There has been an enormous amount of effort placed into optimizing matrix multiplication. The key here is that the methods have impracticably huge constant fac

There has been an enormous amount of effort placed into optimizing matrix multiplication. The key here is that the methods have impracticably huge constant factors, such that for anything beyond Strassen would necessitate matrices larger than we could conceivably use to be more effective. This comment also seems to suggest that legions of scientists and engineers aren't trying to optimize matrix multiplication. They are. Here are a few directions to pay attention to: 1. Specialized matrix multiplications for specific sizes and with reduced precision. For example, Intel's libxsmm, methods for multiplying with reduced precision on [CGT]PU. 2. Structured matrix multiplication Circulant and FFT-like matrices can be multiplied in linearithmic time. If you can set your problem up to use these instead (see Choromanski's Orthogonal Random Features [2016] or Smola's Deep-Fried Convnets [2015]/Fastfood [2014]), you can dramatically improve the speed and memory efficiency of your application. 3. Hardware-specific optimizations. Optimizing algorithms by cache efficiency, using FMA (fused multiply-addition) and SIMD instructions, and prefetching all play a big part in improving these multiplications. I'm commenting specifically on CPUs, but similar issues and methods exist on all hardware backends. In summary, many people are working very hard on improving matrix multiplication. Lower asymptotic complexity algorithms are not the answer for this due to the absurd constant factors they require.