1273 lines
34 KiB
Plaintext
1273 lines
34 KiB
Plaintext
{
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"cells": [
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{
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||
"cell_type": "markdown",
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||
"id": "8e54cd7b",
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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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"# 数据操作\n",
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":label:`sec_ndarray`\n",
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"\n",
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"为了能够完成各种数据操作，我们需要某种方法来存储和操作数据。\n",
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"通常，我们需要做两件重要的事：（1）获取数据；（2）将数据读入计算机后对其进行处理。\n",
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"如果没有某种方法来存储数据，那么获取数据是没有意义的。\n",
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"\n",
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"首先，我们介绍$n$维数组，也称为*张量*（tensor）。\n",
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"使用过Python中NumPy计算包的读者会对本部分很熟悉。\n",
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"无论使用哪个深度学习框架，它的*张量类*（在MXNet中为`ndarray`，\n",
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"在PyTorch和TensorFlow中为`Tensor`）都与Numpy的`ndarray`类似。\n",
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"但深度学习框架又比Numpy的`ndarray`多一些重要功能：\n",
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"首先，GPU很好地支持加速计算，而NumPy仅支持CPU计算；\n",
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"其次，张量类支持自动微分。\n",
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"这些功能使得张量类更适合深度学习。\n",
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"如果没有特殊说明，本书中所说的张量均指的是张量类的实例。\n",
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"\n",
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"## 入门\n",
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"\n",
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"本节的目标是帮助读者了解并运行一些在阅读本书的过程中会用到的基本数值计算工具。\n",
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"如果你很难理解一些数学概念或库函数，请不要担心。\n",
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"后面的章节将通过一些实际的例子来回顾这些内容。\n",
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"如果你已经具有相关经验，想要深入学习数学内容，可以跳过本节。\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a07b55fb",
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"metadata": {
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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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"source": [
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"(**首先，我们导入`torch`。请注意，虽然它被称为PyTorch，但是代码中使用`torch`而不是`pytorch`。**)\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": "278e6d3f",
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||
"metadata": {
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||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:01.545874Z",
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||
"iopub.status.busy": "2023-08-18T07:05:01.545147Z",
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||
"iopub.status.idle": "2023-08-18T07:05:02.992816Z",
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||
"shell.execute_reply": "2023-08-18T07:05:02.991719Z"
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},
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"origin_pos": 5,
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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"
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]
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},
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||
{
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"cell_type": "markdown",
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||
"id": "ad15bff7",
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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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"[**张量表示一个由数值组成的数组，这个数组可能有多个维度**]。\n",
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"具有一个轴的张量对应数学上的*向量*（vector）；\n",
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"具有两个轴的张量对应数学上的*矩阵*（matrix）；\n",
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"具有两个轴以上的张量没有特殊的数学名称。\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0723e844",
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||
"metadata": {
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||
"origin_pos": 10,
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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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"首先，我们可以使用 `arange` 创建一个行向量 `x`。这个行向量包含以0开始的前12个整数，它们默认创建为整数。也可指定创建类型为浮点数。张量中的每个值都称为张量的 *元素*（element）。例如，张量 `x` 中有 12 个元素。除非额外指定，新的张量将存储在内存中，并采用基于CPU的计算。\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": 2,
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||
"id": "b1700627",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:02.997386Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:02.996970Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.007632Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.006483Z"
|
||
},
|
||
"origin_pos": 13,
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||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
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||
"outputs": [
|
||
{
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||
"data": {
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||
"text/plain": [
|
||
"tensor([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])"
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||
