python - 具有广播的稀疏 Scipy 矩阵和向量的元素最大值

Question

我需要一个快速的逐元素最大值，它将 n×m scipy 稀疏矩阵元素的每一行与稀疏的 1×m 矩阵进行比较。这在 Numpy 中使用 np.maximum(mat, vec) 完美运行通过 Numpy 的广播。
然而，Scipy 的 .maximum()没有广播。我的矩阵很大，所以我不能将它转换为一个 numpy 数组。
我目前的解决方法是使用 mat[row,:].maximum(vec) 遍历多行垫子.这个大循环是破坏我的代码效率(必须多次完成)。我的缓慢解决方案在下面的第二个代码片段中 - 有更好的解决方案吗？

# Example
import numpy as np
from scipy import sparse

mat = sparse.csc_matrix(np.arange(12).reshape((4,3)))

vec = sparse.csc_matrix([-1, 5, 100])

# Numpy's np.maximum() gives the **desired result** using broadcasting (but it can't handle sparse matrices):
numpy_result = np.maximum( mat.toarray(), vec.toarray() )
print( numpy_result )
# [[  0   5 100]
#  [  3   5 100]
#  [  6   7 100]
#  [  9  10 100]]

# Scipy only compares the top row of mat to vec (no broadcasting!):
scipy_result = mat.maximum(vec)
print( scipy_result.toarray() )
# [[  0   5 100]
#  [  3   4   5]
#  [  6   7   8]
#  [  9  10  11]]

#Reversing the order of mat and vec in the call to vec.maximum(mat) results in a single row output, and also frequently seg faults (!):

用于速度测试的更大示例和当前解决方案

import numpy as np
from scipy import sparse
import timeit

mat = sparse.csc_matrix(  sparse.random(20000, 4000, density=.01, data_rvs=lambda s: np.random.randint(0, 5000, size=s))  )

vec = sparse.csc_matrix(  sparse.random(1, 4000, density=.01, data_rvs=lambda s: np.random.randint(0, 5000, size=s))  )

def sparse_elementwise_maximum(mat, vec):
    output = sparse.lil_matrix(mat.shape)
    for row_idx in range( mat.shape[0] ):
        output[row_idx] = mat[row_idx,:].maximum(vec)
    return output

# Time it
num_timing_loops = 3.0
starttime = timeit.default_timer()
for _ in range(int(num_timing_loops)):
    sparse_elementwise_maximum(mat, vec)
print('time per call is:', (timeit.default_timer() - starttime)/num_timing_loops, 'seconds')
# 15 seconds per call (way too slow!)

编辑
我接受 Max 的回答，因为该问题专门针对高性能解决方案，而 Max 的解决方案在我尝试的各种输入上提供了 1000x-2500x 的巨大加速，但代价是增加了更多代码行和 Numba 编译。但是，对于一般用途，Daniel F 的 one-liner 是一个很好的解决方案，在我尝试过的示例上提供了 10 到 50 倍的加速——我可能会用于许多其他事情。

Answer 1

低级方法
与往常一样，您可以考虑如何为该操作构建适当的稀疏矩阵格式，对于 csr 矩阵，主要组件是 shape、data_arr、indices 和 ind_ptr。
使用 scipy.sparse.csr 对象的这些部分，使用编译语言(C、C++、Cython、Python-Numba)实现高效算法非常简单，但可能有点耗时。 Int 他的实现我使用了 Numba，但是将它移植到 C++ 应该很容易(语法更改)并且可能避免切片。
实现(初试)

import numpy as np
import numba as nb

# get all needed components of the csr object and create a resulting csr object at the end
def sparse_elementwise_maximum_wrap(mat,vec):
    mat_csr=mat.tocsr()
    vec_csr=vec.tocsr()

    shape_mat=mat_csr.shape
    indices_mat=mat_csr.indices
    indptr_mat=mat_csr.indptr
    data_mat=mat_csr.data
    indices_vec=vec_csr.indices
    data_vec=vec_csr.data

    res=sparse_elementwise_maximum_nb(indices_mat,indptr_mat,data_mat,shape_mat,indices_vec,data_vec)
    res=sparse.csr_matrix(res, shape=shape_mat)
    return res

