最佳答案
我只能部分回答你的问题:
在上面的示例中,您将内核编写为矩阵,将输入编写为向量。如果您可以将输入写为矩阵,则可以使用 torch.nn.Unfold
在 documentation 中明确计算卷积:
# Convolution is equivalent with Unfold + Matrix Multiplication + Fold (or view to output shape)
inp = torch.randn(1, 3, 10, 12)
w = torch.randn(2, 3, 4, 5)
inp_unf = torch.nn.functional.unfold(inp, (4, 5))
out_unf = inp_unf.transpose(1, 2).matmul(w.view(w.size(0), -1).t()).transpose(1, 2)
out = out_unf.view(1, 2, 7, 8)
(torch.nn.functional.conv2d(inp, w) - out).abs().max()
# tensor(1.9073e-06)
但是,如果您需要计算内核的矩阵(较小的矩阵),您可以使用此函数,该函数基于 Warren Weckessers answer :
def toeplitz_1_ch(kernel, input_size):
# shapes
k_h, k_w = kernel.shape
i_h, i_w = input_size
o_h, o_w = i_h-k_h+1, i_w-k_w+1
# construct 1d conv toeplitz matrices for each row of the kernel
toeplitz = []
for r in range(k_h):
toeplitz.append(linalg.toeplitz(c=(kernel[r,0], *np.zeros(i_w-k_w)), r=(*kernel[r], *np.zeros(i_w-k_w))) )
# construct toeplitz matrix of toeplitz matrices (just for padding=0)
h_blocks, w_blocks = o_h, i_h
h_block, w_block = toeplitz[0].shape
W_conv = np.zeros((h_blocks, h_block, w_blocks, w_block))
for i, B in enumerate(toeplitz):
for j in range(o_h):
W_conv[j, :, i+j, :] = B
W_conv.shape = (h_blocks*h_block, w_blocks*w_block)
return W_conv
不在
pytorch
中但在 numpy
.这是给 padding = 0
但可以通过更改 h_blocks
轻松调整和 w_blocks
和 W_conv[i+j, :, j, :]
.更新:多个输出 channel 只是这些矩阵的多个,因为每个输出都有自己的内核。多个输入 channel 也有它们自己的内核——以及它们自己的矩阵——你在卷积后对其进行平均。这可以按如下方式实现:
def conv2d_toeplitz(kernel, input):
"""Compute 2d convolution over multiple channels via toeplitz matrix
Args:
kernel: shape=(n_out, n_in, H_k, W_k)
input: shape=(n_in, H_i, W_i)"""
kernel_size = kernel.shape
input_size = input.shape
output_size = (kernel_size[0], input_size[1] - (kernel_size[1]-1), input_size[2] - (kernel_size[2]-1))
output = np.zeros(output_size)
for i,ks in enumerate(kernel): # loop over output channel
for j,k in enumerate(ks): # loop over input channel
T_k = toeplitz_1_ch(k, input_size[1:])
output[i] += T_k.dot(input[j].flatten()).reshape(output_size[1:]) # sum over input channels
return output
要检查正确性:
k = np.random.randn(4*3*3*3).reshape((4,3,3,3))
i = np.random.randn(3,7,9)
out = conv2d_toeplitz(k, i)
# check correctness of convolution via toeplitz matrix
print(np.sum((out - F.conv2d(torch.tensor(i).view(1,3,7,9), torch.tensor(k)).numpy())**2))
>>> 1.0063523219807736e-28
更新 2:
也可以在不循环一个矩阵的情况下执行此操作:
def toeplitz_mult_ch(kernel, input_size):
"""Compute toeplitz matrix for 2d conv with multiple in and out channels.
Args:
kernel: shape=(n_out, n_in, H_k, W_k)
input_size: (n_in, H_i, W_i)"""
kernel_size = kernel.shape
output_size = (kernel_size[0], input_size[1] - (kernel_size[1]-1), input_size[2] - (kernel_size[2]-1))
T = np.zeros((output_size[0], int(np.prod(output_size[1:])), input_size[0], int(np.prod(input_size[1:]))))
for i,ks in enumerate(kernel): # loop over output channel
for j,k in enumerate(ks): # loop over input channel
T_k = toeplitz_1_ch(k, input_size[1:])
T[i, :, j, :] = T_k
T.shape = (np.prod(output_size), np.prod(input_size))
return T
输入必须变平,乘法后输出必须重新整形。
检查正确性(使用与上面相同的
i
和 k
):T = toeplitz_mult_ch(k, i.shape)
out = T.dot(i.flatten()).reshape((1,4,5,7))
# check correctness of convolution via toeplitz matrix
print(np.sum((out - F.conv2d(torch.tensor(i).view(1,3,7,9), torch.tensor(k)).numpy())**2))
>>> 1.5486060830252635e-28
关于neural-network - PyTorch 中是否有将卷积转换为全连接网络形式的函数?,我们在Stack Overflow上找到一个类似的问题: https://stackoverflow.com/questions/56702873/