]
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||
},
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||
"execution_count": 2,
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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(12)\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",
|
||
"id": "6c8ebb1e",
|
||
"metadata": {
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||
"origin_pos": 16
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||
},
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||
"source": [
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"[**可以通过张量的`shape`属性来访问张量（沿每个轴的长度）的*形状***]\n",
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"(~~和张量中元素的总数~~)。\n"
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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": "b86b6572",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.011628Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.011110Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.017191Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.016193Z"
|
||
},
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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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||
"torch.Size([12])"
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||
]
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||
},
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||
"execution_count": 3,
|
||
"metadata": {},
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||
"output_type": "execute_result"
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||
}
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||
],
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||
"source": [
|
||
"x.shape"
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||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "25cabae8",
|
||
"metadata": {
|
||
"origin_pos": 18
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||
},
|
||
"source": [
|
||
"如果只想知道张量中元素的总数，即形状的所有元素乘积，可以检查它的大小（size）。\n",
|
||
"因为这里在处理的是一个向量，所以它的`shape`与它的`size`相同。\n"
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||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 4,
|
||
"id": "b8b69ca9",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.020938Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.020483Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.026998Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.025752Z"
|
||
},
|
||
"origin_pos": 20,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"12"
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||
]
|
||
},
|
||
"execution_count": 4,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"x.numel()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "af15bcd6",
|
||
"metadata": {
|
||
"origin_pos": 23
|
||
},
|
||
"source": [
|
||
"[**要想改变一个张量的形状而不改变元素数量和元素值，可以调用`reshape`函数。**]\n",
|
||
"例如，可以把张量`x`从形状为（12,）的行向量转换为形状为（3,4）的矩阵。\n",
|
||
"这个新的张量包含与转换前相同的值，但是它被看成一个3行4列的矩阵。\n",
|
||
"要重点说明一下，虽然张量的形状发生了改变，但其元素值并没有变。\n",
|
||
"注意，通过改变张量的形状，张量的大小不会改变。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 5,
|
||
"id": "0f294243",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.031842Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.031448Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.039288Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.038227Z"
|
||
},
|
||
"origin_pos": 24,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[ 0, 1, 2, 3],\n",
|
||
" [ 4, 5, 6, 7],\n",
|
||
" [ 8, 9, 10, 11]])"
|
||
]
|
||
},
|
||
"execution_count": 5,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"X = x.reshape(3, 4)\n",
|
||
"X"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "eae533a0",
|
||
"metadata": {
|
||
"origin_pos": 27
|
||
},
|
||
"source": [
|
||
"我们不需要通过手动指定每个维度来改变形状。\n",
|
||
"也就是说，如果我们的目标形状是（高度,宽度），\n",
|
||
"那么在知道宽度后，高度会被自动计算得出，不必我们自己做除法。\n",
|
||
"在上面的例子中，为了获得一个3行的矩阵，我们手动指定了它有3行和4列。\n",
|
||
"幸运的是，我们可以通过`-1`来调用此自动计算出维度的功能。\n",
|
||
"即我们可以用`x.reshape(-1,4)`或`x.reshape(3,-1)`来取代`x.reshape(3,4)`。\n",
|
||
"\n",
|
||
"有时，我们希望[**使用全0、全1、其他常量，或者从特定分布中随机采样的数字**]来初始化矩阵。\n",
|
||
"我们可以创建一个形状为（2,3,4）的张量，其中所有元素都设置为0。代码如下：\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 6,
|
||
"id": "b23c3056",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.044733Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.043866Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.052195Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.051146Z"
|
||
},
|
||
"origin_pos": 29,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[[0., 0., 0., 0.],\n",
|
||
" [0., 0., 0., 0.],\n",
|
||
" [0., 0., 0., 0.]],\n",
|
||
"\n",
|
||
" [[0., 0., 0., 0.],\n",
|
||
" [0., 0., 0., 0.],\n",
|
||
" [0., 0., 0., 0.]]])"
|
||
]
|
||
},
|
||
"execution_count": 6,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"torch.zeros((2, 3, 4))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "6a180a12",
|
||
"metadata": {
|
||
"origin_pos": 32
|
||
},
|
||
"source": [
|
||
"同样，我们可以创建一个形状为`(2,3,4)`的张量，其中所有元素都设置为1。代码如下：\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 7,
|
||
"id": "25981960",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.057264Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.056578Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.064973Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.063853Z"
|
||
},
|
||
"origin_pos": 34,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[[1., 1., 1., 1.],\n",
|
||
" [1., 1., 1., 1.],\n",
|
||
" [1., 1., 1., 1.]],\n",
|
||
"\n",
|
||
" [[1., 1., 1., 1.],\n",
|
||
" [1., 1., 1., 1.],\n",
|
||
" [1., 1., 1., 1.]]])"
