python - 4 次矩阵乘法的 np.einsum 性能

Question

给定以下 3 个矩阵:

M = np.arange(35 * 37 * 59).reshape([35, 37, 59])
A = np.arange(35 * 51 * 59).reshape([35, 51, 59])
B = np.arange(37 * 51 * 51 * 59).reshape([37, 51, 51, 59])
C = np.arange(59 * 27).reshape([59, 27])

我使用einsum来计算:

D1 = np.einsum('xyf,xtf,ytpf,fr->tpr', M, A, B, C, optimize=True);

但我发现它的性能要低得多:

tmp = np.einsum('xyf,xtf->tfy', A, M, optimize=True)
tmp = np.einsum('ytpf,yft->ftp', B, tmp, optimize=True)
D2 = np.einsum('fr,ftp->tpr', C, tmp, optimize=True)

我不明白为什么。
总的来说，我正在尽力优化这段代码。我已经阅读过有关 np.tensordot 函数的信息，但我似乎无法弄清楚如何将它用于给定的计算。

Answer 1

看起来您偶然发现了一种情况，其中贪婪路径给出了非最佳缩放。

>>> path, desc = np.einsum_path('xyf,xtf,ytpf,fr->tpr', M, A, B, C, optimize="greedy");
>>> print(desc)
  Complete contraction:  xyf,xtf,ytpf,fr->tpr
         Naive scaling:  6
     Optimized scaling:  5
      Naive FLOP count:  3.219e+10
  Optimized FLOP count:  4.165e+08
   Theoretical speedup:  77.299
  Largest intermediate:  5.371e+06 elements
--------------------------------------------------------------------------
scaling                  current                                remaining
--------------------------------------------------------------------------
   5              ytpf,xyf->xptf                         xtf,fr,xptf->tpr
   4               xptf,xtf->ptf                              fr,ptf->tpr
   4                 ptf,fr->tpr                                 tpr->tpr

>>> path, desc = np.einsum_path('xyf,xtf,ytpf,fr->tpr', M, A, B, C, optimize="optimal");
>>> print(desc)
  Complete contraction:  xyf,xtf,ytpf,fr->tpr
         Naive scaling:  6
     Optimized scaling:  4
      Naive FLOP count:  3.219e+10
  Optimized FLOP count:  2.744e+07
   Theoretical speedup:  1173.425
  Largest intermediate:  1.535e+05 elements
--------------------------------------------------------------------------
scaling                  current                                remaining
--------------------------------------------------------------------------
   4                xtf,xyf->ytf                         ytpf,fr,ytf->tpr
   4               ytf,ytpf->ptf                              fr,ptf->tpr
   4                 ptf,fr->tpr                                 tpr->tpr

使用 np.einsum('xyf,xtf,ytpf,fr->tpr', M, A, B, C, Optimize="optimal") 应该可以让您以最佳性能运行。我可以看看这条边，看看贪婪能否捕获它。