424 lines
15 KiB
Plaintext
424 lines
15 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "dda65809",
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"metadata": {
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"origin_pos": 0
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},
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"source": [
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"# 多输入多输出通道\n",
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":label:`sec_channels`\n",
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"\n",
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"虽然我们在 :numref:`subsec_why-conv-channels`中描述了构成每个图像的多个通道和多层卷积层。例如彩色图像具有标准的RGB通道来代表红、绿和蓝。\n",
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"但是到目前为止，我们仅展示了单个输入和单个输出通道的简化例子。\n",
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"这使得我们可以将输入、卷积核和输出看作二维张量。\n",
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"\n",
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"当我们添加通道时，我们的输入和隐藏的表示都变成了三维张量。例如，每个RGB输入图像具有$3\\times h\\times w$的形状。我们将这个大小为$3$的轴称为*通道*（channel）维度。本节将更深入地研究具有多输入和多输出通道的卷积核。\n",
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"\n",
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"## 多输入通道\n",
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"\n",
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"当输入包含多个通道时，需要构造一个与输入数据具有相同输入通道数的卷积核，以便与输入数据进行互相关运算。假设输入的通道数为$c_i$，那么卷积核的输入通道数也需要为$c_i$。如果卷积核的窗口形状是$k_h\\times k_w$，那么当$c_i=1$时，我们可以把卷积核看作形状为$k_h\\times k_w$的二维张量。\n",
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"\n",
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"然而，当$c_i>1$时，我们卷积核的每个输入通道将包含形状为$k_h\\times k_w$的张量。将这些张量$c_i$连结在一起可以得到形状为$c_i\\times k_h\\times k_w$的卷积核。由于输入和卷积核都有$c_i$个通道，我们可以对每个通道输入的二维张量和卷积核的二维张量进行互相关运算，再对通道求和（将$c_i$的结果相加）得到二维张量。这是多通道输入和多输入通道卷积核之间进行二维互相关运算的结果。\n",
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"\n",
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"在 :numref:`fig_conv_multi_in`中，我们演示了一个具有两个输入通道的二维互相关运算的示例。阴影部分是第一个输出元素以及用于计算这个输出的输入和核张量元素：$(1\\times1+2\\times2+4\\times3+5\\times4)+(0\\times0+1\\times1+3\\times2+4\\times3)=56$。\n",
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"\n",
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"\n",
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":label:`fig_conv_multi_in`\n",
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"\n",
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"为了加深理解，我们(**实现一下多输入通道互相关运算**)。\n",
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"简而言之，我们所做的就是对每个通道执行互相关操作，然后将结果相加。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "412ea0b9",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:36.340241Z",
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"iopub.status.busy": "2023-08-18T07:02:36.339505Z",
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"iopub.status.idle": "2023-08-18T07:02:38.335558Z",
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"shell.execute_reply": "2023-08-18T07:02:38.334349Z"
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},
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"origin_pos": 2,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"import torch\n",
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"from d2l import torch as d2l"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "0cff24d4",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:38.339612Z",
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"iopub.status.busy": "2023-08-18T07:02:38.339031Z",
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"iopub.status.idle": "2023-08-18T07:02:38.344485Z",
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"shell.execute_reply": "2023-08-18T07:02:38.343326Z"
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},
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"origin_pos": 4,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"def corr2d_multi_in(X, K):\n",
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" # 先遍历“X”和“K”的第0个维度（通道维度），再把它们加在一起\n",
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" return sum(d2l.corr2d(x, k) for x, k in zip(X, K))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "54507b8a",
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"metadata": {
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"origin_pos": 6
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},
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"source": [
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"我们可以构造与 :numref:`fig_conv_multi_in`中的值相对应的输入张量`X`和核张量`K`，以(**验证互相关运算的输出**)。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "5a60b8f9",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:38.347937Z",
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"iopub.status.busy": "2023-08-18T07:02:38.347463Z",
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"iopub.status.idle": "2023-08-18T07:02:38.380997Z",
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"shell.execute_reply": "2023-08-18T07:02:38.379885Z"
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},
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"origin_pos": 7,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"tensor([[ 56., 72.],\n",
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" [104., 120.]])"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"X = torch.tensor([[[0.0, 1.0, 2.0], [3.0, 4.0, 5.0], [6.0, 7.0, 8.0]],\n",
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" [[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]])\n",
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"K = torch.tensor([[[0.0, 1.0], [2.0, 3.0]], [[1.0, 2.0], [3.0, 4.0]]])\n",
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"\n",
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"corr2d_multi_in(X, K)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "118648d7",
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"metadata": {
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"origin_pos": 8
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},
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"source": [
