alt Hacker NewsFrom the paper, "Notably, for multiplying two 4 × 4 matrices, applying the algorithm of Strassen recursively results in an algorithm with 49 multiplications, which works over any field...AlphaEvolve is the first method to find an algorithm to multiply two 4 × 4 complex-valued matrices using 48 multiplications."
If you do naive matrix multiplication, you get a sense that you're doing similar work multiple times, but it's hard to quantify just what that duplicated work entails. Compare it to, for example, calculating the size of the union of two sets:
Total size = size(A) + size(B) - size(intersection(A, B))
You have to take out that extra intersection amount because you've counted it twice. What if you could avoid counting it twice in the first place? That's easy, you just iterate over each set once, keeping track of the elements you've already seen.
Strassen's algorithm keeps track of calculations that are needed later on. It's all reminiscent of dynamic programming.
What I find interesting is that it seems the extra savings requires complex values. There must be something going on in the complex plane that is again over-counting with the naive approach.
Replies
>What I find interesting is that it seems the extra savings requires complex values. There must be something going on in the complex plane that is again over-counting with the naive approach.
"The rank of a tensor depends on the field over which the tensor is decomposed. It is known that some real tensors may admit a complex decomposition whose rank is strictly less than the rank of a real decomposition of the same tensor."
https://en.wikipedia.org/wiki/Tensor_rank_decomposition#Fiel...
It seems like you have some misconceptions about Strassen's alg:
1. It is a standard example of the divide and conquer approach to algorithm design, not the dynamic programming approach. (I'm not even sure how you'd squint at it to convert it into a dynamic programming problem.)
2. Strassen's does not require complex valued matrices. Everything can be done in the real numbers.
A complex multiplication is "worth" at least 3 real multiplications.
Are you sure the saving needs complex values? I think their algorithm works over any char 0 field. Probably needs to just divide by some divisor of 4!=24 if I had to guess.
By googling "4x4 matrices multiplication 48" I ended up on this discussion on math.stackexchange https://math.stackexchange.com/questions/578342/number-of-el... , where in 2019 someone stated "It is possible to multiply two 4×4 matrix A,B with only 48 multiplications.", with a link to a PhD thesis. This might mean that the result was already known (I still have to check the outline of the algorithm).