mike0609king opened a new pull request, #2196:
URL: https://github.com/apache/systemds/pull/2196
[SYSTEMDS-3778] Introduce determinant computation primitives
This PR introduces the computation of the determinant of a matrix. It
provides different strategies of computing those determinants. Internally
it is possible to switch between calculating the determinant using the
builtin LU decomposition of the commons math library, Gaussian algorithm,
Bareiss algorithm and Laplace expansion. Furthermore, static simplification
rewrites
- `det(t(X)) -> det(X)`
- `det(X%*%Y) -> det(X)*det(Y)`
- `det(lambda*X) -> lambda^nrow(X)*det(X)`
have been implemented.
All strategies have been benchmarked using 12x12-matrices. Very large
matrices
can result in the laplace algorithm to take too long. The numbers consist
of the average execution time of three runs; with different matrix seeds
(7, 233, 4242). Furthermore, it we distinguish between sparse and dense
matrices,
because each algorithm has its own way of optimizing for sparse matrices.
- LU decomposition:
- Total execution time (Dense matrix): 0.403
- Total execution time (Sparse matrix): 0.056
- Gaussian algorithm:
- Total execution time (Dense matrix): 0.364
- Total execution time (Sparse matrix): 0.049
- Bareiss algorithm:
- Total execution time (Dense matrix): 0.370
- Total execution time (Sparse matrix): 0.054
- Laplace expansion:
- Total execution time (Dense matrix): 0.389
- Total execution time (Sparse matrix): 6.553
Further observations:
- The Laplace expansion is slow for big matrices and should not be used.
- Gaussian algorithm is a bit faster than Bareiss algorithm and LU
decomposition.
- Analytical observation: The Bareiss algorithm can be more accurate, if
whole
numbers are used, because the divisions in the algorithm have no remainder in
that case.
- Gaussian and Bareiss algorithm have larger deviations than eps if
the matrix is transposed `(det(t(X)) in R vs det(X) in SystemDS)`. The LU
decomposition does not have this issue. This could be an indicator that
the LU decomposition is more robust against floating-point errors.
- The Gaussian algorithm has higher floating-point error than 1e-8 in
comparison to the equivalent implementation in R. It does not pass the
test with matrix size of around 30x30. LU decomposition and Bareiss
algorithm are suspected to have lower floating-point errors.
It is adviced to never use the laplace expansion, because of its
inefficiency;
the runtime is factorial. Those observations lead to using using the
LU decomposition because of its higher accuracy, despite being a bit slower
than the alternatives.
The reasoning `det(lambda*X) -> lambda^nrow(X)*det(X)` as rule is, that
computing the power of a scalar can be done in logarithmic time, whereas
the multiplication of a scalar with a matrix is quadratic. Another reason
for this simplification are simplifications of examples such as
`det(lambda*t(X)) -> lambda^nrow(X)*det(X)`, which would be harder to
implement
without this simplification.
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