1746 lines
52 KiB
Plaintext
1746 lines
52 KiB
Plaintext
{
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"cells": [
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{
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||
"cell_type": "markdown",
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||
"id": "6af409a8",
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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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"# 线性代数\n",
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":label:`sec_linear-algebra`\n",
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"\n",
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"在介绍完如何存储和操作数据后，接下来将简要地回顾一下部分基本线性代数内容。\n",
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"这些内容有助于读者了解和实现本书中介绍的大多数模型。\n",
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"本节将介绍线性代数中的基本数学对象、算术和运算，并用数学符号和相应的代码实现来表示它们。\n",
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"\n",
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"## 标量\n",
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"\n",
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"\n",
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"如果你曾经在餐厅支付餐费，那么应该已经知道一些基本的线性代数，比如在数字间相加或相乘。\n",
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"例如，北京的温度为$52^{\\circ}F$（华氏度，除摄氏度外的另一种温度计量单位）。\n",
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"严格来说，仅包含一个数值被称为*标量*（scalar）。\n",
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"如果要将此华氏度值转换为更常用的摄氏度，\n",
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"则可以计算表达式$c=\\frac{5}{9}(f-32)$，并将$f$赋为$52$。\n",
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"在此等式中，每一项（$5$、$9$和$32$）都是标量值。\n",
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"符号$c$和$f$称为*变量*（variable），它们表示未知的标量值。\n",
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"\n",
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"本书采用了数学表示法，其中标量变量由普通小写字母表示（例如，$x$、$y$和$z$）。\n",
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"本书用$\\mathbb{R}$表示所有（连续）*实数*标量的空间，之后将严格定义*空间*（space）是什么，\n",
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"但现在只要记住表达式$x\\in\\mathbb{R}$是表示$x$是一个实值标量的正式形式。\n",
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"符号$\\in$称为“属于”，它表示“是集合中的成员”。\n",
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"例如$x, y \\in \\{0,1\\}$可以用来表明$x$和$y$是值只能为$0$或$1$的数字。\n",
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"\n",
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"(**标量由只有一个元素的张量表示**)。\n",
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"下面的代码将实例化两个标量，并执行一些熟悉的算术运算，即加法、乘法、除法和指数。\n"
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]
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||
},
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||
{
|
||
"cell_type": "code",
|
||
"execution_count": 1,
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||
"id": "44889577",
|
||
"metadata": {
|
||
"execution": {
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"iopub.execute_input": "2023-08-18T07:01:42.354657Z",
|
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"iopub.status.busy": "2023-08-18T07:01:42.353844Z",
|
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"iopub.status.idle": "2023-08-18T07:01:43.769394Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.768177Z"
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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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||
{
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||
"data": {
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"text/plain": [
|
||
"(tensor(5.), tensor(6.), tensor(1.5000), tensor(9.))"
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||
]
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||
},
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||
"execution_count": 1,
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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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"import torch\n",
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"\n",
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"x = torch.tensor(3.0)\n",
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"y = torch.tensor(2.0)\n",
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"\n",
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"x + y, x * y, x / y, x**y"
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]
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||
},
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||
{
|
||
"cell_type": "markdown",
|
||
"id": "018bf250",
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||
"metadata": {
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||
"origin_pos": 5
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},
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"source": [
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"## 向量\n",
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"\n",
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"[**向量可以被视为标量值组成的列表**]。\n",
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"这些标量值被称为向量的*元素*（element）或*分量*（component）。\n",
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"当向量表示数据集中的样本时，它们的值具有一定的现实意义。\n",
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"例如，如果我们正在训练一个模型来预测贷款违约风险，可能会将每个申请人与一个向量相关联，\n",
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"其分量与其收入、工作年限、过往违约次数和其他因素相对应。\n",
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"如果我们正在研究医院患者可能面临的心脏病发作风险，可能会用一个向量来表示每个患者，\n",
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"其分量为最近的生命体征、胆固醇水平、每天运动时间等。\n",
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"在数学表示法中，向量通常记为粗体、小写的符号\n",
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"（例如，$\\mathbf{x}$、$\\mathbf{y}$和$\\mathbf{z})$）。\n",
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"\n",
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"人们通过一维张量表示向量。一般来说，张量可以具有任意长度，取决于机器的内存限制。\n"
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]
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||
},
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{
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||
"cell_type": "code",
|
||
"execution_count": 2,
|
||
"id": "e5163ab8",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.774490Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.773987Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.781757Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.780603Z"
|
||
},
|
||
"origin_pos": 7,
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||
"tab": [
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||
"pytorch"
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||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
