• lad
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        54 months ago

        Matrix multiplication is O(n) if you do it in parallel /j

        • @[email protected]
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          4 months ago

          Umm, AKCTSHUALLY it gets more like O(n2) in parallel, assuming you’re using a physically achievable memory. There’s just a lot of criss-crossing the entries have to do.

          Strassen’s algorithm gets O(n2.8) in serial by being terrible, and the weird experimental ones get O(n2.3), but the asymptotic benefits of Coppersmith-Winograd and friends only kick in if you’re a Kardashev III civilisation computing a single big product on a Dyson sphere.

          • @[email protected]
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            34 months ago

            I can’t decide whether this sentence is a joke or not. It has the same tone that triggers my PTSD from my CS degree classes and I also do recognize some of the terms, but it also sounds like it’s just throwing random science terms around as if you asked a LLM to talk about math.

            I love it.

            Also, it’s apparently also real and correct.

            • @[email protected]
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              14 months ago

              Thank you, I’m glad to share the pain of numerical linear algebra with anyone who will listen.

          • lad
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            24 months ago

            Yeah, in fact, I somehow calculated in assumption of n being the amount of elements in matrix, not n² (assuming square matrix)

            But I am impressed to know that there are serial algorithms that approach O(n²), thank you for sharing that info

            • @[email protected]
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              4 months ago

              Yeah, they work by turning the problem into some crazy kind of group theory and attacking it that way. Every once in a while someone shaves the decimal down slightly, just by implementing the deep math in a more efficient way. A new approach will be needed if it is in fact possible to get down to O(n2), though. Strassen’s is a divide and conquer algorithm, and each step of the iteration looks like this:

                  S[1] = B[1, 2] - B[2, 2]
                  S[2] = A[1, 1] + A[1, 2]
                  S[3] = A[2, 1] + A[2, 2]
                  S[4] = B[2, 1] - B[1, 1]
                  S[5] = A[1, 1] + A[2, 2]
                  S[6] = B[1, 1] + B[2, 2]
                  S[7] = A[1, 2] - A[2, 2]
                  S[8] = B[2, 1] + B[2, 2]
                  S[9] = A[1, 1] - A[2, 1]
                  S[10] = B[1, 1] + B[1, 2]
                  P[1] = STRASSEN(A[1, 1], S[1])
                  P[2] = STRASSEN(S[2], B[2, 2])
                  P[3] = STRASSEN(S[3], B[1, 1])
                  P[4] = STRASSEN(A[2, 2], S[4])
                  P[5] = STRASSEN(S[5], S[6])
                  P[6] = STRASSEN(S[7], S[8])
                  P[7] = STRASSEN(S[9], S[10])
                  C[1..n / 2][1..n / 2] = P[5] + P[4] - P[2] + P[6]
                  C[1..n / 2][n / 2 + 1..n] = P[1] + P[2]
                  C[n / 2 + 1..n][1..n / 2] = P[3] + P[4]
                  C[n / 2 + 1..n][n / 2 + 1..n] = P[5] + P[1] - P[3] - P[7]
                  return C
              

              In my copy of Introduction to Algorithms, it says something like “this is the most bullshit algorithm in the book and it’s not close” underneath. You can make it a bit neater by representing the multiplication operation as a 3-dimensional tensor, but at the end of the day it’s still just a stupid arithmetic trick. One that’s built into your GPU.