{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "title: \"期末实验 ipynb 文件\"\n",
    "author: pxz\n",
    "date: 2024-12-26\n",
    "format:\n",
    "  html:\n",
    "    toc: true\n",
    "    html-math-method: katex\n",
    "    css: styles.css\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 准备\n",
    "## debug notes\n",
    "`torch` 当中将向量、矩阵、张量统称为张量，所以下文也不再特意区分；\n",
    "\n",
    "`xxx.data` 访问张量中的数据。如果 `xxx` 是一个 Variable，`.data` 会返回其底层的 Tensor  \n",
    "但是，Variable 是在 PyTorch 0.4 之前版本中使用的旧术语，现在被 Tensor 替代  \n",
    "对于 PyTorch 0.4 及以后的版本，`.data` 已经被弃用  \n",
    "如果有这种需求，可以改用 `.detach()` 方法  \n",
    "但是 `.detach()` 有个副作用，就是会清空这个张量已经计算好的梯度  \n",
    "解决方式是 `x.clone().detach()` 用新建副本绕开原张量"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 环境\n",
    "随机种子选择：34 是我读小学期间的班内学号  \n",
    "保持画图内嵌显示  \n",
    "选择支持中文的字体  \n",
    "解决负号'-'显示为方块的问题"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.5.1+cpu\n"
     ]
    }
   ],
   "source": [
    "import torch\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "torch.manual_seed(34)\n",
    "print(torch.__version__)\n",
    "%matplotlib inline\n",
    "plt.rcParams['font.sans-serif'] = ['Microsoft YaHei']\n",
    "plt.rcParams['axes.unicode_minus'] = False\n",
    "plt.rcParams['figure.figsize'] = (12, 3)\n",
    "plt.rcParams['figure.dpi'] = 300"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 数据\n",
    "\n",
    "$n=1024 \\quad m=512 \\quad \\mu = 0.01 \\quad A \\in \\mathbb{R}^{m \\times n} \\quad b \\in \\mathbb{R}^m$\n",
    "\n",
    "```python\n",
    "A = torch.randn(m,n)  随机生成 m 行 n 列的矩阵\n",
    "u = torch.randn(n)*(torch.rand(n)<0.1)\n",
    "b = A @ u\n",
    "```\n",
    "`rand` 是均匀分布，`randn` 是正态分布  \n",
    "`(torch.rand(n)<0.1)` 作为乘子意思是随机保留 10% 的数  \n",
    "前后乘起来得到 `u` 就是正态分布随机保留 10%"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 1024; m = 512; mu = 0.01\n",
    "A = torch.randn(m,n)\n",
    "u = torch.randn(n)*(torch.rand(n)<0.1)\n",
    "b = A @ u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 规定超参\n",
    "### 回溯线搜索\n",
    "$\\alpha$ 是我们可以接受函数每次下降至少要陡到梯度的多少倍（介于 $0$ 到 $1$ 之间）\n",
    "\n",
    "如果步长导致下降不够陡，不可接受，缩减步长，新步长和旧步长之比记作 $\\beta$\n",
    "\n",
    "针对标准正态分布生成的数据，初始步长 $t$ 设为 $1$ 在数量级上是合理的"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "alpha = 0.05; beta = 0.7; t=1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 停机准则\n",
    "`max_iter` 是最大迭代次数，达到该次数则无条件结束迭代\n",
    "\n",
    "$\\varepsilon$ 是用新旧点二范数之差来衡量是否已足够接近收敛的指标，低过此阈值则停机"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "max_iter=800; epsilon=1e-6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ADMM 参数\n",
    "针对标准正态分布生成的数据，罚因子 $\\rho$ 设为 $1$ 在数量级上是合理的\n",
    "\n",
    "Lagrange 乘子更新的步长 $\\tau$ 满足 $0<\\tau<\\dfrac{1+\\sqrt{5}}{2}$ 即可收敛，为方便运算和表示，取此范围内唯一整数 $1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "rho=1; tau=1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "{{< pagebreak >}}\n",
    "\n",
    "## 规定函数\n",
    "\n",
    "### 特殊形式软阈值算子\n",
    "\n",
    "**具有步长和一次惩罚项实际意义**\n",
    "\n",
    "根据 **Handout 4**，既然 $\\mu>0$ 且 $h(x)=\\mu \\|x\\|_1$ 那么\n",
    "$$\n",
    "\\operatorname{prox}_{h,t}(x)=S_\\mu (x)\n",
    "$$\n",
    "**但**这只是步长 $t=1$ **不变**时的情况  \n",
    "在步长动态变化（例如精确线搜索和回溯线搜索）时，$h(x)=\\|\\mu x\\|_1$  \n",
    "从 <u>中文辅助教材（文再文等）例 8.1</u> 进一步推论有：\n",
    "\n",
    "邻近算子 $u=\\operatorname{prox}_{t,h}(x)$ 的最优性条件为\n",
    "$$\n",
    "\\begin{aligned}\n",
    "x-u &\\in t \\partial h(u)\n",
    "\\\\\n",
    "\\text{ 即 } x-u &\\in t \\partial\\|\\mu x\\|_1\n",
    "\\\\\n",
    "\\text{ 即 } x-u &\\in t \\mu \\partial\\| x\\|_1\n",
    "\\begin{cases}\n",
    "\\{t\\} &u>0\n",
    "\\\\\n",
    "[-t,t] \\quad &u=0\n",
    "\\\\\n",
    "\\{-t\\} &u<0\n",
    "\\end{cases}\n",
    "\\end{aligned}\n",
    "$$\n",
    "这里用 $u$ 是为了和书上的 notation 一致，和实验题目中的 `u` 没有关系\n",
    "$$\n",
    "\\text{所以 }\n",
    "[S_{t\\mu} (x)]_i=\n",