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||
"text/plain": [
|
||
"tensor([0, 1, 2, 3])"
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||
]
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||
},
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||
"execution_count": 2,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
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||
}
|
||
],
|
||
"source": [
|
||
"x = torch.arange(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": "7fc8cd94",
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||
"metadata": {
|
||
"origin_pos": 10
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||
},
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||
"source": [
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||
"我们可以使用下标来引用向量的任一元素，例如可以通过$x_i$来引用第$i$个元素。\n",
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"注意，元素$x_i$是一个标量，所以我们在引用它时不会加粗。\n",
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"大量文献认为列向量是向量的默认方向，在本书中也是如此。\n",
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||
"在数学中，向量$\\mathbf{x}$可以写为：\n",
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"\n",
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"$$\\mathbf{x} =\\begin{bmatrix}x_{1} \\\\x_{2} \\\\ \\vdots \\\\x_{n}\\end{bmatrix},$$\n",
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||
":eqlabel:`eq_vec_def`\n",
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"\n",
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||
"其中$x_1,\\ldots,x_n$是向量的元素。在代码中，我们(**通过张量的索引来访问任一元素**)。\n"
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||
]
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||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 3,
|
||
"id": "34dd7630",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.786346Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.785939Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.793065Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.791986Z"
|
||
},
|
||
"origin_pos": 12,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor(3)"
|
||
]
|
||
},
|
||
"execution_count": 3,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"x[3]"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "59e98e89",
|
||
"metadata": {
|
||
"origin_pos": 15
|
||
},
|
||
"source": [
|
||
"### 长度、维度和形状\n",
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||
"\n",
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||
"向量只是一个数字数组，就像每个数组都有一个长度一样，每个向量也是如此。\n",
|
||
"在数学表示法中，如果我们想说一个向量$\\mathbf{x}$由$n$个实值标量组成，\n",
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||
"可以将其表示为$\\mathbf{x}\\in\\mathbb{R}^n$。\n",
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||
"向量的长度通常称为向量的*维度*（dimension）。\n",
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||
"\n",
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||
"与普通的Python数组一样，我们可以通过调用Python的内置`len()`函数来[**访问张量的长度**]。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 4,
|
||
"id": "d469059b",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.798087Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.797197Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.804049Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.802867Z"
|
||
},
|
||
"origin_pos": 17,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"4"
|
||
]
|
||
},
|
||
"execution_count": 4,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"len(x)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "32d0ccc4",
|
||
"metadata": {
|
||
"origin_pos": 20
|
||
},
|
||
"source": [
|
||
"当用张量表示一个向量（只有一个轴）时，我们也可以通过`.shape`属性访问向量的长度。\n",
|
||
"形状（shape）是一个元素组，列出了张量沿每个轴的长度（维数）。\n",
|
||
"对于(**只有一个轴的张量，形状只有一个元素。**)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 5,
|
||
"id": "bf9bf15e",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.809543Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.808709Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.815762Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.814675Z"
|
||
},
|
||
"origin_pos": 22,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"torch.Size([4])"
|
||
]
|
||
},
|
||
"execution_count": 5,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"x.shape"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "1237aeca",
|
||
"metadata": {
|
||
"origin_pos": 25
|
||
},
|
||
"source": [
|
||
"请注意，*维度*（dimension）这个词在不同上下文时往往会有不同的含义，这经常会使人感到困惑。\n",
|
||
"为了清楚起见，我们在此明确一下：\n",
|
||
"*向量*或*轴*的维度被用来表示*向量*或*轴*的长度，即向量或轴的元素数量。\n",
|
||
"然而，张量的维度用来表示张量具有的轴数。\n",
|
||
"在这个意义上，张量的某个轴的维数就是这个轴的长度。\n",
|
||
"\n",
|
||
"## 矩阵\n",
|
||
"\n",
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||
"正如向量将标量从零阶推广到一阶，矩阵将向量从一阶推广到二阶。\n",
|
||
"矩阵，我们通常用粗体、大写字母来表示\n",
|
||
"（例如，$\\mathbf{X}$、$\\mathbf{Y}$和$\\mathbf{Z}$），\n",
|
||
"在代码中表示为具有两个轴的张量。\n",
|
||
"\n",
|
||
"数学表示法使用$\\mathbf{A} \\in \\mathbb{R}^{m \\times n}$\n",
|
||
"来表示矩阵$\\mathbf{A}$，其由$m$行和$n$列的实值标量组成。\n",
|
||
"我们可以将任意矩阵$\\mathbf{A} \\in \\mathbb{R}^{m \\times n}$视为一个表格，\n",
|
||
"其中每个元素$a_{ij}$属于第$i$行第$j$列：\n",
|
||
"\n",
|
||
"$$\\mathbf{A}=\\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\\\ \\end{bmatrix}.$$\n",
|
||
":eqlabel:`eq_matrix_def`\n",
|
||
"\n",
|
||
"对于任意$\\mathbf{A} \\in \\mathbb{R}^{m \\times n}$，\n",
|
||
"$\\mathbf{A}$的形状是（$m$,$n$）或$m \\times n$。\n",
|
||
"当矩阵具有相同数量的行和列时，其形状将变为正方形；\n",
|
||
"因此，它被称为*方阵*（square matrix）。\n",
|
||
"\n",
|
||
"当调用函数来实例化张量时，\n",
|
||
"我们可以[**通过指定两个分量$m$和$n$来创建一个形状为$m \\times n$的矩阵**]。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 6,
|
||
"id": "b1eac085",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.820985Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.820088Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.828057Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.826957Z"
|
||
},
|
||
"origin_pos": 27,
|
||
"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",
|
||
" [16, 17, 18, 19]])"
|
||
]
|
||
},
|
||
"execution_count": 6,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A = torch.arange(20).reshape(5, 4)\n",
|
||
"A"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "15b5181b",
|
||
"metadata": {
|
||
"origin_pos": 30
|
||
},
|
||
"source": [
|
||
"我们可以通过行索引（$i$）和列索引（$j$）来访问矩阵中的标量元素$a_{ij}$，\n",
|
||
"例如$[\\mathbf{A}]_{ij}$。\n",
|
||
"如果没有给出矩阵$\\mathbf{A}$的标量元素，如在 :eqref:`eq_matrix_def`那样，\n",
|
||
"我们可以简单地使用矩阵$\\mathbf{A}$的小写字母索引下标$a_{ij}$\n",
|
||
"来引用$[\\mathbf{A}]_{ij}$。\n",
|
||
"为了表示起来简单，只有在必要时才会将逗号插入到单独的索引中，\n",
|
||
"例如$a_{2,3j}$和$[\\mathbf{A}]_{2i-1,3}$。\n",
|
||
"\n",
|
||
"当我们交换矩阵的行和列时，结果称为矩阵的*转置*（transpose）。\n",
|
||
"通常用$\\mathbf{a}^\\top$来表示矩阵的转置，如果$\\mathbf{B}=\\mathbf{A}^\\top$，\n",
|
||
"则对于任意$i$和$j$，都有$b_{ij}=a_{ji}$。\n",
|
||
"因此，在 :eqref:`eq_matrix_def`中的转置是一个形状为$n \\times m$的矩阵：\n",
|
||