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"## 多输出通道\n",
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"\n",
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"到目前为止，不论有多少输入通道，我们还只有一个输出通道。然而，正如我们在 :numref:`subsec_why-conv-channels`中所讨论的，每一层有多个输出通道是至关重要的。在最流行的神经网络架构中，随着神经网络层数的加深，我们常会增加输出通道的维数，通过减少空间分辨率以获得更大的通道深度。直观地说，我们可以将每个通道看作对不同特征的响应。而现实可能更为复杂一些，因为每个通道不是独立学习的，而是为了共同使用而优化的。因此，多输出通道并不仅是学习多个单通道的检测器。\n",
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"\n",
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"用$c_i$和$c_o$分别表示输入和输出通道的数目，并让$k_h$和$k_w$为卷积核的高度和宽度。为了获得多个通道的输出，我们可以为每个输出通道创建一个形状为$c_i\\times k_h\\times k_w$的卷积核张量，这样卷积核的形状是$c_o\\times c_i\\times k_h\\times k_w$。在互相关运算中，每个输出通道先获取所有输入通道，再以对应该输出通道的卷积核计算出结果。\n",
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"\n",
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"如下所示，我们实现一个[**计算多个通道的输出的互相关函数**]。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "aa2e4e5f",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:38.384845Z",
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"iopub.status.busy": "2023-08-18T07:02:38.384104Z",
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"iopub.status.idle": "2023-08-18T07:02:38.389279Z",
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"shell.execute_reply": "2023-08-18T07:02:38.388126Z"
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},
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"origin_pos": 9,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"def corr2d_multi_in_out(X, K):\n",
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" # 迭代“K”的第0个维度，每次都对输入“X”执行互相关运算。\n",
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" # 最后将所有结果都叠加在一起\n",
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" return torch.stack([corr2d_multi_in(X, k) for k in K], 0)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f5677efa",
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"metadata": {
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"origin_pos": 10
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},
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"source": [
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"通过将核张量`K`与`K+1`（`K`中每个元素加$1$）和`K+2`连接起来，构造了一个具有$3$个输出通道的卷积核。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "6dde7543",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:38.392733Z",
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"iopub.status.busy": "2023-08-18T07:02:38.392298Z",
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"iopub.status.idle": "2023-08-18T07:02:38.399310Z",
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"shell.execute_reply": "2023-08-18T07:02:38.398211Z"
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},
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"origin_pos": 11,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"torch.Size([3, 2, 2, 2])"
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]
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"K = torch.stack((K, K + 1, K + 2), 0)\n",
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"K.shape"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c7e08b44",
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"metadata": {
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"origin_pos": 12
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},
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"source": [
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"下面，我们对输入张量`X`与卷积核张量`K`执行互相关运算。现在的输出包含$3$个通道，第一个通道的结果与先前输入张量`X`和多输入单输出通道的结果一致。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "86b2b71f",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:38.403159Z",
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"iopub.status.busy": "2023-08-18T07:02:38.402457Z",
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"iopub.status.idle": "2023-08-18T07:02:38.410409Z",
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"shell.execute_reply": "2023-08-18T07:02:38.409310Z"
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},
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"origin_pos": 13,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"tensor([[[ 56., 72.],\n",
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" [104., 120.]],\n",
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"\n",
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" [[ 76., 100.],\n",
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" [148., 172.]],\n",
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"\n",
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" [[ 96., 128.],\n",
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" [192., 224.]]])"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"corr2d_multi_in_out(X, K)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "285e9413",
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"metadata": {
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"origin_pos": 14
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},
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"source": [
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"## $1\\times 1$ 卷积层\n",
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"\n",
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"[~~1x1卷积~~]\n",
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"\n",
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"$1 \\times 1$卷积，即$k_h = k_w = 1$，看起来似乎没有多大意义。\n",
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"毕竟，卷积的本质是有效提取相邻像素间的相关特征，而$1 \\times 1$卷积显然没有此作用。\n",
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"尽管如此，$1 \\times 1$仍然十分流行，经常包含在复杂深层网络的设计中。下面，让我们详细地解读一下它的实际作用。\n",
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"\n",
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"因为使用了最小窗口，$1\\times 1$卷积失去了卷积层的特有能力——在高度和宽度维度上，识别相邻元素间相互作用的能力。\n",
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"其实$1\\times 1$卷积的唯一计算发生在通道上。\n",
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"\n",
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" :numref:`fig_conv_1x1`展示了使用$1\\times 1$卷积核与$3$个输入通道和$2$个输出通道的互相关计算。\n",