|
||
]
|
||
},
|
||
"execution_count": 7,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"torch.ones((2, 3, 4))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "672f1257",
|
||
"metadata": {
|
||
"origin_pos": 37
|
||
},
|
||
"source": [
|
||
"有时我们想通过从某个特定的概率分布中随机采样来得到张量中每个元素的值。\n",
|
||
"例如，当我们构造数组来作为神经网络中的参数时，我们通常会随机初始化参数的值。\n",
|
||
"以下代码创建一个形状为（3,4）的张量。\n",
|
||
"其中的每个元素都从均值为0、标准差为1的标准高斯分布（正态分布）中随机采样。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 8,
|
||
"id": "2493f09a",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.069946Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.069231Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.077304Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.076139Z"
|
||
},
|
||
"origin_pos": 39,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[-0.0135, 0.0665, 0.0912, 0.3212],\n",
|
||
" [ 1.4653, 0.1843, -1.6995, -0.3036],\n",
|
||
" [ 1.7646, 1.0450, 0.2457, -0.7732]])"
|
||
]
|
||
},
|
||
"execution_count": 8,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"torch.randn(3, 4)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "cc2299f0",
|
||
"metadata": {
|
||
"origin_pos": 42
|
||
},
|
||
"source": [
|
||
"我们还可以[**通过提供包含数值的Python列表（或嵌套列表），来为所需张量中的每个元素赋予确定值**]。\n",
|
||
"在这里，最外层的列表对应于轴0，内层的列表对应于轴1。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 9,
|
||
"id": "708be494",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.082360Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.081424Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.090148Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.088973Z"
|
||
},
|
||
"origin_pos": 44,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[2, 1, 4, 3],\n",
|
||
" [1, 2, 3, 4],\n",
|
||
" [4, 3, 2, 1]])"
|
||
]
|
||
},
|
||
"execution_count": 9,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"torch.tensor([[2, 1, 4, 3], [1, 2, 3, 4], [4, 3, 2, 1]])"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "561a2ec6",
|
||
"metadata": {
|
||
"origin_pos": 47
|
||
},
|
||
"source": [
|
||
"## 运算符\n",
|
||
"\n",
|
||
"我们的兴趣不仅限于读取数据和写入数据。\n",
|
||
"我们想在这些数据上执行数学运算，其中最简单且最有用的操作是*按元素*（elementwise）运算。\n",
|
||
"它们将标准标量运算符应用于数组的每个元素。\n",
|
||
"对于将两个数组作为输入的函数，按元素运算将二元运算符应用于两个数组中的每对位置对应的元素。\n",
|
||
"我们可以基于任何从标量到标量的函数来创建按元素函数。\n",
|
||
"\n",
|
||
"在数学表示法中，我们将通过符号$f: \\mathbb{R} \\rightarrow \\mathbb{R}$\n",
|
||
"来表示*一元*标量运算符（只接收一个输入）。\n",
|
||
"这意味着该函数从任何实数（$\\mathbb{R}$）映射到另一个实数。\n",
|
||
"同样，我们通过符号$f: \\mathbb{R}, \\mathbb{R} \\rightarrow \\mathbb{R}$\n",
|
||
"表示*二元*标量运算符，这意味着该函数接收两个输入，并产生一个输出。\n",
|
||
"给定同一形状的任意两个向量$\\mathbf{u}$和$\\mathbf{v}$和二元运算符$f$，\n",
|
||
"我们可以得到向量$\\mathbf{c} = F(\\mathbf{u},\\mathbf{v})$。\n",
|
||
"具体计算方法是$c_i \\gets f(u_i, v_i)$，\n",
|
||
"其中$c_i$、$u_i$和$v_i$分别是向量$\\mathbf{c}$、$\\mathbf{u}$和$\\mathbf{v}$中的元素。\n",
|
||
"在这里，我们通过将标量函数升级为按元素向量运算来生成向量值\n",
|
||
"$F: \\mathbb{R}^d, \\mathbb{R}^d \\rightarrow \\mathbb{R}^d$。\n",
|
||
"\n",
|
||
"对于任意具有相同形状的张量，\n",
|
||
"[**常见的标准算术运算符（`+`、`-`、`*`、`/`和`**`）都可以被升级为按元素运算**]。\n",
|
||
"我们可以在同一形状的任意两个张量上调用按元素操作。\n",
|
||
"在下面的例子中，我们使用逗号来表示一个具有5个元素的元组，其中每个元素都是按元素操作的结果。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 10,
|
||
"id": "99b28553",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.095504Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.094688Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.106084Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.104976Z"
|
||
},
|
||
"origin_pos": 49,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([ 3., 4., 6., 10.]),\n",
|
||
" tensor([-1., 0., 2., 6.]),\n",
|
||
" tensor([ 2., 4., 8., 16.]),\n",
|
||
" tensor([0.5000, 1.0000, 2.0000, 4.0000]),\n",
|
||
" tensor([ 1., 4., 16., 64.]))"