"\n",
|
||
"$$\n",
|
||
"\\mathbf{A}^\\top =\n",
|
||
"\\begin{bmatrix}\n",
|
||
" a_{11} & a_{21} & \\dots & a_{m1} \\\\\n",
|
||
" a_{12} & a_{22} & \\dots & a_{m2} \\\\\n",
|
||
" \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
|
||
" a_{1n} & a_{2n} & \\dots & a_{mn}\n",
|
||
"\\end{bmatrix}.\n",
|
||
"$$\n",
|
||
"\n",
|
||
"现在在代码中访问(**矩阵的转置**)。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 7,
|
||
"id": "289523ed",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.833285Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.832377Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.839757Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.838656Z"
|
||
},
|
||
"origin_pos": 32,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[ 0, 4, 8, 12, 16],\n",
|
||
" [ 1, 5, 9, 13, 17],\n",
|
||
" [ 2, 6, 10, 14, 18],\n",
|
||
" [ 3, 7, 11, 15, 19]])"
|
||
]
|
||
},
|
||
"execution_count": 7,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A.T"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "18ce004f",
|
||
"metadata": {
|
||
"origin_pos": 35
|
||
},
|
||
"source": [
|
||
"作为方阵的一种特殊类型，[***对称矩阵*（symmetric matrix）$\\mathbf{A}$等于其转置：$\\mathbf{A} = \\mathbf{A}^\\top$**]。\n",
|
||
"这里定义一个对称矩阵$\\mathbf{B}$：\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 8,
|
||
"id": "f0eb414b",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.845394Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.844475Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.852725Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.851678Z"
|
||
},
|
||
"origin_pos": 37,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[1, 2, 3],\n",
|
||
" [2, 0, 4],\n",
|
||
" [3, 4, 5]])"
|
||
]
|
||
},
|
||
"execution_count": 8,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"B = torch.tensor([[1, 2, 3], [2, 0, 4], [3, 4, 5]])\n",
|
||
"B"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "9d810164",
|
||
"metadata": {
|
||
"origin_pos": 40
|
||
},
|
||
"source": [
|
||
"现在我们将`B`与它的转置进行比较。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 9,
|
||
"id": "44cc700c",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.857930Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.856978Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.864388Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.863329Z"
|
||
},
|
||
"origin_pos": 42,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[True, True, True],\n",
|
||
" [True, True, True],\n",
|
||
" [True, True, True]])"
|
||
]
|
||
},
|
||
"execution_count": 9,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"B == B.T"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "faa5d327",
|
||
"metadata": {
|
||
"origin_pos": 45
|
||
},
|
||
"source": [
|
||
"矩阵是有用的数据结构：它们允许我们组织具有不同模式的数据。\n",
|
||
"例如，我们矩阵中的行可能对应于不同的房屋（数据样本），而列可能对应于不同的属性。\n",
|
||
"曾经使用过电子表格软件或已阅读过 :numref:`sec_pandas`的人，应该对此很熟悉。\n",
|
||
"因此，尽管单个向量的默认方向是列向量，但在表示表格数据集的矩阵中，\n",
|
||
"将每个数据样本作为矩阵中的行向量更为常见。\n",
|
||
"后面的章节将讲到这点，这种约定将支持常见的深度学习实践。\n",
|
||
"例如，沿着张量的最外轴，我们可以访问或遍历小批量的数据样本。\n",
|
||
"\n",
|
||
"\n",
|
||
"## 张量\n",
|
||
"\n",
|
||
"[**就像向量是标量的推广，矩阵是向量的推广一样，我们可以构建具有更多轴的数据结构**]。\n",
|
||
"张量（本小节中的“张量”指代数对象）是描述具有任意数量轴的$n$维数组的通用方法。\n",
|
||
"例如，向量是一阶张量，矩阵是二阶张量。\n",
|
||
"张量用特殊字体的大写字母表示（例如，$\\mathsf{X}$、$\\mathsf{Y}$和$\\mathsf{Z}$），\n",
|
||
"它们的索引机制（例如$x_{ijk}$和$[\\mathsf{X}]_{1,2i-1,3}$）与矩阵类似。\n",
|
||
"\n",
|
||
"当我们开始处理图像时，张量将变得更加重要，图像以$n$维数组形式出现，\n",
|
||
"其中3个轴对应于高度、宽度，以及一个*通道*（channel）轴，\n",
|
||
"用于表示颜色通道（红色、绿色和蓝色）。\n",
|
||
"现在先将高阶张量暂放一边，而是专注学习其基础知识。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 10,
|
||
"id": "8f7227a9",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.869592Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.868624Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.876563Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.875497Z"
|
||
},
|
||
"origin_pos": 47,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[[ 0, 1, 2, 3],\n",
|
||
" [ 4, 5, 6, 7],\n",
|
||
" [ 8, 9, 10, 11]],\n",
|
||
"\n",
|
||
" [[12, 13, 14, 15],\n",
|
||
" [16, 17, 18, 19],\n",
|
||
" [20, 21, 22, 23]]])"
|
||
]
|
||
},
|
||
"execution_count": 10,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"X = torch.arange(24).reshape(2, 3, 4)\n",
|
||
"X"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "e003e028",
|
||
"metadata": {
|
||
"origin_pos": 50
|
||
},
|
||
"source": [
|
||
"## 张量算法的基本性质\n",
|
||
"\n",
|
||
"标量、向量、矩阵和任意数量轴的张量（本小节中的“张量”指代数对象）有一些实用的属性。\n",
|
||
"例如，从按元素操作的定义中可以注意到，任何按元素的一元运算都不会改变其操作数的形状。\n",
|
||
"同样，[**给定具有相同形状的任意两个张量，任何按元素二元运算的结果都将是相同形状的张量**]。\n",
|
||
"例如，将两个相同形状的矩阵相加，会在这两个矩阵上执行元素加法。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 11,
|
||
"id": "d6c89bd2",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.881686Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.880912Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.891206Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.890082Z"
|
||
},
|
||
"origin_pos": 52,
|
||
"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",
|
||
" [16., 17., 18., 19.]]),\n",
|
||
" tensor([[ 0., 2., 4., 6.],\n",
|
||
" [ 8., 10., 12., 14.],\n",
|
||
" [16., 18., 20., 22.],\n",
|
||
" [24., 26., 28., 30.],\n",
|
||
" [32., 34., 36., 38.]]))"
|
||
]
|
||
},
|
||
"execution_count": 11,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A = torch.arange(20, dtype=torch.float32).reshape(5, 4)\n",
|
||
"B = A.clone() # 通过分配新内存，将A的一个副本分配给B\n",
|
||
"A, A + B"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "5f634ff0",
|
||
"metadata": {
|
||
"origin_pos": 55
|
||
},
|
||
"source": [
|
||
"具体而言，[**两个矩阵的按元素乘法称为*Hadamard积*（Hadamard product）（数学符号$\\odot$）**]。\n",
|
||
"对于矩阵$\\mathbf{B} \\in \\mathbb{R}^{m \\times n}$，\n",
|
||
"其中第$i$行和第$j$列的元素是$b_{ij}$。\n",
|
||
"矩阵$\\mathbf{A}$（在 :eqref:`eq_matrix_def`中定义）和$\\mathbf{B}$的Hadamard积为：\n",
|
||
"$$\n",
|
||
"\\mathbf{A} \\odot \\mathbf{B} =\n",
|
||
"\\begin{bmatrix}\n",
|
||
" a_{11} b_{11} & a_{12} b_{12} & \\dots & a_{1n} b_{1n} \\\\\n",
|
||
" a_{21} b_{21} & a_{22} b_{22} & \\dots & a_{2n} b_{2n} \\\\\n",
|
||
" \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
|
||
" a_{m1} b_{m1} & a_{m2} b_{m2} & \\dots & a_{mn} b_{mn}\n",
|
||
"\\end{bmatrix}.\n",
|
||
"$$\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 12,
|
||
"id": "1efe4855",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.896102Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.895401Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.903331Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.902251Z"
|
||
},
|
||
"origin_pos": 57,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[ 0., 1., 4., 9.],\n",
|
||
" [ 16., 25., 36., 49.],\n",
|
||
" [ 64., 81., 100., 121.],\n",
|
||
" [144., 169., 196., 225.],\n",
|
||
" [256., 289., 324., 361.]])"