@nb.njit(cache=True)
def sparse_elementwise_maximum_nb(indices_mat,indptr_mat,data_mat,shape_mat,vec_row_ind,vec_row_data):
    data_res=[]
    indices_res=[]
    indptr_mat_res=[]

    indptr_mat_=0
    indptr_mat_res.append(indptr_mat_)

    for row_idx in range(shape_mat[0]):
        mat_row_ind=indices_mat[indptr_mat[row_idx]:indptr_mat[row_idx+1]]
        mat_row_data=data_mat[indptr_mat[row_idx]:indptr_mat[row_idx+1]]

        mat_ptr=0
        vec_ptr=0
        while mat_ptr<mat_row_ind.shape[0] and vec_ptr<vec_row_ind.shape[0]:
            ind_mat=mat_row_ind[mat_ptr]
            ind_vec=vec_row_ind[vec_ptr]

            #value for both matrix and vector is present
            if ind_mat==ind_vec:
                data_res.append(max(mat_row_data[mat_ptr],vec_row_data[vec_ptr]))
                indices_res.append(ind_mat)
                mat_ptr+=1
                vec_ptr+=1
                indptr_mat_+=1

            #only value for the matrix is present vector is assumed 0
            elif ind_mat<ind_vec:
                if mat_row_data[mat_ptr] >0:
                    data_res.append(mat_row_data[mat_ptr])
                    indices_res.append(ind_mat)
                    indptr_mat_+=1
                mat_ptr+=1

            #only value for the vector is present matrix is assumed 0
            else:
                if vec_row_data[vec_ptr] >0:
                    data_res.append(vec_row_data[vec_ptr])
                    indices_res.append(ind_vec)
                    indptr_mat_+=1
                vec_ptr+=1

        for i in range(mat_ptr,mat_row_ind.shape[0]):
            if mat_row_data[i] >0:
                data_res.append(mat_row_data[i])
                indices_res.append(mat_row_ind[i])
                indptr_mat_+=1
        for i in range(vec_ptr,vec_row_ind.shape[0]):
            if vec_row_data[i] >0:
                data_res.append(vec_row_data[i])
                indices_res.append(vec_row_ind[i])
                indptr_mat_+=1
        indptr_mat_res.append(indptr_mat_)

    return np.array(data_res),np.array(indices_res),np.array(indptr_mat_res)

实现(优化)
在这种方法中，列表被动态调整大小的数组替换。我以 60 MB 的步长增加了输出的大小。在创建 csr 对象时，也没有复制的数据，只有引用。如果你想避免内存开销，你必须最后复制数组。

@nb.njit(cache=True)
def sparse_elementwise_maximum_nb(indices_mat,indptr_mat,data_mat,shape_mat,vec_row_ind,vec_row_data):
    mem_step=5_000_000
    #preallocate memory for 5M non-zero elements (60 MB in this example)
    data_res=np.empty(mem_step,dtype=data_mat.dtype)
    indices_res=np.empty(mem_step,dtype=np.int32)
    data_res_p=0

    indptr_mat_res=np.empty((shape_mat[0]+1),dtype=np.int32)
    indptr_mat_res[0]=0
    indptr_mat_res_p=1
    indptr_mat_=0

    for row_idx in range(shape_mat[0]):
        mat_row_ind=indices_mat[indptr_mat[row_idx]:indptr_mat[row_idx+1]]
        mat_row_data=data_mat[indptr_mat[row_idx]:indptr_mat[row_idx+1]]