|
||
]
|
||
},
|
||
"execution_count": 10,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"x = torch.tensor([1.0, 2, 4, 8])\n",
|
||
"y = torch.tensor([2, 2, 2, 2])\n",
|
||
"x + y, x - y, x * y, x / y, x ** y # **运算符是求幂运算"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "1d37bed5",
|
||
"metadata": {
|
||
"origin_pos": 52
|
||
},
|
||
"source": [
|
||
"(**“按元素”方式可以应用更多的计算**)，包括像求幂这样的一元运算符。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 11,
|
||
"id": "ef07c995",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.110973Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.110221Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.120389Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.119471Z"
|
||
},
|
||
"origin_pos": 54,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([2.7183e+00, 7.3891e+00, 5.4598e+01, 2.9810e+03])"
|
||
]
|
||
},
|
||
"execution_count": 11,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"torch.exp(x)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "cc76ca21",
|
||
"metadata": {
|
||
"origin_pos": 57
|
||
},
|
||
"source": [
|
||
"除了按元素计算外，我们还可以执行线性代数运算，包括向量点积和矩阵乘法。\n",
|
||
"我们将在 :numref:`sec_linear-algebra`中解释线性代数的重点内容。\n",
|
||
"\n",
|
||
"[**我们也可以把多个张量*连结*（concatenate）在一起**]，\n",
|
||
"把它们端对端地叠起来形成一个更大的张量。\n",
|
||
"我们只需要提供张量列表，并给出沿哪个轴连结。\n",
|
||
"下面的例子分别演示了当我们沿行（轴-0，形状的第一个元素）\n",
|
||
"和按列（轴-1，形状的第二个元素）连结两个矩阵时，会发生什么情况。\n",
|
||
"我们可以看到，第一个输出张量的轴-0长度（$6$）是两个输入张量轴-0长度的总和（$3 + 3$）；\n",
|
||
"第二个输出张量的轴-1长度（$8$）是两个输入张量轴-1长度的总和（$4 + 4$）。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 12,
|
||
"id": "a583b891",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.125263Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.124477Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.136328Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.135199Z"
|
||
},
|
||
"origin_pos": 59,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([[ 0., 1., 2., 3.],\n",
|
||
" [ 4., 5., 6., 7.],\n",
|
||
" [ 8., 9., 10., 11.],\n",
|
||
" [ 2., 1., 4., 3.],\n",
|
||
" [ 1., 2., 3., 4.],\n",
|
||
" [ 4., 3., 2., 1.]]),\n",
|
||
" tensor([[ 0., 1., 2., 3., 2., 1., 4., 3.],\n",
|
||
" [ 4., 5., 6., 7., 1., 2., 3., 4.],\n",
|
||
" [ 8., 9., 10., 11., 4., 3., 2., 1.]]))"
