三维numpy阵列的卷积

Question

我从我的主管那里得到了一个代码来实现MDCT多相分析和综合。不幸的是，这段代码包含一个非常慢的函数，有两个循环。如果有人能帮助我简化这个功能，使它更快，我将感谢你的帮助。这是代码的一部分：

def polmatmult(A, B):
    """polmatmult(A,B)
    multiplies two polynomial matrices (arrays) A and B, where each matrix entry is a polynomial.
    Those polynomial entries are in the 3rd dimension
    The third dimension can also be interpreted as containing the (2D) coefficient matrices of exponent of z^-1.
    Result is C=A*B;"""

    print("np.shape(A)", np.shape(A))
    print("np.shape(B)", np.shape(B))
    [NAx, NAy, NAz] = np.shape(A);
    [NBx, NBy, NBz] = np.shape(B);


    "Degree +1 of resulting polynomial, with NAz-1 and NBz-1 being the degree of the input  polynomials:"

    Deg = NAz + NBz - 1;
    print("Deg", Deg)
    C = np.zeros((NAx, NBy, Deg));

    "Convolution of matrices:"
    for n in range(0, (Deg)):
        for m in range(0, n + 1):
            if ((n - m) < NAz and m < NBz):
                C[:, :, n] = C[:, :, n] + np.dot(A[:, :, (n - m)], B[:, :, m]);      

    return C

Answer 1

首先我很惊讶美国运输部那里没有np.乘法. 卷积已经在for循环中发生了，它应该被广播到前两个维度，对吗？不管怎样，我会和np.乘法而不是美国运输部如果我错了，你可以把它改回来。在

如果这个函数是一个真正的瓶颈，我将使用Cython来提高速度。以下是代码示例：

在myconvolve.pyx公司在

import numpy as np
cimport numpy as np
cimport cython

@cython.boundscheck(False)
@cython.wraparound(False)
def myconvolve(np.ndarray[np.float64_t, ndim=3] A,
               np.ndarray[np.float64_t, ndim=3] B):
    cdef:
        int n, m, i, j
        int NAx = A.shape[0], NAy = A.shape[1], NAz = A.shape[2]
        int NBx = A.shape[0], NBy = A.shape[1], NBz = A.shape[2]
        int Deg = NAz + NBz - 1;
        np.ndarray[np.float64_t, ndim=3] C = np.zeros((NAx, NBy, Deg));
    assert((NAx == NBx) and (NAy == NBy))

    for n in range(0, (Deg)):
        for m in range(0, n + 1):
            if ((n - m) < NAz and m < NBz):
                for i in range(0, NAx):
                    for j in range(0, NAy):
                        C[i, j, n] = C[i, j, n] + A[i, j, (n - m)] * B[i, j, m]

    return C

这必须被编译，我用它

^{pr2}$

然后用下面的比较脚本

import timeit
import numpy as np
from myconvolve import myconvolve

def original_convolution(A, B):
    [NAx, NAy, NAz] = np.shape(A);
    [NBx, NBy, NBz] = np.shape(B);

    Deg = NAz + NBz - 1;
    C = np.zeros((NAx, NBy, Deg));

    for n in range(0, (Deg)):
        for m in range(0, n + 1):
            if ((n - m) < NAz and m < NBz):
                C[:, :, n] = C[:, :, n] + np.multiply(A[:, :, (n - m)], B[:, :, m])

    return C

print "Checking that implementations produce identical results."
A = np.random.rand(20, 20, 20)
B = np.random.rand(20, 20, 20)
C1 = original_convolution(A, B)
C2 = myconvolve(A, B)
assert(np.abs((C1 - C2).sum()) < 1.e-6)

mysetup = '''
import numpy as np
np.random.seed(0)
from myconvolve import myconvolve
from __main__ import A, B
from __main__ import original_convolution
'''

print 'Numpy implementation time [s]: ', min(timeit.Timer('original_convolution(A, B)', setup=mysetup).repeat(7, 100))
print 'Cython implementation time [s]: ', min(timeit.Timer('myconvolve(A, B)', setup=mysetup).repeat(7, 100))

我得到：

Numpy implementation time [s]:  0.494730949402
Cython implementation time [s]:  0.0905570983887

Answer 2

编辑：我现在意识到poly1d比原来的解决方案效率要低得多，主要是因为poly1d是用Python而不是C实现的

    %prun np.dot(mod_A, mod_B)
          888804 function calls (872804 primitive calls) in 0.436 seconds