        #check if resizing is necessary
        if data_res.shape[0]<data_res_p+shape_mat[1]:
            #add at least memory for another mem_step elements
            size_to_add=mem_step
            if shape_mat[1] >size_to_add:
                size_to_add=shape_mat[1]

            data_res_2   =np.empty(data_res.shape[0]   +size_to_add,data_res.dtype)
            indices_res_2=np.empty(indices_res.shape[0]+size_to_add,indices_res.dtype)
            for i in range(data_res_p):
                data_res_2[i]=data_res[i]
                indices_res_2[i]=indices_res[i]
            data_res=data_res_2
            indices_res=indices_res_2

        mat_ptr=0
        vec_ptr=0
        while mat_ptr<mat_row_ind.shape[0] and vec_ptr<vec_row_ind.shape[0]:
            ind_mat=mat_row_ind[mat_ptr]
            ind_vec=vec_row_ind[vec_ptr]

            #value for both matrix and vector is present
            if ind_mat==ind_vec:
                data_res[data_res_p]=max(mat_row_data[mat_ptr],vec_row_data[vec_ptr])
                indices_res[data_res_p]=ind_mat
                data_res_p+=1
                mat_ptr+=1
                vec_ptr+=1
                indptr_mat_+=1

            #only value for the matrix is present vector is assumed 0
            elif ind_mat<ind_vec:
                if mat_row_data[mat_ptr] >0:
                    data_res[data_res_p]=mat_row_data[mat_ptr]
                    indices_res[data_res_p]=ind_mat
                    data_res_p+=1
                    indptr_mat_+=1
                mat_ptr+=1

            #only value for the vector is present matrix is assumed 0
            else:
                if vec_row_data[vec_ptr] >0:
                    data_res[data_res_p]=vec_row_data[vec_ptr]
                    indices_res[data_res_p]=ind_vec
                    data_res_p+=1
                    indptr_mat_+=1
                vec_ptr+=1

        for i in range(mat_ptr,mat_row_ind.shape[0]):
            if mat_row_data[i] >0:
                data_res[data_res_p]=mat_row_data[i]
                indices_res[data_res_p]=mat_row_ind[i]
                data_res_p+=1
                indptr_mat_+=1
        for i in range(vec_ptr,vec_row_ind.shape[0]):
            if vec_row_data[i] >0:
                data_res[data_res_p]=vec_row_data[i]
                indices_res[data_res_p]=vec_row_ind[i]
                data_res_p+=1
                indptr_mat_+=1
        indptr_mat_res[indptr_mat_res_p]=indptr_mat_
        indptr_mat_res_p+=1

    return data_res[:data_res_p],indices_res[:data_res_p],indptr_mat_res

开始时分配的最大内存
这种方法的性能和可用性在很大程度上取决于输入。在这种方法中分配了最大内存(这很容易导致内存不足错误)。

@nb.njit(cache=True)
def sparse_elementwise_maximum_nb(indices_mat,indptr_mat,data_mat,shape_mat,vec_row_ind,vec_row_data,shrink_to_fit):
    max_non_zero=shape_mat[0]*vec_row_data.shape[0]+data_mat.shape[0]
    data_res=np.empty(max_non_zero,dtype=data_mat.dtype)
    indices_res=np.empty(max_non_zero,dtype=np.int32)
    data_res_p=0

    indptr_mat_res=np.empty((shape_mat[0]+1),dtype=np.int32)
    indptr_mat_res[0]=0
    indptr_mat_res_p=1
    indptr_mat_=0

    for row_idx in range(shape_mat[0]):
        mat_row_ind=indices_mat[indptr_mat[row_idx]:indptr_mat[row_idx+1]]
        mat_row_data=data_mat[indptr_mat[row_idx]:indptr_mat[row_idx+1]]

        mat_ptr=0
        vec_ptr=0
        while mat_ptr<mat_row_ind.shape[0] and vec_ptr<vec_row_ind.shape[0]:
            ind_mat=mat_row_ind[mat_ptr]
            ind_vec=vec_row_ind[vec_ptr]

            #value for both matrix and vector is present
            if ind_mat==ind_vec:
                data_res[data_res_p]=max(mat_row_data[mat_ptr],vec_row_data[vec_ptr])
                indices_res[data_res_p]=ind_mat
                data_res_p+=1
                mat_ptr+=1
                vec_ptr+=1
                indptr_mat_+=1