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"这里输入和输出具有相同的高度和宽度，输出中的每个元素都是从输入图像中同一位置的元素的线性组合。\n",
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"我们可以将$1\\times 1$卷积层看作在每个像素位置应用的全连接层，以$c_i$个输入值转换为$c_o$个输出值。\n",
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"因为这仍然是一个卷积层，所以跨像素的权重是一致的。\n",
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"同时，$1\\times 1$卷积层需要的权重维度为$c_o\\times c_i$，再额外加上一个偏置。\n",
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"\n",
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"\n",
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":label:`fig_conv_1x1`\n",
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"\n",
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"下面，我们使用全连接层实现$1 \\times 1$卷积。\n",
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"请注意，我们需要对输入和输出的数据形状进行调整。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "f5be69b4",
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"metadata": {
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"execution": {
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||
"iopub.execute_input": "2023-08-18T07:02:38.413874Z",
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"iopub.status.busy": "2023-08-18T07:02:38.413425Z",
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"iopub.status.idle": "2023-08-18T07:02:38.419141Z",
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"shell.execute_reply": "2023-08-18T07:02:38.418037Z"
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},
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"origin_pos": 15,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"def corr2d_multi_in_out_1x1(X, K):\n",
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" c_i, h, w = X.shape\n",
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" c_o = K.shape[0]\n",
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" X = X.reshape((c_i, h * w))\n",
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" K = K.reshape((c_o, c_i))\n",
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" # 全连接层中的矩阵乘法\n",
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" Y = torch.matmul(K, X)\n",
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" return Y.reshape((c_o, h, w))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0685d9f1",
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"metadata": {
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"origin_pos": 16
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},
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"source": [
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"当执行$1\\times 1$卷积运算时，上述函数相当于先前实现的互相关函数`corr2d_multi_in_out`。让我们用一些样本数据来验证这一点。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "420f0d54",
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"metadata": {
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"execution": {
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||
"iopub.execute_input": "2023-08-18T07:02:38.422499Z",
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"iopub.status.busy": "2023-08-18T07:02:38.422070Z",
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||
"iopub.status.idle": "2023-08-18T07:02:38.427214Z",
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||
"shell.execute_reply": "2023-08-18T07:02:38.426115Z"
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},
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"origin_pos": 17,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"X = torch.normal(0, 1, (3, 3, 3))\n",
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"K = torch.normal(0, 1, (2, 3, 1, 1))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"id": "7250eae2",
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"metadata": {
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"execution": {
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||
"iopub.execute_input": "2023-08-18T07:02:38.430613Z",
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||
"iopub.status.busy": "2023-08-18T07:02:38.430184Z",
|
||
"iopub.status.idle": "2023-08-18T07:02:38.438715Z",
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"shell.execute_reply": "2023-08-18T07:02:38.437662Z"
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},
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"origin_pos": 19,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"Y1 = corr2d_multi_in_out_1x1(X, K)\n",
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"Y2 = corr2d_multi_in_out(X, K)\n",
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"assert float(torch.abs(Y1 - Y2).sum()) < 1e-6"
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]
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},
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{
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"cell_type": "markdown",
|
||
"id": "8ba378bd",
|
||
"metadata": {
|
||
"origin_pos": 20
|
||
},
|
||
"source": [
|
||
"## 小结\n",
|
||
"\n",
|
||
"* 多输入多输出通道可以用来扩展卷积层的模型。\n",
|
||
"* 当以每像素为基础应用时，$1\\times 1$卷积层相当于全连接层。\n",
|
||
"* $1\\times 1$卷积层通常用于调整网络层的通道数量和控制模型复杂性。\n",
|
||
"\n",
|
||
"## 练习\n",
|
||
"\n",
|
||
"1. 假设我们有两个卷积核，大小分别为$k_1$和$k_2$（中间没有非线性激活函数）。\n",
|
||
" 1. 证明运算可以用单次卷积来表示。\n",
|
||
" 1. 这个等效的单个卷积核的维数是多少呢？\n",
|
||
" 1. 反之亦然吗？\n",
|
||
"1. 假设输入为$c_i\\times h\\times w$，卷积核大小为$c_o\\times c_i\\times k_h\\times k_w$，填充为$(p_h, p_w)$，步幅为$(s_h, s_w)$。\n",
|
||
" 1. 前向传播的计算成本（乘法和加法）是多少？\n",
|
||
" 1. 内存占用是多少？\n",
|
||
" 1. 反向传播的内存占用是多少？\n",
|
||
" 1. 反向传播的计算成本是多少？\n",
|
||
"1. 如果我们将输入通道$c_i$和输出通道$c_o$的数量加倍，计算数量会增加多少？如果我们把填充数量翻一番会怎么样？\n",
|
||
"1. 如果卷积核的高度和宽度是$k_h=k_w=1$，前向传播的计算复杂度是多少？\n",
|
||
"1. 本节最后一个示例中的变量`Y1`和`Y2`是否完全相同？为什么？\n",
|
||
"1. 当卷积窗口不是$1\\times 1$时，如何使用矩阵乘法实现卷积？\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "0167237f",
|
||
"metadata": {
|
||
"origin_pos": 22,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"source": [
|
||
"[Discussions](https://discuss.d2l.ai/t/1854)\n"
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"language_info": {
|
||
"name": "python"
|
||
},
|
||
"required_libs": []
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 5
|
||
} |