Difference between revisions of "Todo"

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(Sparse matrices/vectors)
(Geometry module)
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== Geometry module ==
 
== Geometry module ==
  
Started by gael. Still need:
+
The main part is there: Transform, rotations (Quaternion, EulerAngles, Angle-Axis), cross product. Still need:
* Projective geometry / homographies
+
* API review
* Rotation matrices: reuse Eigen 1 code.
+
* Extend EulerAngles to support any convention ?
* Need a fixed-size Gram-Schmidt here, so it's not clear whether it belongs here or in SVD. Is this a case for having it in a separate module?
+
* Add methods to create perspective/orthogonal projection matrix ?
 +
* Need a fixed-size Gram-Schmidt here, so it's not clear whether it belongs here or in SVD (or QR !). Is this a case for having it in a separate module?
  
 
== Sparse matrices/vectors ==
 
== Sparse matrices/vectors ==

Revision as of 14:16, 22 June 2008

Eigen Owl Todo.jpg

Core module

This module is essentially complete and working, but needs improvement in these areas:

  • Optimizing executable code size
    • When an xpr evaluates its own arguments, we should use that to make it depend only on the resulting matrix type, not on the xpr type. This way, code would be generated only once per actual matrix type, not once per xpr type.
      • Can that be done systematically? Theoretically, any xpr may evaluate its arguments, this is handled in ei_nested.
      • In practice, the main xpr for which this matters is Product, for two reasons. First, it is one of the few xprs using ei_nested with parameter n>1, which means that it is much more likely to evaluate its arguments. Second, it generates a lot of code.
  • Vectorization
    • Evaluate adding vectorization support to (certain) block expressions. Related to adding support for specifying startRow/startCol parameters as template parameters. => WIP via unaligned load and store
  • BLAS/LAPACK backend?
    • BLAS: apparently that would only be useful for matrix product and backward substitution. However, Gael already optimized the matrix product enough to beat the best free BLAS out there (though some proprietary BLAS such as MKL are still faster)... so perhaps a BLAS backend is not the top priority.
    • LAPACK: letting big algorithms (such as inversion, QR...) use a LAPACK backend should be easier (because here we deal with plain matrices not expressions) and more useful (because there are too many different algorithms to optimize for us to do it ourselves).


  • Fast product for large matrices
    • See EigenInternals for an explanation of our cache friendly algorithm.
    • Issues/todo:
      • B gets evaluated if it is not column-major => offer too different algorithms ?
      • the algorithm actually computes C+=A*B, currently C+=(A*B).lazy() gets indeed optimized via a specialization of operator+=. However D=C+(A*B).lazy() is not optimized (extra temporary + set to zero) while it could be if rewritten like this: (D+=C)+=(A*B).lazy().
      • some trivial expressions get evaluated like negate (take care to write c = -(a*b)) or conjugate (only for complex)
        • more generally, if A as to be evaluated then it could be directly evaluated to a blocky form.


  • Misc Features
    • Improve current meta unroller to the more general concept of partial loop unrolling.
    • Consider the use of alloca (dynamic allocation on the stack for dynamic-sized temporary). After some benchmarks, it seems this really pay off for matrices smaller than 16x16, for larger matrices the gain is not so obvious. In practice alloca cannot be called in MatrixStorage, and it as to be called in the function creating the temporary. This might be implemented using Matrix::map().
    • Another fine tuning idea (very low priority): reuse a previously allocated temporary when possible. For instance, theoretically, the Cholesky decomposition could be done "in place", my working directly on the input matrix. Currently, Cholesky stores its own data. However, a typical use case is: (m.adjoint() * m).cholesky()... In that case, if m is large enough, the matrix product creates a temporary and Cholesky too while the latter could simply "pick" the data allocated by the product since we know this data cannot be reused by any other expression. I guess this concept could be extended to a more general mechanism. (this is only applicable for dynamic-size matrix and in the case we don't use alloca, cf. the previous point)

LU module

  • Inversion: it's done, but more work is needed:
    • fixed-size 4x4: either merge the vectorization code by Markos (hopefully making it work with SSE as well as AltiVec) or improve Eigen's generic vectorization code enough to achieve comparable performance.
    • general case: evaluate doing at least one step of divide-and-conquer as in the size 4 case, to reduce overall complexity and improve parallelizability.
  • Determinant: only fixed-size <= 4 is done, the general case remains to do. Perhaps make it just use a LU decomposition, as large determinants are seldom used anyway.
  • LU decomposition: to do. Should have optimized paths for fixed-size <= 4. Very important, as this will be our main device for exact solving.
    • provide isInvertible(), inverse(), determinant(). It's not redundant to have it also here. Once the LU decomposition is computed these are much faster to compute.
    • provide rank(), kernelBasis(), imageBasis(), someAntecedent(vector).