|
||
]
|
||
},
|
||
"execution_count": 12,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A * B"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "dcd1666f",
|
||
"metadata": {
|
||
"origin_pos": 60
|
||
},
|
||
"source": [
|
||
"将张量乘以或加上一个标量不会改变张量的形状，其中张量的每个元素都将与标量相加或相乘。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 13,
|
||
"id": "587335a3",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.908593Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.907694Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.916299Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.915117Z"
|
||
},
|
||
"origin_pos": 62,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([[[ 2, 3, 4, 5],\n",
|
||
" [ 6, 7, 8, 9],\n",
|
||
" [10, 11, 12, 13]],\n",
|
||
" \n",
|
||
" [[14, 15, 16, 17],\n",
|
||
" [18, 19, 20, 21],\n",
|
||
" [22, 23, 24, 25]]]),\n",
|
||
" torch.Size([2, 3, 4]))"
|
||
]
|
||
},
|
||
"execution_count": 13,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"a = 2\n",
|
||
"X = torch.arange(24).reshape(2, 3, 4)\n",
|
||
"a + X, (a * X).shape"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "30fee7fa",
|
||
"metadata": {
|
||
"origin_pos": 65
|
||
},
|
||
"source": [
|
||
"## 降维\n",
|
||
"\n",
|
||
":label:`subseq_lin-alg-reduction`\n",
|
||
"\n",
|
||
"我们可以对任意张量进行的一个有用的操作是[**计算其元素的和**]。\n",
|
||
"数学表示法使用$\\sum$符号表示求和。\n",
|
||
"为了表示长度为$d$的向量中元素的总和，可以记为$\\sum_{i=1}^dx_i$。\n",
|
||
"在代码中可以调用计算求和的函数：\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 14,
|
||
"id": "32507943",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.921298Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.920499Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.929213Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.928096Z"
|
||
},
|
||
"origin_pos": 67,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([0., 1., 2., 3.]), tensor(6.))"
|
||
]
|
||
},
|
||
"execution_count": 14,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"x = torch.arange(4, dtype=torch.float32)\n",
|
||
"x, x.sum()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "ae2f651b",
|
||
"metadata": {
|
||
"origin_pos": 70
|
||
},
|
||
"source": [
|
||
"我们可以(**表示任意形状张量的元素和**)。\n",
|
||
"例如，矩阵$\\mathbf{A}$中元素的和可以记为$\\sum_{i=1}^{m} \\sum_{j=1}^{n} a_{ij}$。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 15,
|
||
"id": "3e0cd60f",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.934058Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.933342Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.940936Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.939832Z"
|
||
},
|
||
"origin_pos": 72,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(torch.Size([5, 4]), tensor(190.))"
|
||
]
|
||
},
|
||
"execution_count": 15,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A.shape, A.sum()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "d62dab82",
|
||
"metadata": {
|
||
"origin_pos": 75
|
||
},
|
||
"source": [
|
||
"默认情况下，调用求和函数会沿所有的轴降低张量的维度，使它变为一个标量。\n",
|
||
"我们还可以[**指定张量沿哪一个轴来通过求和降低维度**]。\n",
|
||
"以矩阵为例，为了通过求和所有行的元素来降维（轴0），可以在调用函数时指定`axis=0`。\n",
|
||
"由于输入矩阵沿0轴降维以生成输出向量，因此输入轴0的维数在输出形状中消失。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 16,
|
||
"id": "9420cc92",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.946290Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.945345Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.953195Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.952092Z"
|
||
},
|
||
"origin_pos": 77,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([40., 45., 50., 55.]), torch.Size([4]))"
|
||
]
|
||
},
|
||
"execution_count": 16,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A_sum_axis0 = A.sum(axis=0)\n",
|
||
"A_sum_axis0, A_sum_axis0.shape"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "f166972e",
|
||
"metadata": {
|
||
"origin_pos": 80
|
||
},
|
||
"source": [
|
||
"指定`axis=1`将通过汇总所有列的元素降维（轴1）。因此，输入轴1的维数在输出形状中消失。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 17,
|
||
"id": "50e59a41",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.958180Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.957431Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.965338Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.964267Z"
|
||
},
|
||
"origin_pos": 82,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([ 6., 22., 38., 54., 70.]), torch.Size([5]))"
|
||
]
|
||
},
|
||
"execution_count": 17,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A_sum_axis1 = A.sum(axis=1)\n",
|
||
"A_sum_axis1, A_sum_axis1.shape"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "1c6fac47",
|
||
"metadata": {
|
||
"origin_pos": 85
|
||
},
|
||
"source": [
|
||
"沿着行和列对矩阵求和，等价于对矩阵的所有元素进行求和。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 18,
|
||
"id": "e1ba976a",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.970587Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.969706Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.977405Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.976340Z"
|
||
},
|
||
"origin_pos": 87,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor(190.)"
|
||
]
|
||
},
|
||
"execution_count": 18,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A.sum(axis=[0, 1]) # 结果和A.sum()相同"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "38057dab",
|
||
"metadata": {
|
||
"origin_pos": 90
|
||
},
|
||
"source": [
|
||
"[**一个与求和相关的量是*平均值*（mean或average）**]。\n",
|
||
"我们通过将总和除以元素总数来计算平均值。\n",
|
||
"在代码中，我们可以调用函数来计算任意形状张量的平均值。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 19,
|
||
"id": "1d901892",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.982674Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.981742Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:43.990067Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:43.988981Z"
|
||
},
|
||
"origin_pos": 92,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor(9.5000), tensor(9.5000))"
|
||
]
|
||
},
|
||
"execution_count": 19,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A.mean(), A.sum() / A.numel()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "2d311917",
|
||
"metadata": {
|
||
"origin_pos": 95
|
||
},
|
||
"source": [
|
||
"同样，计算平均值的函数也可以沿指定轴降低张量的维度。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 20,
|
||
"id": "65c39834",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:43.995223Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:43.994254Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.003242Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.002162Z"
|
||
},
|
||
"origin_pos": 97,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([ 8., 9., 10., 11.]), tensor([ 8., 9., 10., 11.]))"