|
||
]
|
||
},
|
||
"execution_count": 12,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"X = torch.arange(12, dtype=torch.float32).reshape((3,4))\n",
|
||
"Y = torch.tensor([[2.0, 1, 4, 3], [1, 2, 3, 4], [4, 3, 2, 1]])\n",
|
||
"torch.cat((X, Y), dim=0), torch.cat((X, Y), dim=1)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "44965a95",
|
||
"metadata": {
|
||
"origin_pos": 62
|
||
},
|
||
"source": [
|
||
"有时，我们想[**通过*逻辑运算符*构建二元张量**]。\n",
|
||
"以`X == Y`为例：\n",
|
||
"对于每个位置，如果`X`和`Y`在该位置相等，则新张量中相应项的值为1。\n",
|
||
"这意味着逻辑语句`X == Y`在该位置处为真，否则该位置为0。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 13,
|
||
"id": "6405ec63",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.141449Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.140776Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.148692Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.147491Z"
|
||
},
|
||
"origin_pos": 63,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[False, True, False, True],\n",
|
||
" [False, False, False, False],\n",
|
||
" [False, False, False, False]])"
|
||
]
|
||
},
|
||
"execution_count": 13,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"X == Y"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "ffab6a71",
|
||
"metadata": {
|
||
"origin_pos": 64
|
||
},
|
||
"source": [
|
||
"[**对张量中的所有元素进行求和，会产生一个单元素张量。**]\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 14,
|
||
"id": "a13cb291",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.153907Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.152814Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.160277Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.159188Z"
|
||
},
|
||
"origin_pos": 65,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor(66.)"
|
||
]
|
||
},
|
||
"execution_count": 14,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"X.sum()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "e4420b99",
|
||
"metadata": {
|
||
"origin_pos": 67
|
||
},
|
||
"source": [
|
||
"## 广播机制\n",
|
||
":label:`subsec_broadcasting`\n",
|
||
"\n",
|
||
"在上面的部分中，我们看到了如何在相同形状的两个张量上执行按元素操作。\n",
|
||
"在某些情况下，[**即使形状不同，我们仍然可以通过调用\n",
|
||
"*广播机制*（broadcasting mechanism）来执行按元素操作**]。\n",
|
||
"这种机制的工作方式如下：\n",
|
||
"\n",
|
||
"1. 通过适当复制元素来扩展一个或两个数组，以便在转换之后，两个张量具有相同的形状；\n",
|
||
"2. 对生成的数组执行按元素操作。\n",
|
||
"\n",
|
||
"在大多数情况下，我们将沿着数组中长度为1的轴进行广播，如下例子：\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 15,
|
||
"id": "a1de79a2",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.165305Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.164274Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.172771Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.171692Z"
|
||
},
|
||
"origin_pos": 69,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([[0],\n",
|
||
" [1],\n",
|
||
" [2]]),\n",
|
||
" tensor([[0, 1]]))"
|
||
]
|
||
},
|
||
"execution_count": 15,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"a = torch.arange(3).reshape((3, 1))\n",
|
||
"b = torch.arange(2).reshape((1, 2))\n",
|
||
"a, b"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "e4f53984",
|
||
"metadata": {
|
||
"origin_pos": 72
|
||
},
|
||
"source": [
|
||
"由于`a`和`b`分别是$3\\times1$和$1\\times2$矩阵，如果让它们相加，它们的形状不匹配。\n",
|
||
"我们将两个矩阵*广播*为一个更大的$3\\times2$矩阵，如下所示：矩阵`a`将复制列，\n",
|
||
"矩阵`b`将复制行，然后再按元素相加。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 16,
|
||
"id": "4d8904b1",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.177900Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.176935Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.184212Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.183156Z"
|
||
},
|
||
"origin_pos": 73,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[0, 1],\n",
|
||
" [1, 2],\n",
|
||
" [2, 3]])"
|
||
]
|
||
},
|
||
"execution_count": 16,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"a + b"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "68b62722",
|
||
"metadata": {
|
||
"origin_pos": 74
|
||
},
|
||
"source": [
|
||
"## 索引和切片\n",
|
||
"\n",
|
||
"就像在任何其他Python数组中一样，张量中的元素可以通过索引访问。\n",
|
||
"与任何Python数组一样：第一个元素的索引是0，最后一个元素索引是-1；\n",
|
||
"可以指定范围以包含第一个元素和最后一个之前的元素。\n",
|
||
"\n",
|
||
"如下所示，我们[**可以用`[-1]`选择最后一个元素，可以用`[1:3]`选择第二个和第三个元素**]：\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 17,
|
||
"id": "b62b00c7",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.189786Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.188961Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.197712Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.196559Z"
|
||
},
|
||
"origin_pos": 75,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([ 8., 9., 10., 11.]),\n",
|
||
" tensor([[ 4., 5., 6., 7.],\n",
|
||
" [ 8., 9., 10., 11.]]))"
