527 lines
16 KiB
Plaintext
527 lines
16 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "3406a2db",
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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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"# 汇聚层\n",
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":label:`sec_pooling`\n",
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"\n",
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"通常当我们处理图像时，我们希望逐渐降低隐藏表示的空间分辨率、聚集信息，这样随着我们在神经网络中层叠的上升，每个神经元对其敏感的感受野（输入）就越大。\n",
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"\n",
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"而我们的机器学习任务通常会跟全局图像的问题有关（例如，“图像是否包含一只猫呢？”），所以我们最后一层的神经元应该对整个输入的全局敏感。通过逐渐聚合信息，生成越来越粗糙的映射，最终实现学习全局表示的目标，同时将卷积图层的所有优势保留在中间层。\n",
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"\n",
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"此外，当检测较底层的特征时（例如 :numref:`sec_conv_layer`中所讨论的边缘），我们通常希望这些特征保持某种程度上的平移不变性。例如，如果我们拍摄黑白之间轮廓清晰的图像`X`，并将整个图像向右移动一个像素，即`Z[i, j] = X[i, j + 1]`，则新图像`Z`的输出可能大不相同。而在现实中，随着拍摄角度的移动，任何物体几乎不可能发生在同一像素上。即使用三脚架拍摄一个静止的物体，由于快门的移动而引起的相机振动，可能会使所有物体左右移动一个像素（除了高端相机配备了特殊功能来解决这个问题）。\n",
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"\n",
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"本节将介绍*汇聚*（pooling）层，它具有双重目的：降低卷积层对位置的敏感性，同时降低对空间降采样表示的敏感性。\n",
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"\n",
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"## 最大汇聚层和平均汇聚层\n",
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"\n",
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"与卷积层类似，汇聚层运算符由一个固定形状的窗口组成，该窗口根据其步幅大小在输入的所有区域上滑动，为固定形状窗口（有时称为*汇聚窗口*）遍历的每个位置计算一个输出。\n",
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"然而，不同于卷积层中的输入与卷积核之间的互相关计算，汇聚层不包含参数。\n",
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"相反，池运算是确定性的，我们通常计算汇聚窗口中所有元素的最大值或平均值。这些操作分别称为*最大汇聚层*（maximum pooling）和*平均汇聚层*（average pooling）。\n",
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"\n",
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"在这两种情况下，与互相关运算符一样，汇聚窗口从输入张量的左上角开始，从左往右、从上往下的在输入张量内滑动。在汇聚窗口到达的每个位置，它计算该窗口中输入子张量的最大值或平均值。计算最大值或平均值是取决于使用了最大汇聚层还是平均汇聚层。\n",
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"\n",
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"\n",
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":label:`fig_pooling`\n",
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"\n",
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" :numref:`fig_pooling`中的输出张量的高度为$2$，宽度为$2$。这四个元素为每个汇聚窗口中的最大值：\n",
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"\n",
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"$$\n",
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"\\max(0, 1, 3, 4)=4,\\\\\n",
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"\\max(1, 2, 4, 5)=5,\\\\\n",
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"\\max(3, 4, 6, 7)=7,\\\\\n",
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"\\max(4, 5, 7, 8)=8.\\\\\n",
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"$$\n",
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"\n",
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"汇聚窗口形状为$p \\times q$的汇聚层称为$p \\times q$汇聚层，汇聚操作称为$p \\times q$汇聚。\n",
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"\n",
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"回到本节开头提到的对象边缘检测示例，现在我们将使用卷积层的输出作为$2\\times 2$最大汇聚的输入。\n",
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"设置卷积层输入为`X`，汇聚层输出为`Y`。\n",
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"无论`X[i, j]`和`X[i, j + 1]`的值相同与否，或`X[i, j + 1]`和`X[i, j + 2]`的值相同与否，汇聚层始终输出`Y[i, j] = 1`。\n",
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"也就是说，使用$2\\times 2$最大汇聚层，即使在高度或宽度上移动一个元素，卷积层仍然可以识别到模式。\n",
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"\n",
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"在下面的代码中的`pool2d`函数，我们(**实现汇聚层的前向传播**)。\n",
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"这类似于 :numref:`sec_conv_layer`中的`corr2d`函数。\n",
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"然而，这里我们没有卷积核，输出为输入中每个区域的最大值或平均值。\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": "292e979e",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:18.192662Z",
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"iopub.status.busy": "2023-08-18T07:02:18.191844Z",
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"iopub.status.idle": "2023-08-18T07:02:20.224371Z",
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"shell.execute_reply": "2023-08-18T07:02:20.223413Z"
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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 torch import nn\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": "fe35adac",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:20.228639Z",
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"iopub.status.busy": "2023-08-18T07:02:20.227964Z",
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"iopub.status.idle": "2023-08-18T07:02:20.234155Z",
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"shell.execute_reply": "2023-08-18T07:02:20.233266Z"
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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 pool2d(X, pool_size, mode='max'):\n",
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" p_h, p_w = pool_size\n",
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" Y = torch.zeros((X.shape[0] - p_h + 1, X.shape[1] - p_w + 1))\n",
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" for i in range(Y.shape[0]):\n",
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" for j in range(Y.shape[1]):\n",
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" if mode == 'max':\n",
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" Y[i, j] = X[i: i + p_h, j: j + p_w].max()\n",
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" elif mode == 'avg':\n",
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" Y[i, j] = X[i: i + p_h, j: j + p_w].mean()\n",
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" return Y"
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]
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},
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{
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"cell_type": "markdown",
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"id": "27b51b5e",
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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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"我们可以构建 :numref:`fig_pooling`中的输入张量`X`，[**验证二维最大汇聚层的输出**]。\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": "3a781c85",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:20.237767Z",
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"iopub.status.busy": "2023-08-18T07:02:20.237211Z",
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"iopub.status.idle": "2023-08-18T07:02:20.268065Z",
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"shell.execute_reply": "2023-08-18T07:02:20.267212Z"
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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([[4., 5.],\n",
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" [7., 8.]])"