|
||
]
|
||
},
|
||
"execution_count": 20,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A.mean(axis=0), A.sum(axis=0) / A.shape[0]"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "7d24d4e3",
|
||
"metadata": {
|
||
"origin_pos": 100
|
||
},
|
||
"source": [
|
||
"### 非降维求和\n",
|
||
"\n",
|
||
":label:`subseq_lin-alg-non-reduction`\n",
|
||
"\n",
|
||
"但是，有时在调用函数来[**计算总和或均值时保持轴数不变**]会很有用。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 21,
|
||
"id": "2cc17274",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.008471Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.007568Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.016007Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.014845Z"
|
||
},
|
||
"origin_pos": 102,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[ 6.],\n",
|
||
" [22.],\n",
|
||
" [38.],\n",
|
||
" [54.],\n",
|
||
" [70.]])"
|
||
]
|
||
},
|
||
"execution_count": 21,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"sum_A = A.sum(axis=1, keepdims=True)\n",
|
||
"sum_A"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "eae08c05",
|
||
"metadata": {
|
||
"origin_pos": 105
|
||
},
|
||
"source": [
|
||
"例如，由于`sum_A`在对每行进行求和后仍保持两个轴，我们可以(**通过广播将`A`除以`sum_A`**)。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 22,
|
||
"id": "63a5b49d",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.020992Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.020591Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.028726Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.027663Z"
|
||
},
|
||
"origin_pos": 107,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[0.0000, 0.1667, 0.3333, 0.5000],\n",
|
||
" [0.1818, 0.2273, 0.2727, 0.3182],\n",
|
||
" [0.2105, 0.2368, 0.2632, 0.2895],\n",
|
||
" [0.2222, 0.2407, 0.2593, 0.2778],\n",
|
||
" [0.2286, 0.2429, 0.2571, 0.2714]])"
|
||
]
|
||
},
|
||
"execution_count": 22,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A / sum_A"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "fcb2a480",
|
||
"metadata": {
|
||
"origin_pos": 110
|
||
},
|
||
"source": [
|
||
"如果我们想沿[**某个轴计算`A`元素的累积总和**]，\n",
|
||
"比如`axis=0`（按行计算），可以调用`cumsum`函数。\n",
|
||
"此函数不会沿任何轴降低输入张量的维度。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 23,
|
||
"id": "27eb9655",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.033849Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.033115Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.041281Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.040150Z"
|
||
},
|
||
"origin_pos": 112,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[ 0., 1., 2., 3.],\n",
|
||
" [ 4., 6., 8., 10.],\n",
|
||
" [12., 15., 18., 21.],\n",
|
||
" [24., 28., 32., 36.],\n",
|
||
" [40., 45., 50., 55.]])"
|
||
]
|
||
},
|
||
"execution_count": 23,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A.cumsum(axis=0)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "f4dec732",
|
||
"metadata": {
|
||
"origin_pos": 115
|
||
},
|
||
"source": [
|
||
"## 点积（Dot Product）\n",
|
||
"\n",
|
||
"我们已经学习了按元素操作、求和及平均值。\n",
|
||
"另一个最基本的操作之一是点积。\n",
|
||
"给定两个向量$\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^d$，\n",
|
||
"它们的*点积*（dot product）$\\mathbf{x}^\\top\\mathbf{y}$\n",
|
||
"（或$\\langle\\mathbf{x},\\mathbf{y}\\rangle$）\n",
|
||
"是相同位置的按元素乘积的和：$\\mathbf{x}^\\top \\mathbf{y} = \\sum_{i=1}^{d} x_i y_i$。\n",
|
||
"\n",
|
||
"[~~点积是相同位置的按元素乘积的和~~]\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 24,
|
||
"id": "7840d740",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.045914Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.045514Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.058183Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.057040Z"
|
||
},
|
||
"origin_pos": 117,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(tensor([0., 1., 2., 3.]), tensor([1., 1., 1., 1.]), tensor(6.))"
|
||
]
|
||
},
|
||
"execution_count": 24,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"y = torch.ones(4, dtype = torch.float32)\n",
|
||
"x, y, torch.dot(x, y)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "18953e09",
|
||
"metadata": {
|
||
"origin_pos": 120
|
||
},
|
||
"source": [
|
||
"注意，(**我们可以通过执行按元素乘法，然后进行求和来表示两个向量的点积**)：\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 25,
|
||
"id": "dadc2a45",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.062812Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.062422Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.070070Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.068907Z"
|
||
},
|
||
"origin_pos": 122,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor(6.)"
