How to figure out the procedure of Axis reordering and Weight permutation for sequential memory access

[Ruixin Huang (HRXWEB)]

Axis reordering

I understand the meaning and motivation of axis rearrangement, but I want to know the procedure of this method.

According to the paper, loop among the spatial axes 𝑁 and 𝑀 first and then temporal axis 𝐾 is natural. The pseudo code I write is:

for _ in range(K):
    for _ in range(N):
        for _ in range(M):

It is weird for me. Naturally, the procedure should be?

for _ in range(N):
    for _ in range(M):
        for _ in range(K):

So it will loop among the temporal axis K first. Why paper claims

For GEMM𝐶[𝑁,𝑀]= 𝐴[𝑁,𝐾]×𝑊[𝑀,𝐾], it is natural to loop among the spatial axes 𝑁 and 𝑀, and then the temporal axis 𝐾.

And how to figure out

But if we swap the axis order from spatial first to temporal first, it will only maintain a small lookup table [1,𝐾].

Because the LUT was built among the axis K, as for the k in K, the weight will share the same LUT table entries. The size of lookup table for the specific n in N, m in M should be [ K//g, 2^g]? why [1, K] in paper?

Weight permutation for sequential memory access

I can't find the corresponding code for this design. Can someone explain the process in detail using words and code? And the final effect.

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This may involve a lot of hardware knowledge. I would be very grateful for anyone can explain it in detail.

Thanks advance.