Okay, the title of this post is getting longer and sillier, but this is the 3rd continuation of my last two post on comparing different libraries for everyday matrix operations. The last two posts compared basic operations such as multiplication, transposition, inversion etc. etc. in isolation, which is probably not a good reflection of real life usage. So I decided to come up with a new test that would combine different matrix operations together. I chose the pseudoinverse because it is something I use every now and then and it combines multiplication, transposition and inversion, which seems like a good test.
For benchmarking I’m going to be solving the following over determined linear system:
and solve for X using
A is a NxM matrix, where N is much larger than M. I’ll be using N=1,000,000 data points and M (dimensions of the data) varying from 2 to 16.
B is a Nx1 matrix.
The matrix values will be randomly generated from 0 to 1 with uniform noise of [-1,1] added to B. They values are kept to a small range to avoid any significant numerical problems that can come about doing the pseudoinverse this way, not that I care too much for this benchmark. Each test is performed for 10 iterations, but not averaged out since I’m not interested in absolute time but relative to the other libraries.
Just to make the benchmark more interesting I’ve added GSL and OpenBLAS to the test, since they were just an apt-get away on Ubuntu.
The following libraries were used
- OpenCV 2.4.3 (compiled from source)
- Eigen 3.1.2 (C++ headers from website)
- Armadillo 3.4.4 (compiled from source)
- GSL 1.15 (Ubuntu 12.10 package)
- OpenBLAS 1.13 (Ubuntu 12.10 package)
- Atlas 3.8.4 (Ubuntu 12.10 package)
My laptop has an Intel i7 1.60GHz with 6GB of RAM.
All values reported are in milliseconds. Each psuedoinverse test is performed 10 times but NOT averaged out. Lower is better. Just as a reminder each test is dealing with 1,000,000 data points of varying dimensions.
|Armadillo + Atlas||162.332||184.834||273.822||396.629||528.831||706.238||848.51||1088.47|
|Armadillo + OpenBLAS||79.803||118.718||147.714||298.839||372.235||484.864||411.337||507.84|
|Armadillo + Atlas||1297.01||1519.04||1792.74||2064.77||1438.16||1720.64||1906.79|
|Armadillo + OpenBLAS||534.947||581.294||639.175||772.382||824.971||825.79||893.771|
Ranking from best to worse
- Armadillo + OpenBLAS
- Armadillo + Atlas (no multi-core support out of the box???)
All I can say is, holly smokes Batman! Armadillo + OpenBLAS wins out for every single dimension! Last is GSL, okay no surprise there for me. It never boasted being the fastest car on the track.
The cool thing about Armadillo is switching the BLAS engine only requires a different library to be linked, no recompilation of Armadillo. What is surprising is the Atlas library doesn’t seem to support multi-core by default. I’m probably not doing it right. Maybe I’m missing an environmental variable setting?
OpenBLAS is based on GotoBLAS and is actually a ‘made in China’ product, except this time I don’t get to make any jokes about the quality. It is fast because it takes advantage of multi-core CPU, while the others appear to only use 1 CPU core.
I’m rather sad OpenCV is not that fast since I use it heavily for computer vision tasks. My compiled version actually uses Eigen, but that doesn’t explain why it’s slower than Eigen! Back in the old days OpenCV used to use BLAS/LAPACK, something they might need to consider bringing back.
test_matrix_pseudoinverse.cpp (right click save as)
Edit the code to #define in the libraries you want to test. Make sure you don’t turn on Armadillo + GSL, because they have conflicting enums. Instructions for compiling is at the top of the cpp file, but here it is again for reference.
To compile using ATLAS:
g++ test_matrix_pseudoinverse.cpp -o test_matrix_pseudoinverse -L/usr/lib/atlas-base -L/usr/lib/openblas-base -lopencv_core -larmadillo -lgomp -fopenmp -lcblas -llapack_atlas -lgsl -lgslcblas -march=native -O3 -DARMA_NO_DEBUG -DNDEBUG -DHAVE_INLINE -DGSL_RANGE_CHECK_OFF
To compile with OpenBLAS:
g++ test_matrix_pseudoinverse.cpp -o test_matrix_pseudoinverse -L/usr/lib/atlas-base -L/usr/lib/openblas-base -lopencv_core -larmadillo -lgomp -fopenmp -lopenblas -llapack_atlas -lgsl -lgslcblas -march=native -O3 -DARMA_NO_DEBUG -DNDEBUG -DHAVE_INLINE -DGSL_RANGE_CHECK_OFF