|
||
]
|
||
},
|
||
"execution_count": 17,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"X[-1], X[1:3]"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "49b18803",
|
||
"metadata": {
|
||
"origin_pos": 76,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"source": [
|
||
"[**除读取外，我们还可以通过指定索引来将元素写入矩阵。**]\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 18,
|
||
"id": "56a8261a",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.203157Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.202390Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.210176Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.209097Z"
|
||
},
|
||
"origin_pos": 78,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[ 0., 1., 2., 3.],\n",
|
||
" [ 4., 5., 9., 7.],\n",
|
||
" [ 8., 9., 10., 11.]])"
|
||
]
|
||
},
|
||
"execution_count": 18,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"X[1, 2] = 9\n",
|
||
"X"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "fdec2856",
|
||
"metadata": {
|
||
"origin_pos": 80
|
||
},
|
||
"source": [
|
||
"如果我们想[**为多个元素赋值相同的值，我们只需要索引所有元素，然后为它们赋值。**]\n",
|
||
"例如，`[0:2, :]`访问第1行和第2行，其中“:”代表沿轴1（列）的所有元素。\n",
|
||
"虽然我们讨论的是矩阵的索引，但这也适用于向量和超过2个维度的张量。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 19,
|
||
"id": "bd48bae9",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.214118Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.213430Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.221215Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.220084Z"
|
||
},
|
||
"origin_pos": 81,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[12., 12., 12., 12.],\n",
|
||
" [12., 12., 12., 12.],\n",
|
||
" [ 8., 9., 10., 11.]])"
|
||
]
|
||
},
|
||
"execution_count": 19,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"X[0:2, :] = 12\n",
|
||
"X"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "79660f82",
|
||
"metadata": {
|
||
"origin_pos": 83
|
||
},
|
||
"source": [
|
||
"## 节省内存\n",
|
||
"\n",
|
||
"[**运行一些操作可能会导致为新结果分配内存**]。\n",
|
||
"例如，如果我们用`Y = X + Y`，我们将取消引用`Y`指向的张量，而是指向新分配的内存处的张量。\n",
|
||
"\n",
|
||
"在下面的例子中，我们用Python的`id()`函数演示了这一点，\n",
|
||
"它给我们提供了内存中引用对象的确切地址。\n",
|
||
"运行`Y = Y + X`后，我们会发现`id(Y)`指向另一个位置。\n",
|
||
"这是因为Python首先计算`Y + X`，为结果分配新的内存，然后使`Y`指向内存中的这个新位置。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 20,
|
||
"id": "6bcd6d07",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.225106Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.224353Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.231715Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.230626Z"
|
||
},
|
||
"origin_pos": 84,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"False"
|
||
]
|
||
},
|
||
"execution_count": 20,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"before = id(Y)\n",
|
||
"Y = Y + X\n",
|
||
"id(Y) == before"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "fdb5a8e4",
|
||
"metadata": {
|
||
"origin_pos": 85
|
||
},
|
||
"source": [
|
||
"这可能是不可取的，原因有两个：\n",
|
||
"\n",
|
||
"1. 首先，我们不想总是不必要地分配内存。在机器学习中，我们可能有数百兆的参数，并且在一秒内多次更新所有参数。通常情况下，我们希望原地执行这些更新；\n",
|
||
"2. 如果我们不原地更新，其他引用仍然会指向旧的内存位置，这样我们的某些代码可能会无意中引用旧的参数。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "47e2e97c",
|
||
"metadata": {
|
||
"origin_pos": 86,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"source": [
|
||
"幸运的是，(**执行原地操作**)非常简单。\n",
|
||
"我们可以使用切片表示法将操作的结果分配给先前分配的数组，例如`Y[:] = <expression>`。\n",
|
||
"为了说明这一点，我们首先创建一个新的矩阵`Z`，其形状与另一个`Y`相同，\n",
|
||
"使用`zeros_like`来分配一个全$0$的块。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 21,
|
||
"id": "13b7fdf6",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.236933Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.236016Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.243252Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.242153Z"
