SparseMatrix
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> >. It is simple to implement and easy to use but it is also damn slow ! would be really nice if we could find a better alternative for efficient random writes.
- 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.
- efficient code for T^-1 * V, with T a sparse triangular matrix, and V a dense vector.
Write/Update access patterns
When dealing with sparse matrix, a critical operation is the efficient set/update of the coefficients. In order to offer optimal performance it is necessary to propose different solutions according to the required access pattern. For instance, a temporary sparse matrix based on a std::map can handle any kind of access pattern, but it also performs very poorly if you are able to fill a matrix in a coherent order. We can identify four different access pattern schemes with their respective technical solutions, ranging from the most efficient to the most flexible.
Fully coherent access
Here we assume the matrix is set in a fully coherent order, i.e., such that the coefficients (i,j) are set with increasing i + j*outersize where i is the inner coordinates and j the outer coordinate. In such a case we can directly set a compressed sparse matrix as we would fill a dynamic vector. In order to reduce memory allocations/memory copies, it is important to be able to give a hint about the expected number of non-zeros such that we are able to preallocate enough memory. Of course using a more dynamic data structure like a linked list of small array would probably performs better compared to resizing a single large buffer but since this not a standard storage format and that the standard compressed scheme works pretty well it's probably better to use it directly.
Updating a sparse matrix using this access pattern can be done by filling a new temporary matrix followed by an efficient shallow copy.
Inner coherent access
Here we assume the coefficients (i,j) are set with increasing inner coordinate i for each outer vector j. For instance the following sequence of coordinates is valid:
(2,10) (4,7) (1,12) (4,10) (3,12) (2,1)
On the other hand, at that point it is forbidden to set the coefficient (2,12) since (3,12) has already been set. A typical use case of such an access pattern is to copy a row-major matrix B to a column-major matrix A: since B has to be traversed in a row major order the sequence of coordinates (i,j) won't fit the requirements of the fully coherent access pattern but perfectly match the current one.
To implement such a behavior, we need a dynamic data structure per inner vector. Since there is no such standard storage scheme, we are free to choose whatever we want. Currently, this access pattern is implemented by mean of small linked array in the LinkedVectorMatrix class. Still to do:
- make the granularity of the chunks configurable at runtime ?
- write a memory allocator for the chunks shared at the matrix level ?
- write a clean linked vector class for reuse.
Outer coherent access
Here each inner vector j is filled randomly, but once we have started to fill the inner vector j, it is forbidden to update any inner vector k with k<j. This scheme occurs in matrix product and triangular solver. The current solution is to allocate a dense vector, fill it, and push it into the sparse matrix once we are done with this inner vector. Currently this strategy is implemented manually when needed, i.e., there is no special class for that yet (unlike LinkedVectorMatrix for inner coherent access).
Random access
=> currently by mean of an array of std::map but I'd like to try:
- array of a custom sorted linked lists (if the number of non zero per inner vector is low, i.e. < 24, this might be much faster)
- use a custom binary tree impl. See also the "elastic binary tree" in haproxy (http://haproxy.1wt.eu/) which seems to be 5 times faster than std::map for insertions without being slower for lookups.
- hm... actually haproxy's ebt is faster only if elements are not inserted too randomly. anyways this optimization if of very very low priority for now !
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).
Generic API to fill a sparse/dense matrices
The current SparseMatrix can be filled like that:
SparseMatrix m(rows,cols); m.startFill(2*cols); // 2*cols is a hint on the number of nonzero entries // note that startFill() delete all previous elements in the matrix m.fill(2,0) = rand(); m.fill(3,0) = rand(); m.fill(0,2) = rand(); m.fill(7,2) = rand(); m.fill(12,9) = rand(); m.fill(8,11) = rand(); m.endFill();
At that point m.nonZeros() == 6 and you cannot add any other nonzero entries. For instance x=m(0,0); will returns zero, while m(0,0)=x; will issue an assert. Of course you can still update an existing nonzero: m(7,2) += 1;.
To allow to treat any matrix as a sparse matrix we could define dummy startFill(), fill() and endFill() members to MatrixBase but maybe we could find some better API ? Indeed, while efficient there are two major limitations:
- we cannot update the matrix, only set it from zero
- to be consistent, for dense matrices startFill() should do a setZero() that is not really nice
For instance we could imagine something based on the facade design pattern where you could request for a random setter or coherent setter or an updater or whatever else is needed ? For dense matrices these facade objects would degenerate to a simple references. This last solution is currently experimented in SparseSetter.
FullyCoherentAccessPattern
Requirements:
- notify the start of the filling
- special insertion mechanism
- notify the end of the filling
Current API:
SparseMatrix<float> m; { SparseSetter<MatrixType, FullyCoherentAccessPattern> w(m); w->startFill(); for (int j=0; j<cols; ++j) for (int i=0; i<rows; ++i) if (nonzero) w->fill(i,j) = some_non_zero_value; w->endFill(); }
However, the startFill and endFill could be hidden by SparseSetter, thus:
SparseMatrix<float> m; { SparseSetter<MatrixType, FullyCoherentAccessPattern> w(m); for (int j=0; j<cols; ++j) for (int i=0; i<rows; ++i) if (nonzero) w->fill(i,j) = some_non_zero_value; }
InnerCoherentAccessPattern
Requirements:
- notify the start of the filling (clear)
- special insertion mechanism
Current API:
SparseMatrix<float> m; { SparseSetter<MatrixType, InnerCoherentAccessPattern> w(m); w->startFill(); for (int i=0; i<rows; ++i) for () if (nonzero) w->fill(i,rand()) = some_non_zero_value; }
OuterCoherentAccessPattern
Requirements:
- notify the start of a column
- special insertion function
- notify the end of a column
Proposed API:
SparseMatrix<float> m; { SparseSetter<MatrixType, OuterCoherentAccessPattern> w(m); for (int j=0; j<cols; ++j) { w->startFillInner(j); for () if (nonzero) w->fill(rand(),j) = some_non_zero_value; w->endFillInner(j); } }
RandomAccessPattern
Current API:
{ SparseSetter<MatrixType, RandomAccessPattern> w(m); for (...) w->coeffRef(rand(),rand()) = some_non_zero_value; }
Comments
Issues:
- currently the SparseSetter works as a pointer to the actual matrix or a temporary matrix, maybe it would be better to only access functions defined in SparseSetter, e.g.:
SparseMatrix<float> m; {
SparseSetter<MatrixType, FullyCoherentAccessPattern> w(m); for (int j=0; j<cols; ++j) for (int i=0; i<rows; ++i) if (nonzero) w(i,j) = some_non_zero_value;
}
- in above we only discussed about filling a matrix from scratch, what about updates ?
- when you update you want to either erase previous values or accumulate
- the API should allow to use sub-matrix expressions, e.g.:
SparseMatrix<float> m; { SparseSetter<MatrixType, OuterCoherentAccessPattern> w(m); for (int j=0; j<cols; ++j) { w->startFillInner(j); for (k) w->col(j) += alpha * m2.col(k); w->endFillInner(j); } }
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 | col | if transpose is fast then evaluate rhs to a column major format, otherwise use a dynamic temporary for the results (array of sorted linked lists) |
col | row | row | like col row col |
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/