and iterative solvers, depend on lapack and metis
Links
- Common storage formats: http://netlib.org/linalg/html_templates/node90.html
- A comparison of free libs: http://www.netlib.org/utk/people/JackDongarra/la-sw.html
- A collection of sparse matrices: http://www.cise.ufl.edu/research/sparse/matrices/