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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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"pool2d(X, (2, 2))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "8cc88d86",
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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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"此外，我们还可以(**验证平均汇聚层**)。\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": "4f9a1ffd",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:20.272001Z",
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"iopub.status.busy": "2023-08-18T07:02:20.271411Z",
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"iopub.status.idle": "2023-08-18T07:02:20.277849Z",
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"shell.execute_reply": "2023-08-18T07:02:20.276928Z"
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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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{
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"data": {
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"text/plain": [
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"tensor([[2., 3.],\n",
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" [5., 6.]])"
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]
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},
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"execution_count": 4,
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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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"pool2d(X, (2, 2), 'avg')"
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]
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},
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{
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"cell_type": "markdown",
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"id": "447c6999",
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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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"## [**填充和步幅**]\n",
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"\n",
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"与卷积层一样，汇聚层也可以改变输出形状。和以前一样，我们可以通过填充和步幅以获得所需的输出形状。\n",
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"下面，我们用深度学习框架中内置的二维最大汇聚层，来演示汇聚层中填充和步幅的使用。\n",
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"我们首先构造了一个输入张量`X`，它有四个维度，其中样本数和通道数都是1。\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": "140d08f5",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:20.281458Z",
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"iopub.status.busy": "2023-08-18T07:02:20.280874Z",
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"iopub.status.idle": "2023-08-18T07:02:20.287391Z",
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"shell.execute_reply": "2023-08-18T07:02:20.286578Z"
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},
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"origin_pos": 12,
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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([[[[ 0., 1., 2., 3.],\n",
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" [ 4., 5., 6., 7.],\n",
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" [ 8., 9., 10., 11.],\n",
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" [12., 13., 14., 15.]]]])"
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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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"X = torch.arange(16, dtype=torch.float32).reshape((1, 1, 4, 4))\n",
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"X"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f95f2492",
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"metadata": {
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"origin_pos": 15
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},
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"source": [
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"默认情况下，(**深度学习框架中的步幅与汇聚窗口的大小相同**)。\n",
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"因此，如果我们使用形状为`(3, 3)`的汇聚窗口，那么默认情况下，我们得到的步幅形状为`(3, 3)`。\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": "a3cc01e3",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:02:20.291052Z",
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"iopub.status.busy": "2023-08-18T07:02:20.290402Z",
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"iopub.status.idle": "2023-08-18T07:02:20.296276Z",
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"shell.execute_reply": "2023-08-18T07:02:20.295476Z"
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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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{
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"data": {
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"text/plain": [
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"tensor([[[[10.]]]])"
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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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"pool2d = nn.MaxPool2d(3)\n",
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"pool2d(X)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0b19d625",
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"metadata": {
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"origin_pos": 20
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},
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"source": [
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"[**填充和步幅可以手动设定**]。\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": "9c247428",
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"metadata": {
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"execution": {
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||
"iopub.execute_input": "2023-08-18T07:02:20.299965Z",
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||
"iopub.status.busy": "2023-08-18T07:02:20.299310Z",
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||
"iopub.status.idle": "2023-08-18T07:02:20.307455Z",
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||
"shell.execute_reply": "2023-08-18T07:02:20.306477Z"
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},
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"origin_pos": 22,
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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([[[[ 5., 7.],\n",
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" [13., 15.]]]])"
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]
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},
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"execution_count": 7,
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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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"pool2d = nn.MaxPool2d(3, padding=1, stride=2)\n",
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"pool2d(X)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "635b4034",
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||
"metadata": {
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||
"origin_pos": 26,
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"tab": [
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"pytorch"
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]
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||
},
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"source": [
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"当然，我们可以(**设定一个任意大小的矩形汇聚窗口，并分别设定填充和步幅的高度和宽度**)。\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": "7c169b2f",
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||
"metadata": {
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||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:02:20.311794Z",
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||
"iopub.status.busy": "2023-08-18T07:02:20.311492Z",
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||
"iopub.status.idle": "2023-08-18T07:02:20.320399Z",
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||
"shell.execute_reply": "2023-08-18T07:02:20.319108Z"
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},
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"origin_pos": 30,
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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([[[[ 5., 7.],\n",
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" [13., 15.]]]])"
