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Optimizing Matrix Multiplication
August 28, 2016 by attractivechaos
Vector and matrix arithmetic (e.g. vector dot and matrix multiplication) are the basic to linear algebra and are also widely used in other fields such as deep learning. It is easy to implement vector/matrix arithmetic, but when performance is needed, we often resort to a highly optimized BLAS implementation, such as ATLAS and OpenBLAS. Are these libraries much faster than our own implementations? Is it worth introducing a dependency to BLAS if you only need basic vector/matrix arithmetic? The following post may give you some hints.
Results
In this github repository, I implemented matrix multiplication in seven different ways, including a naive implementation, several optimized implementations with cache miss reduction, SSE and loop blocking, and two implementations on top of OpenBLAS. The following table shows the timing of multiplying two 2000×2000 or 4000×4000 random matrices on my personal Mac laptop and a remote linux server (please see the source code repo for details):
Implementation  a  Linux,n2000  Linux,n4000  Mac,n2000 

Naive  0  7.53 sec  188.85 sec  77.45 sec 
Transposed  1  6.66 sec  55.48 sec  9.73 sec 
sdot w/o hints  4  6.66 sec  55.04 sec  9.70 sec 
sdot with hints  3  2.41 sec  29.47 sec  2.92 sec 
SSE sdot  2  1.36 sec  21.79 sec  2.92 sec 
SSE+tiling sdot  7  1.11 sec  10.84 sec  1.90 sec 
OpenBLAS sdot  5  2.69 sec  28.87 sec  5.61 sec 
OpenBLAS sgemm  6  0.63 sec  4.91 sec  0.86 sec 
uBLAS  7.43 sec  165.74 sec  
Eigen  0.61 sec  4.76 sec 
You can see that a naive implementation of matrix multiplication is quite slow. Simply transposing the second matrix may greatly improve the performance when the second matrix does not fit to the CPU cache (the linux server has a 35MB cache, which can hold a 2000×2000 float matrix in cache, but not a 4000×4000 matrix). Transpose also enables vectorization of the inner loop. This leads to significant performance boost (SSE sdot vs Transposed). Loop blocking further reduces cache misses and timing for large matrices. However, OpenBLAS’ matrix multiplication (sgemm) is still the king of performance, twice as fast as my best handwritten implementation and tens of times faster than a naive implementation. OpenBLAS is fast mostly due to its advanced techniques to minimize cache misses.
As side notes, “sdot with hints” partially unrolls the inner loop. It gives a hint to the compiler that the loop may be vectorized. Clang on Mac can fully vectorize this loop, achieving the same speed of explicit vectorization. Gcc4.4 seems not as good. The Intel compiler vectorizes the loop even without this hint (see the full table in README). Interestingly, the OpenBLAS sdot implementation is slower than my explicit vectorization on both Linux and Mac. I haven’t figured out the reason. I speculate that it may be related to cache optimization.
As to C++ libraries, Eigen has similar performance to OpenBLAS. The native uBLAS implementation in Boost is quite primitive, nearly as slow as the most naive implementation. Boost should ditch uBLAS. Even in the old days, it was badly implemented.
Conclusions
 For multiplying two large matrices, sophisticated BLAS libraries, such as OpenBLAS, are tens of times faster than the most naive implementation.
 With transposing, SSE (x86 only) and loop blocking, we can achieve half of the speed of OpenBLAS’ sgemm while still maintaining relatively simple code. If you want to avoid a BLAS dependency, this is the way to go.
 For BLAS level1 routines (vector arithmetic), an implementation with SSE vectorization may match or sometimes exceeds the performance of OpenBLAS.
 If you prefer a C++ interface and are serious about performance, don’t use uBLAS; use Eigen instead.
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