Cholesky module

This would provide nice exact-solving capabilities for positive matrices.

  • The code for Cholesky decomposition and positive solver is done
  • still need to check the API
  • applications: positive exact solving, generalized eigen problems, etc.
  • should linear regression use that, or SVD? I heard SVD is ideal for regression, but still Cholesky seems good too. At least for now one could implement linear regression with Cholesky, as this is needed soon (required for libavogadro to port to eigen2).

QR module

We have QR, eigenvalues/vectors in selfadjoint case, eigenvalues/vector in real non symmetric matrix, and matrix norm. Todo:

  • implement eigensolver in non-selfadjoint case (complex case)
  • generalized eigen problem using Cholesky decomposition (here or in Cholesky ?), spectral radius, matrix exponential and maybe more operator calculus by applying a cwiseUnaryOp to the eigenvalues vector.

SVD module

To do. Should provide SVD decomposition, pseudoinverse, what the vision folks need, and least squares / linear regression.

  • Maybe Gram-Schmidt goes here? It's really similar to SVD.

Geometry module

The main part is there: Transform, rotations (Quaternion, EulerAngles, Angle-Axis), cross product. Still need:

  • API review
  • Extend EulerAngles to support any convention ?
  • Add methods to create perspective/orthogonal projection matrix ?
  • Need a fixed-size Gram-Schmidt here, so it's not clear whether it belongs here or in SVD (or QR !). Is this a case for having it in a separate module?

Sparse matrices/vectors

  • Since GMM++ seems to be best lib for sparse matrix, let's do a brief review:
    • native support of basics (add, mul)
    • native support of various iterative linear solvers including various pre-conditioners for selfadjoint and non-selfadjoint matrix
    • 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 features to such matrices), one dimension is dense
      • block matrix (not documented)
  • Current state:
    • we already have some proof of concept code for CCS matrix:
      • supports any binary ops with any matrix using generic iterators over the non zero coefficients of a column
      • unlike GMM++, our CCS matrix can be filled dynamically in a coherent order, i.e. with increasing i+j*rows (no random write)
  • What's the plan ? ideally:
    • use our own classes for storage and basic algebra (sum, product)
    • a correct support of CCS (and maybe 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++ to directly use our matrix class (btw, those algo have already been adapted from ITL)
    • provide the ability to use more optimized backends like TAUCS or SuperLU and backends for eigen value solvers
  • The first step is therefore to implement good support for sparse matrix and basic linear algebra. In particular, since there is no ideal storage format, we probably need to provide several sparse matrix classes:
    • SparseMatrix: would implement the CCS format (maybe the CRS as well via a template argument)
    • HashMatrix: would provide easy random read/write access. Two major options for the implementation:
      • implemented as a column (or row) array of std::map<row (or col), value> (or whatever hash map), this is the GMM++ way.
      • implemented as a plain std::map<hash(row,col), value> (this is the ublas way)
        • pro: sparse on both direction
        • cons: the key of the map must be a int64, need some operations to extract the coordinates, seems to be more difficult to mix with other matrix types which are processed by slice (unless we add linear access feature, but that's still an issue for sub matrix and matrix product).
    • BlockMatrix: a sparse array of small dense Matrix. The algorithms to exploit such matrices are probably quite different than for the two previous one.

Unit-tests

  • Keep adding more!

Documentation

  • Keep updating
  • all our issues with doxygen are now solved (i.e. we have workarounds)
  • Write special pages on certain topics, illustrate them with examples:
    • Compiler support/options
    • Vectorization
    • Parallelization
    • Fixed-size matrices/vectors
      • in particular, usage with OpenGL
    • Dynamic-size matrices/vectors