   Ordered by: internal time

   ncalls  tottime  percall  cumtime  percall filename:lineno(function)
    47200    0.069    0.000    0.168    0.000 polynomial.py:1076(__init__)
78800/62800    0.053    0.000    0.085    0.000 {built-in method numpy.core.multiarray.array}
    46800    0.036    0.000    0.063    0.000 shape_base.py:11(atleast_1d)
    31600    0.030    0.000    0.036    0.000 function_base.py:2209(trim_zeros)
    47200    0.024    0.000    0.024    0.000 {method 'copy' of 'numpy.ndarray' objects}
     8000    0.023    0.000    0.023    0.000 {built-in method numpy.core.multiarray.correlate}
    47200    0.020    0.000    0.050    0.000 polynomial.py:1041(coeffs)
     8000    0.020    0.000    0.304    0.000 polynomial.py:1183(__mul__)
     7600    0.019    0.000    0.041    0.000 polynomial.py:683(polyadd)
     8000    0.016    0.000    0.123    0.000 numeric.py:978(convolve)
     8000    0.014    0.000    0.204    0.000 polynomial.py:790(polymul)
     7600    0.014    0.000    0.119    0.000 polynomial.py:1197(__add__)
        1    0.013    0.013    0.436    0.436 {built-in method numpy.core.multiarray.dot}
    94400    0.012    0.000    0.012    0.000 {built-in method builtins.isinstance}
   157200    0.011    0.000    0.011    0.000 {built-in method builtins.len}
    16000    0.011    0.000    0.034    0.000 polynomial.py:1103(__array__)
    46800    0.010    0.000    0.019    0.000 numeric.py:534(asanyarray)
    62800    0.009    0.000    0.009    0.000 polynomial.py:1064(_coeffs)
    47200    0.009    0.000    0.009    0.000 polynomial.py:1067(_coeffs)
     8000    0.008    0.000    0.009    0.000 numeric.py:2135(isscalar)
     8000    0.005    0.000    0.007    0.000 numeric.py:904(_mode_from_name)
    46800    0.004    0.000    0.004    0.000 {method 'append' of 'list' objects}
    16000    0.004    0.000    0.007    0.000 numeric.py:463(asarray)
    31600    0.003    0.000    0.003    0.000 {method 'upper' of 'str' objects}
     8000    0.001    0.000    0.001    0.000 {method 'lower' of 'str' objects}
        1    0.000    0.000    0.436    0.436 <string>:1(<module>)
        1    0.000    0.000    0.436    0.436 {built-in method builtins.exec}
        1    0.000    0.000    0.000    0.000 {method 'disable' of '_lsprof.Profiler' objects}

%prun polymatmult(A, B)
          7 function calls in 0.004 seconds

   Ordered by: internal time

   ncalls  tottime  percall  cumtime  percall filename:lineno(function)
        1    0.004    0.004    0.004    0.004 <ipython-input-1053-a9f13893aa45>:1(original_convolution)
        1    0.000    0.000    0.000    0.000 {built-in method numpy.core.multiarray.zeros}
        1    0.000    0.000    0.004    0.004 {built-in method builtins.exec}
        1    0.000    0.000    0.004    0.004 <string>:1(<module>)
        2    0.000    0.000    0.000    0.000 fromnumeric.py:1565(shape)
        1    0.000    0.000    0.000    0.000 {method 'disable' of '_lsprof.Profiler' objects}

不过，我会坚持，至少要更容易些。在

---

如果您使用poly1d类型，这将非常容易：

^{pr2}$

Answer 3

一个小的操作，使点积变得更简单，ufuc_at操作可以删除for循环：

def polmatmult_(A, B):
    print("np.shape(A)", np.shape(A))
    print("np.shape(B)", np.shape(B))
    [NAx, NAy, NAz] = np.shape(A)
    [NBx, NBy, NBz] = np.shape(B)

    Deg = NAz + NBz - 1
    print("Deg", Deg)
    C = np.zeros((Deg, NAx, NBy))

    m, n = np.triu_indices(NBz, 0, Deg)
    m, n = m[n - m < NAz], n[n - m < NAz]
    np.add.at(C, n,  np.moveaxis(A[:, :, (n - m)], -1, 0) @ np.moveaxis(B[:, :, m], -1, 0))

    return np.moveaxis(C, 0, -1)

一般来说，您希望您的索引轴（z）是第一个维度，而不是最后一个维度。这允许您使用ufunc技巧（如add.at）、@而不是{}，并进行广播。因此所有的np.moveaxis。在