|
||
]
|
||
},
|
||
"execution_count": 25,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"torch.sum(x * y)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "92f4b767",
|
||
"metadata": {
|
||
"origin_pos": 125
|
||
},
|
||
"source": [
|
||
"点积在很多场合都很有用。\n",
|
||
"例如，给定一组由向量$\\mathbf{x} \\in \\mathbb{R}^d$表示的值，\n",
|
||
"和一组由$\\mathbf{w} \\in \\mathbb{R}^d$表示的权重。\n",
|
||
"$\\mathbf{x}$中的值根据权重$\\mathbf{w}$的加权和，\n",
|
||
"可以表示为点积$\\mathbf{x}^\\top \\mathbf{w}$。\n",
|
||
"当权重为非负数且和为1（即$\\left(\\sum_{i=1}^{d}{w_i}=1\\right)$）时，\n",
|
||
"点积表示*加权平均*（weighted average）。\n",
|
||
"将两个向量规范化得到单位长度后，点积表示它们夹角的余弦。\n",
|
||
"本节后面的内容将正式介绍*长度*（length）的概念。\n",
|
||
"\n",
|
||
"## 矩阵-向量积\n",
|
||
"\n",
|
||
"现在我们知道如何计算点积，可以开始理解*矩阵-向量积*（matrix-vector product）。\n",
|
||
"回顾分别在 :eqref:`eq_matrix_def`和 :eqref:`eq_vec_def`中定义的矩阵$\\mathbf{A} \\in \\mathbb{R}^{m \\times n}$和向量$\\mathbf{x} \\in \\mathbb{R}^n$。\n",
|
||
"让我们将矩阵$\\mathbf{A}$用它的行向量表示：\n",
|
||
"\n",
|
||
"$$\\mathbf{A}=\n",
|
||
"\\begin{bmatrix}\n",
|
||
"\\mathbf{a}^\\top_{1} \\\\\n",
|
||
"\\mathbf{a}^\\top_{2} \\\\\n",
|
||
"\\vdots \\\\\n",
|
||
"\\mathbf{a}^\\top_m \\\\\n",
|
||
"\\end{bmatrix},$$\n",
|
||
"\n",
|
||
"其中每个$\\mathbf{a}^\\top_{i} \\in \\mathbb{R}^n$都是行向量，表示矩阵的第$i$行。\n",
|
||
"[**矩阵向量积$\\mathbf{A}\\mathbf{x}$是一个长度为$m$的列向量，\n",
|
||
"其第$i$个元素是点积$\\mathbf{a}^\\top_i \\mathbf{x}$**]：\n",
|
||
"\n",
|
||
"$$\n",
|
||
"\\mathbf{A}\\mathbf{x}\n",
|
||
"= \\begin{bmatrix}\n",
|
||
"\\mathbf{a}^\\top_{1} \\\\\n",
|
||
"\\mathbf{a}^\\top_{2} \\\\\n",
|
||
"\\vdots \\\\\n",
|
||
"\\mathbf{a}^\\top_m \\\\\n",
|
||
"\\end{bmatrix}\\mathbf{x}\n",
|
||
"= \\begin{bmatrix}\n",
|
||
" \\mathbf{a}^\\top_{1} \\mathbf{x} \\\\\n",
|
||
" \\mathbf{a}^\\top_{2} \\mathbf{x} \\\\\n",
|
||
"\\vdots\\\\\n",
|
||
" \\mathbf{a}^\\top_{m} \\mathbf{x}\\\\\n",
|
||
"\\end{bmatrix}.\n",
|
||
"$$\n",
|
||
"\n",
|
||
"我们可以把一个矩阵$\\mathbf{A} \\in \\mathbb{R}^{m \\times n}$乘法看作一个从$\\mathbb{R}^{n}$到$\\mathbb{R}^{m}$向量的转换。\n",
|
||
"这些转换是非常有用的，例如可以用方阵的乘法来表示旋转。\n",
|
||
"后续章节将讲到，我们也可以使用矩阵-向量积来描述在给定前一层的值时，\n",
|
||
"求解神经网络每一层所需的复杂计算。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "2bff356a",
|
||
"metadata": {
|
||
"origin_pos": 127,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"source": [
|
||
"在代码中使用张量表示矩阵-向量积，我们使用`mv`函数。\n",
|
||
"当我们为矩阵`A`和向量`x`调用`torch.mv(A, x)`时，会执行矩阵-向量积。\n",
|
||
"注意，`A`的列维数（沿轴1的长度）必须与`x`的维数（其长度）相同。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 26,
|
||
"id": "62c6809c",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.075294Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.074579Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.082607Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.081496Z"
|
||
},
|
||
"origin_pos": 130,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(torch.Size([5, 4]), torch.Size([4]), tensor([ 14., 38., 62., 86., 110.]))"
|
||
]
|
||
},
|
||
"execution_count": 26,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A.shape, x.shape, torch.mv(A, x)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "363d1be1",
|
||
"metadata": {
|
||
"origin_pos": 133
|
||
},
|
||
"source": [
|
||
"## 矩阵-矩阵乘法\n",
|
||
"\n",
|
||
"在掌握点积和矩阵-向量积的知识后，\n",
|
||
"那么**矩阵-矩阵乘法**（matrix-matrix multiplication）应该很简单。\n",
|
||
"\n",
|
||
"假设有两个矩阵$\\mathbf{A} \\in \\mathbb{R}^{n \\times k}$和$\\mathbf{B} \\in \\mathbb{R}^{k \\times m}$：\n",
|
||
"\n",
|
||
"$$\\mathbf{A}=\\begin{bmatrix}\n",
|
||
" a_{11} & a_{12} & \\cdots & a_{1k} \\\\\n",
|
||
" a_{21} & a_{22} & \\cdots & a_{2k} \\\\\n",
|
||
"\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
|
||
" a_{n1} & a_{n2} & \\cdots & a_{nk} \\\\\n",
|
||
"\\end{bmatrix},\\quad\n",
|
||
"\\mathbf{B}=\\begin{bmatrix}\n",
|
||
" b_{11} & b_{12} & \\cdots & b_{1m} \\\\\n",
|
||
" b_{21} & b_{22} & \\cdots & b_{2m} \\\\\n",
|
||
"\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
|
||
" b_{k1} & b_{k2} & \\cdots & b_{km} \\\\\n",
|
||
"\\end{bmatrix}.$$\n",
|
||
"\n",
|
||
"用行向量$\\mathbf{a}^\\top_{i} \\in \\mathbb{R}^k$表示矩阵$\\mathbf{A}$的第$i$行，并让列向量$\\mathbf{b}_{j} \\in \\mathbb{R}^k$作为矩阵$\\mathbf{B}$的第$j$列。要生成矩阵积$\\mathbf{C} = \\mathbf{A}\\mathbf{B}$，最简单的方法是考虑$\\mathbf{A}$的行向量和$\\mathbf{B}$的列向量:\n",
|
||
"\n",
|
||
"$$\\mathbf{A}=\n",
|
||
"\\begin{bmatrix}\n",
|
||
"\\mathbf{a}^\\top_{1} \\\\\n",
|
||
"\\mathbf{a}^\\top_{2} \\\\\n",
|
||
"\\vdots \\\\\n",
|
||
"\\mathbf{a}^\\top_n \\\\\n",
|
||
"\\end{bmatrix},\n",
|
||
"\\quad \\mathbf{B}=\\begin{bmatrix}\n",
|
||
" \\mathbf{b}_{1} & \\mathbf{b}_{2} & \\cdots & \\mathbf{b}_{m} \\\\\n",
|
||
"\\end{bmatrix}.\n",
|
||
"$$\n",
|
||