            #only value for the matrix is present vector is assumed 0
            elif ind_mat<ind_vec:
                if mat_row_data[mat_ptr] >0:
                    data_res[data_res_p]=mat_row_data[mat_ptr]
                    indices_res[data_res_p]=ind_mat
                    data_res_p+=1
                    indptr_mat_+=1
                mat_ptr+=1

            #only value for the vector is present matrix is assumed 0
            else:
                if vec_row_data[vec_ptr] >0:
                    data_res[data_res_p]=vec_row_data[vec_ptr]
                    indices_res[data_res_p]=ind_vec
                    data_res_p+=1
                    indptr_mat_+=1
                vec_ptr+=1

        for i in range(mat_ptr,mat_row_ind.shape[0]):
            if mat_row_data[i] >0:
                data_res[data_res_p]=mat_row_data[i]
                indices_res[data_res_p]=mat_row_ind[i]
                data_res_p+=1
                indptr_mat_+=1
        for i in range(vec_ptr,vec_row_ind.shape[0]):
            if vec_row_data[i] >0:
                data_res[data_res_p]=vec_row_data[i]
                indices_res[data_res_p]=vec_row_ind[i]
                data_res_p+=1
                indptr_mat_+=1
        indptr_mat_res[indptr_mat_res_p]=indptr_mat_
        indptr_mat_res_p+=1

    if shrink_to_fit==True:
        data_res=np.copy(data_res[:data_res_p])
        indices_res=np.copy(indices_res[:data_res_p])
    else:
        data_res=data_res[:data_res_p]
        indices_res=indices_res[:data_res_p]

    return data_res,indices_res,indptr_mat_res

# get all needed components of the csr object and create a resulting csr object at the end
def sparse_elementwise_maximum_wrap(mat,vec,shrink_to_fit=True):
    mat_csr=mat.tocsr()
    vec_csr=vec.tocsr()

    shape_mat=mat_csr.shape
    indices_mat=mat_csr.indices
    indptr_mat=mat_csr.indptr
    data_mat=mat_csr.data
    indices_vec=vec_csr.indices
    data_vec=vec_csr.data

    res=sparse_elementwise_maximum_nb(indices_mat,indptr_mat,data_mat,shape_mat,indices_vec,data_vec,shrink_to_fit)
    res=sparse.csr_matrix(res, shape=shape_mat)
    return res

计时
Numba 有编译开销或一些开销来从缓存加载函数。如果要获取运行时而不是编译+运行时，请不要考虑第一次调用。

import numpy as np
from scipy import sparse

mat = sparse.csr_matrix(  sparse.random(20000, 4000, density=.01, data_rvs=lambda s: np.random.randint(0, 5000, size=s))  )
vec = sparse.csr_matrix(  sparse.random(1, 4000, density=.01, data_rvs=lambda s: np.random.randint(0, 5000, size=s))  )

%timeit output=sparse_elementwise_maximum(mat, vec)
#for csc input
37.9 s ± 224 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
#for csr input
10.7 s ± 90.8 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

#Daniel F
%timeit sparse_maximum(mat, vec)
164 ms ± 1.74 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

#low level implementation (first try)
%timeit res=sparse_elementwise_maximum_wrap(mat,vec)
89.7 ms ± 2.51 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

#low level implementation (optimized, csr)
%timeit res=sparse_elementwise_maximum_wrap(mat,vec)
16.5 ms ± 122 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

#low level implementation (preallocation, without copying at the end)
%timeit res=sparse_elementwise_maximum_wrap(mat,vec)
16.5 ms ± 122 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

#low level implementation (preallocation, with copying at the end)
%timeit res=sparse_elementwise_maximum_wrap(mat,vec)
16.5 ms ± 122 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

%timeit res=sparse_elementwise_maximum_wrap(mat,vec,shrink_to_fit=False)
14.9 ms ± 110 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
%timeit res=sparse_elementwise_maximum_wrap(mat,vec,shrink_to_fit=True)
21.7 ms ± 399 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)

#For comparison, copying the result takes
%%timeit
np.copy(res.data)
np.copy(res.indices)
np.copy(res.indptr)
7.8 ms ± 47.8 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)