Fair point! A single complex multiplication `(a+bi)(c+di)` indeed requires at least 3 real multiplications to be implemented.
However, when researchers (and systems like AlphaEvolve in this context) analyze fast matrix multiplication algorithms like Strassen's, the primary goal is usually to improve the asymptotic complexity (and understand the space of these algorithms better). This complexity is determined by the number of multiplications in the field over which the matrices are defined.
* For real matrices, we count real scalar multiplications.
* For complex-valued matrices (as in the 4x4 example where AlphaEvolve found a solution with 48 scalar multiplications), "scalar multiplication" refers to a complex scalar multiplication.
The key is that these are the operations you recurse on. Additions, or the constant factor cost of implementing one field's multiplication, don't change the exponent in the `N^(log_base(multiplications))` complexity. They are constant factors.
Of course, for practical performance on a specific 4x4 matrix, one would absolutely dive into the real operations, additions, memory layout, etc., but making 4x4 matrix multiplication practically faster on some particular hardware was not the focus in this section. (We do improve practical implementation of large matrix multiplications on the target hardware in the “Enhancing AI training and inference” section of the blog post.)
alt Hacker News
Fair point! A single complex multiplication `(a+bi)(c+di)` indeed requires at least 3 real multiplications to be implemented.
However, when researchers (and systems like AlphaEvolve in this context) analyze fast matrix multiplication algorithms like Strassen's, the primary goal is usually to improve the asymptotic complexity (and understand the space of these algorithms better). This complexity is determined by the number of multiplications in the field over which the matrices are defined. * For real matrices, we count real scalar multiplications. * For complex-valued matrices (as in the 4x4 example where AlphaEvolve found a solution with 48 scalar multiplications), "scalar multiplication" refers to a complex scalar multiplication.
The key is that these are the operations you recurse on. Additions, or the constant factor cost of implementing one field's multiplication, don't change the exponent in the `N^(log_base(multiplications))` complexity. They are constant factors.
Of course, for practical performance on a specific 4x4 matrix, one would absolutely dive into the real operations, additions, memory layout, etc., but making 4x4 matrix multiplication practically faster on some particular hardware was not the focus in this section. (We do improve practical implementation of large matrix multiplications on the target hardware in the “Enhancing AI training and inference” section of the blog post.)
(Disclaimer: one of the authors.)