"当我们简单地将每个元素$c_{ij}$计算为点积$\\mathbf{a}^\\top_i \\mathbf{b}_j$:\n",
|
||
"\n",
|
||
"$$\\mathbf{C} = \\mathbf{AB} = \\begin{bmatrix}\n",
|
||
"\\mathbf{a}^\\top_{1} \\\\\n",
|
||
"\\mathbf{a}^\\top_{2} \\\\\n",
|
||
"\\vdots \\\\\n",
|
||
"\\mathbf{a}^\\top_n \\\\\n",
|
||
"\\end{bmatrix}\n",
|
||
"\\begin{bmatrix}\n",
|
||
" \\mathbf{b}_{1} & \\mathbf{b}_{2} & \\cdots & \\mathbf{b}_{m} \\\\\n",
|
||
"\\end{bmatrix}\n",
|
||
"= \\begin{bmatrix}\n",
|
||
"\\mathbf{a}^\\top_{1} \\mathbf{b}_1 & \\mathbf{a}^\\top_{1}\\mathbf{b}_2& \\cdots & \\mathbf{a}^\\top_{1} \\mathbf{b}_m \\\\\n",
|
||
" \\mathbf{a}^\\top_{2}\\mathbf{b}_1 & \\mathbf{a}^\\top_{2} \\mathbf{b}_2 & \\cdots & \\mathbf{a}^\\top_{2} \\mathbf{b}_m \\\\\n",
|
||
" \\vdots & \\vdots & \\ddots &\\vdots\\\\\n",
|
||
"\\mathbf{a}^\\top_{n} \\mathbf{b}_1 & \\mathbf{a}^\\top_{n}\\mathbf{b}_2& \\cdots& \\mathbf{a}^\\top_{n} \\mathbf{b}_m\n",
|
||
"\\end{bmatrix}.\n",
|
||
"$$\n",
|
||
"\n",
|
||
"[**我们可以将矩阵-矩阵乘法$\\mathbf{AB}$看作简单地执行$m$次矩阵-向量积，并将结果拼接在一起，形成一个$n \\times m$矩阵**]。\n",
|
||
"在下面的代码中，我们在`A`和`B`上执行矩阵乘法。\n",
|
||
"这里的`A`是一个5行4列的矩阵，`B`是一个4行3列的矩阵。\n",
|
||
"两者相乘后，我们得到了一个5行3列的矩阵。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 27,
|
||
"id": "1e3efc16",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.087651Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.086870Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.095375Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.094329Z"
|
||
},
|
||
"origin_pos": 135,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor([[ 6., 6., 6.],\n",
|
||
" [22., 22., 22.],\n",
|
||
" [38., 38., 38.],\n",
|
||
" [54., 54., 54.],\n",
|
||
" [70., 70., 70.]])"
|
||
]
|
||
},
|
||
"execution_count": 27,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"B = torch.ones(4, 3)\n",
|
||
"torch.mm(A, B)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "2fab0ddd",
|
||
"metadata": {
|
||
"origin_pos": 138
|
||
},
|
||
"source": [
|
||
"矩阵-矩阵乘法可以简单地称为**矩阵乘法**，不应与\"Hadamard积\"混淆。\n",
|
||
"\n",
|
||
"## 范数\n",
|
||
":label:`subsec_lin-algebra-norms`\n",
|
||
"\n",
|
||
"线性代数中最有用的一些运算符是*范数*（norm）。\n",
|
||
"非正式地说，向量的*范数*是表示一个向量有多大。\n",
|
||
"这里考虑的*大小*（size）概念不涉及维度，而是分量的大小。\n",
|
||
"\n",
|
||
"在线性代数中，向量范数是将向量映射到标量的函数$f$。\n",
|
||
"给定任意向量$\\mathbf{x}$，向量范数要满足一些属性。\n",
|
||
"第一个性质是：如果我们按常数因子$\\alpha$缩放向量的所有元素，\n",
|
||
"其范数也会按相同常数因子的*绝对值*缩放：\n",
|
||
"\n",
|
||
"$$f(\\alpha \\mathbf{x}) = |\\alpha| f(\\mathbf{x}).$$\n",
|
||
"\n",
|
||
"第二个性质是熟悉的三角不等式:\n",
|
||
"\n",
|
||
"$$f(\\mathbf{x} + \\mathbf{y}) \\leq f(\\mathbf{x}) + f(\\mathbf{y}).$$\n",
|
||
"\n",
|
||
"第三个性质简单地说范数必须是非负的:\n",
|
||
"\n",
|
||
"$$f(\\mathbf{x}) \\geq 0.$$\n",
|
||
"\n",
|
||
"这是有道理的。因为在大多数情况下，任何东西的最小的*大小*是0。\n",
|
||
"最后一个性质要求范数最小为0，当且仅当向量全由0组成。\n",
|
||
"\n",
|
||
"$$\\forall i, [\\mathbf{x}]_i = 0 \\Leftrightarrow f(\\mathbf{x})=0.$$\n",
|
||
"\n",
|
||
"范数听起来很像距离的度量。\n",
|
||
"欧几里得距离和毕达哥拉斯定理中的非负性概念和三角不等式可能会给出一些启发。\n",
|
||
"事实上，欧几里得距离是一个$L_2$范数：\n",
|
||
"假设$n$维向量$\\mathbf{x}$中的元素是$x_1,\\ldots,x_n$，其[**$L_2$*范数*是向量元素平方和的平方根：**]\n",
|
||
"\n",
|
||
"(**$$\\|\\mathbf{x}\\|_2 = \\sqrt{\\sum_{i=1}^n x_i^2},$$**)\n",
|
||
"\n",
|
||
"其中，在$L_2$范数中常常省略下标$2$，也就是说$\\|\\mathbf{x}\\|$等同于$\\|\\mathbf{x}\\|_2$。\n",
|
||
"在代码中，我们可以按如下方式计算向量的$L_2$范数。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 28,
|
||
"id": "f829c100",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.100377Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.099628Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.107745Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.106642Z"
|
||
},
|
||
"origin_pos": 140,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor(5.)"
|
||
]
|
||
},
|
||
"execution_count": 28,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"u = torch.tensor([3.0, -4.0])\n",
|
||
"torch.norm(u)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "c9608c4c",
|
||
"metadata": {
|
||
"origin_pos": 143
|
||
},
|
||
"source": [
|
||
"深度学习中更经常地使用$L_2$范数的平方，也会经常遇到[**$L_1$范数，它表示为向量元素的绝对值之和：**]\n",
|
||
"\n",
|
||
"(**$$\\|\\mathbf{x}\\|_1 = \\sum_{i=1}^n \\left|x_i \\right|.$$**)\n",
|
||
"\n",
|
||
"与$L_2$范数相比，$L_1$范数受异常值的影响较小。\n",
|
||
"为了计算$L_1$范数，我们将绝对值函数和按元素求和组合起来。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 29,
|
||
"id": "01356584",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.143775Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.142900Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.151418Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.150335Z"
|
||
},
|
||
"origin_pos": 145,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor(7.)"