|
||
},
|
||
"origin_pos": 89,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"id(Z): 140327634811696\n",
|
||
"id(Z): 140327634811696\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"Z = torch.zeros_like(Y)\n",
|
||
"print('id(Z):', id(Z))\n",
|
||
"Z[:] = X + Y\n",
|
||
"print('id(Z):', id(Z))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "36a38425",
|
||
"metadata": {
|
||
"origin_pos": 92,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"source": [
|
||
"[**如果在后续计算中没有重复使用`X`，\n",
|
||
"我们也可以使用`X[:] = X + Y`或`X += Y`来减少操作的内存开销。**]\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 22,
|
||
"id": "c8a97d75",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.248290Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.247521Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.255046Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.253935Z"
|
||
},
|
||
"origin_pos": 94,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"True"
|
||
]
|
||
},
|
||
"execution_count": 22,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"before = id(X)\n",
|
||
"X += Y\n",
|
||
"id(X) == before"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "3185bd60",
|
||
"metadata": {
|
||
"origin_pos": 96
|
||
},
|
||
"source": [
|
||
"## 转换为其他Python对象\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "d871a6fa",
|
||
"metadata": {
|
||
"origin_pos": 98,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"source": [
|
||
"将深度学习框架定义的张量[**转换为NumPy张量（`ndarray`）**]很容易，反之也同样容易。\n",
|
||
"torch张量和numpy数组将共享它们的底层内存，就地操作更改一个张量也会同时更改另一个张量。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 23,
|
||
"id": "7386f580",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.259655Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.259273Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.266501Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.265738Z"
|
||
},
|
||
"origin_pos": 100,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(numpy.ndarray, torch.Tensor)"
|
||
]
|
||
},
|
||
"execution_count": 23,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A = X.numpy()\n",
|
||
"B = torch.tensor(A)\n",
|
||
"type(A), type(B)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "74f0ae4e",
|
||
"metadata": {
|
||
"origin_pos": 103
|
||
},
|
||
"source": [
|
||
"要(**将大小为1的张量转换为Python标量**)，我们可以调用`item`函数或Python的内置函数。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 24,
|
||
"id": "10a429bd",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:05:03.270566Z",
|
||
"iopub.status.busy": "2023-08-18T07:05:03.270102Z",
|
||
"iopub.status.idle": "2023-08-18T07:05:03.276982Z",
|
||
"shell.execute_reply": "2023-08-18T07:05:03.276051Z"
|
||
},
|
||
"origin_pos": 105,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([3.5000]), 3.5, 3.5, 3)"
|
||
]
|
||
},
|
||
"execution_count": 24,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"a = torch.tensor([3.5])\n",
|
||
"a, a.item(), float(a), int(a)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "ed98db5b",
|
||
"metadata": {
|
||
"origin_pos": 108
|
||
},
|
||
"source": [
|
||
"## 小结\n",
|
||
"\n",
|
||
"* 深度学习存储和操作数据的主要接口是张量（$n$维数组）。它提供了各种功能，包括基本数学运算、广播、索引、切片、内存节省和转换其他Python对象。\n",
|
||
"\n",
|
||
"## 练习\n",
|
||
"\n",
|
||
"1. 运行本节中的代码。将本节中的条件语句`X == Y`更改为`X < Y`或`X > Y`，然后看看你可以得到什么样的张量。\n",
|
||
"1. 用其他形状（例如三维张量）替换广播机制中按元素操作的两个张量。结果是否与预期相同？\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "161ce063",
|
||
"metadata": {
|
||
"origin_pos": 110,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"source": [
|
||
"[Discussions](https://discuss.d2l.ai/t/1747)\n"
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"language_info": {
|
||
"name": "python"
|
||
},
|
||
"required_libs": []
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 5
|
||
} |