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]
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},
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"execution_count": 8,
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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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"pool2d = nn.MaxPool2d((2, 3), stride=(2, 3), padding=(0, 1))\n",
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"pool2d(X)"
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]
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},
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{
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"cell_type": "markdown",
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||
"id": "a893596a",
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"metadata": {
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"origin_pos": 33
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},
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"source": [
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"## 多个通道\n",
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"\n",
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"在处理多通道输入数据时，[**汇聚层在每个输入通道上单独运算**]，而不是像卷积层一样在通道上对输入进行汇总。\n",
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"这意味着汇聚层的输出通道数与输入通道数相同。\n",
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"下面，我们将在通道维度上连结张量`X`和`X + 1`，以构建具有2个通道的输入。\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": 9,
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"id": "c0a30a7f",
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||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:02:20.325617Z",
|
||
"iopub.status.busy": "2023-08-18T07:02:20.324879Z",
|
||
"iopub.status.idle": "2023-08-18T07:02:20.335303Z",
|
||
"shell.execute_reply": "2023-08-18T07:02:20.334055Z"
|
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},
|
||
"origin_pos": 35,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[[[ 0., 1., 2., 3.],\n",
|
||
" [ 4., 5., 6., 7.],\n",
|
||
" [ 8., 9., 10., 11.],\n",
|
||
" [12., 13., 14., 15.]],\n",
|
||
"\n",
|
||
" [[ 1., 2., 3., 4.],\n",
|
||
" [ 5., 6., 7., 8.],\n",
|
||
" [ 9., 10., 11., 12.],\n",
|
||
" [13., 14., 15., 16.]]]])"
|
||
]
|
||
},
|
||
"execution_count": 9,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"X = torch.cat((X, X + 1), 1)\n",
|
||
"X"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "45add004",
|
||
"metadata": {
|
||
"origin_pos": 37
|
||
},
|
||
"source": [
|
||
"如下所示，汇聚后输出通道的数量仍然是2。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 10,
|
||
"id": "e534c8f3",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:02:20.340529Z",
|
||
"iopub.status.busy": "2023-08-18T07:02:20.339767Z",
|
||
"iopub.status.idle": "2023-08-18T07:02:20.349365Z",
|
||
"shell.execute_reply": "2023-08-18T07:02:20.348159Z"
|
||
},
|
||
"origin_pos": 39,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[[[ 5., 7.],\n",
|
||
" [13., 15.]],\n",
|
||
"\n",
|
||
" [[ 6., 8.],\n",
|
||
" [14., 16.]]]])"
|
||
]
|
||
},
|
||
"execution_count": 10,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"pool2d = nn.MaxPool2d(3, padding=1, stride=2)\n",
|
||
"pool2d(X)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "0a91fd9f",
|
||
"metadata": {
|
||
"origin_pos": 43
|
||
},
|
||
"source": [
|
||
"## 小结\n",
|
||
"\n",
|
||
"* 对于给定输入元素，最大汇聚层会输出该窗口内的最大值，平均汇聚层会输出该窗口内的平均值。\n",
|
||
"* 汇聚层的主要优点之一是减轻卷积层对位置的过度敏感。\n",
|
||
"* 我们可以指定汇聚层的填充和步幅。\n",
|
||
"* 使用最大汇聚层以及大于1的步幅，可减少空间维度（如高度和宽度）。\n",
|
||
"* 汇聚层的输出通道数与输入通道数相同。\n",
|
||
"\n",
|
||
"## 练习\n",
|
||
"\n",
|
||
"1. 尝试将平均汇聚层作为卷积层的特殊情况实现。\n",
|
||
"1. 尝试将最大汇聚层作为卷积层的特殊情况实现。\n",
|
||
"1. 假设汇聚层的输入大小为$c\\times h\\times w$，则汇聚窗口的形状为$p_h\\times p_w$，填充为$(p_h, p_w)$，步幅为$(s_h, s_w)$。这个汇聚层的计算成本是多少？\n",
|
||
"1. 为什么最大汇聚层和平均汇聚层的工作方式不同？\n",
|
||
"1. 我们是否需要最小汇聚层？可以用已知函数替换它吗？\n",
|
||
"1. 除了平均汇聚层和最大汇聚层，是否有其它函数可以考虑（提示：回想一下`softmax`）？为什么它不流行？\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "f53a8320",
|
||
"metadata": {
|
||
"origin_pos": 45,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"source": [
|
||
"[Discussions](https://discuss.d2l.ai/t/1857)\n"
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"language_info": {
|
||
"name": "python"
|
||
},
|
||
"required_libs": []
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 5
|
||
} |