|
||
]
|
||
},
|
||
"execution_count": 29,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"torch.abs(u).sum()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "a9454ae0",
|
||
"metadata": {
|
||
"origin_pos": 148
|
||
},
|
||
"source": [
|
||
"$L_2$范数和$L_1$范数都是更一般的$L_p$范数的特例：\n",
|
||
"\n",
|
||
"$$\\|\\mathbf{x}\\|_p = \\left(\\sum_{i=1}^n \\left|x_i \\right|^p \\right)^{1/p}.$$\n",
|
||
"\n",
|
||
"类似于向量的$L_2$范数，[**矩阵**]$\\mathbf{X} \\in \\mathbb{R}^{m \\times n}$(**的*Frobenius范数*（Frobenius norm）是矩阵元素平方和的平方根：**)\n",
|
||
"\n",
|
||
"(**$$\\|\\mathbf{X}\\|_F = \\sqrt{\\sum_{i=1}^m \\sum_{j=1}^n x_{ij}^2}.$$**)\n",
|
||
"\n",
|
||
"Frobenius范数满足向量范数的所有性质，它就像是矩阵形向量的$L_2$范数。\n",
|
||
"调用以下函数将计算矩阵的Frobenius范数。\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 30,
|
||
"id": "0a8792ee",
|
||
"metadata": {
|
||
"execution": {
|
||
"iopub.execute_input": "2023-08-18T07:01:44.156452Z",
|
||
"iopub.status.busy": "2023-08-18T07:01:44.155694Z",
|
||
"iopub.status.idle": "2023-08-18T07:01:44.163608Z",
|
||
"shell.execute_reply": "2023-08-18T07:01:44.162540Z"
|
||
},
|
||
"origin_pos": 150,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"tensor(6.)"
|
||
]
|
||
},
|
||
"execution_count": 30,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"torch.norm(torch.ones((4, 9)))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "4b7470df",
|
||
"metadata": {
|
||
"origin_pos": 153
|
||
},
|
||
"source": [
|
||
"### 范数和目标\n",
|
||
"\n",
|
||
":label:`subsec_norms_and_objectives`\n",
|
||
"\n",
|
||
"在深度学习中，我们经常试图解决优化问题：\n",
|
||
"*最大化*分配给观测数据的概率;\n",
|
||
"*最小化*预测和真实观测之间的距离。\n",
|
||
"用向量表示物品（如单词、产品或新闻文章），以便最小化相似项目之间的距离，最大化不同项目之间的距离。\n",
|
||
"目标，或许是深度学习算法最重要的组成部分（除了数据），通常被表达为范数。\n",
|
||
"\n",
|
||
"## 关于线性代数的更多信息\n",
|
||
"\n",
|
||
"仅用一节，我们就教会了阅读本书所需的、用以理解现代深度学习的线性代数。\n",
|
||
"线性代数还有很多，其中很多数学对于机器学习非常有用。\n",
|
||
"例如，矩阵可以分解为因子，这些分解可以显示真实世界数据集中的低维结构。\n",
|
||
"机器学习的整个子领域都侧重于使用矩阵分解及其向高阶张量的泛化，来发现数据集中的结构并解决预测问题。\n",
|
||
"当开始动手尝试并在真实数据集上应用了有效的机器学习模型，你会更倾向于学习更多数学。\n",
|
||
"因此，这一节到此结束，本书将在后面介绍更多数学知识。\n",
|
||
"\n",
|
||
"如果渴望了解有关线性代数的更多信息，可以参考[线性代数运算的在线附录](https://d2l.ai/chapter_appendix-mathematics-for-deep-learning/geometry-linear-algebraic-ops.html)或其他优秀资源 :cite:`Strang.1993,Kolter.2008,Petersen.Pedersen.ea.2008`。\n",
|
||
"\n",
|
||
"## 小结\n",
|
||
"\n",
|
||
"* 标量、向量、矩阵和张量是线性代数中的基本数学对象。\n",
|
||
"* 向量泛化自标量，矩阵泛化自向量。\n",
|
||
"* 标量、向量、矩阵和张量分别具有零、一、二和任意数量的轴。\n",
|
||
"* 一个张量可以通过`sum`和`mean`沿指定的轴降低维度。\n",
|
||
"* 两个矩阵的按元素乘法被称为他们的Hadamard积。它与矩阵乘法不同。\n",
|
||
"* 在深度学习中，我们经常使用范数，如$L_1$范数、$L_2$范数和Frobenius范数。\n",
|
||
"* 我们可以对标量、向量、矩阵和张量执行各种操作。\n",
|
||
"\n",
|
||
"## 练习\n",
|
||
"\n",
|
||
"1. 证明一个矩阵$\\mathbf{A}$的转置的转置是$\\mathbf{A}$，即$(\\mathbf{A}^\\top)^\\top = \\mathbf{A}$。\n",
|
||
"1. 给出两个矩阵$\\mathbf{A}$和$\\mathbf{B}$，证明“它们转置的和”等于“它们和的转置”，即$\\mathbf{A}^\\top + \\mathbf{B}^\\top = (\\mathbf{A} + \\mathbf{B})^\\top$。\n",
|
||
"1. 给定任意方阵$\\mathbf{A}$，$\\mathbf{A} + \\mathbf{A}^\\top$总是对称的吗?为什么?\n",
|
||
"1. 本节中定义了形状$(2,3,4)$的张量`X`。`len(X)`的输出结果是什么？\n",
|
||
"1. 对于任意形状的张量`X`,`len(X)`是否总是对应于`X`特定轴的长度?这个轴是什么?\n",
|
||
"1. 运行`A/A.sum(axis=1)`，看看会发生什么。请分析一下原因？\n",
|
||
"1. 考虑一个具有形状$(2,3,4)$的张量，在轴0、1、2上的求和输出是什么形状?\n",
|
||
"1. 为`linalg.norm`函数提供3个或更多轴的张量，并观察其输出。对于任意形状的张量这个函数计算得到什么?\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"id": "1ca6f271",
|
||
"metadata": {
|
||
"origin_pos": 155,
|
||
"tab": [
|
||
"pytorch"
|
||
]
|
||
},
|
||
"source": [
|
||
"[Discussions](https://discuss.d2l.ai/t/1751)\n"
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"language_info": {
|
||
"name": "python"
|
||
},
|
||
"required_libs": []
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 5
|
||
} |