    "\\begin{cases}\n",
    "x_i-t\\mu \\quad &\\text{if }\\quad x_i>t\\mu\n",
    "\\\\\n",
    "0 \\quad &\\text{if }\\quad -t\\mu \\leq x_i \\leq t\\mu\n",
    "\\\\\n",
    "x_i+t\\mu \\quad &\\text{if }\\quad x_i<-t\\mu\n",
    "\\end{cases}\n",
    "$$\n",
    "$$\n",
    "G_t(x)\n",
    "=\\dfrac{x-\\operatorname{prox}_{h,t}(x-t\\nabla g(x))}{t}\n",
    "=\\dfrac{x-S_{t\\mu} [x-t\\nabla g(x)]_i^{\\text{用分量表示向量}}}{t}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "def S(x,t):\n",
    "    return torch.sign(x) * torch.clamp(torch.abs(x) - t*mu, min=0)\n",
    "def G(x,t):\n",
    "    return ( x-S(x-t*x.grad,t) )/t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这个通过符号函数和设置下限来实现软阈值逻辑的想法**可从**<u>中文辅助教材（文再文等）例 8.1</u> **找到**\n",
    "\n",
    "`torch.sign(z)` 返回输入张量 $z$ 中每个元素的符号\n",
    "\n",
    "如果 $z_i$ 是正数，返回 $1$；如果是负数，返回 $-1$；如果是 $0$，则返回 $0$\n",
    "\n",
    "`torch.abs(z)` 返回输入张量 $z$ 中每个元素的绝对值，这样用 `torch.abs(z) - mu` 就可以保证得到的差值肯定是个正的数值，差值是 $z_i$ 和 $\\mu_i$ 的绝对值之差的绝对值\n",
    "\n",
    "`torch.clamp` 函数将输入张量中的元素限制在指定的范围内。在这里，它将所有小于 $0$ 的值设置为 $0$，所以它是以 $0$ 为下限的非负值\n",
    "\n",
    "再乘以符号函数，实现了 $x_i$ 正归正，负归负"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 一般形式软阈值算子"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于 $S_\\lambda (\\beta)=\\operatorname{argmin}_z \\dfrac{1}{2} \\|\\beta-z\\|^2_2 +\\lambda \\|z\\|_1$ 有\n",
    "\n",
    "$$\n",
    "[S_\\lambda (\\beta)]_i=\n",
    "\\begin{cases}\n",
    "\\beta_i -\\lambda    &\\text{if} \\quad \\beta_i > \\lambda\n",
    "\\\\\n",
    "0               &\\text{if} \\quad -\\lambda \\leq \\beta_i \\leq \\lambda\n",
    "\\\\\n",
    "\\beta_i +\\lambda    &\\text{if} \\quad \\beta_i < -\\lambda\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "以下代码解释参考上文《特殊形式软阈值算子》"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def SS(beta,lmbd):\n",
    "    return torch.sign(beta) * torch.clamp(torch.abs(beta) - lmbd, min=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 目标函数\n",
    "\n",
    "$$\n",
    "\\min_x\\mu\\|x\\|_1+\\|Ax-b\\|_2^2\n",
    "$$\n",
    "\n",
    "`x.requires_grad_(True)` 以免求解时遭遇各种各样意想不到的「无梯度」报错"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "def g(x):  # 可微的项\n",
    "    x.requires_grad_(True)\n",
    "    return torch.norm(A@x-b,p=2)**2\n",
    "def h(x):  # 不可微的项\n",
    "    return mu*torch.norm(x, p=1)\n",
    "def f(x):  # 完整的损失函数\n",
    "    return g(x)+h(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 余弦相似度"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cosine_similarity(a, b):\n",
    "    dot_product = np.dot(a, b)\n",
    "    norm_a = np.linalg.norm(a)\n",
    "    norm_b = np.linalg.norm(b)\n",
    "    return dot_product / (norm_a * norm_b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 画图模块"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def draw(flist):\n",
    "     # 将列表中的每个张量转换为NumPy数组\n",
    "    flist_numpy = np.array([t.detach().numpy() for t in flist])\n",
    "     # 对数组中的每个元素取自然对数\n",
    "    log_flist = np.log(flist_numpy)\n",
    "    plt.figure()\n",
    "    plt.plot(log_flist)\n",
    "    plt.xlabel('迭代次数')\n",
    "    plt.ylabel('目标函数值的自然对数')\n",
    "    plt.title('Log Scale 上的收敛速度')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 求解模块\n",
    "## 近端梯度下降\n",
    "也即 **ISTA**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "def PGD(x,max_iter=max_iter,alpha=alpha,beta=beta,eps=epsilon,t=t):\n",
    "    f_all=[]\n",
    "    x_all=[]\n",
    "    x_all.append( x.clone().detach().numpy() )\n",
    "    for i in range( int(max_iter) ):\n",
    "        gg=g(x); ff=f(x)\n",
    "        f_all.append(ff)\n",
    "        gg.backward()\n",
    "\n",
    "        while True:\n",
    "            with torch.no_grad():\n",
    "             # 这是个上下文管理器，表示在这个缩进块内不需要计算梯度\n",
    "             # 更新参数不需要梯度\n",
    "                x_new = x - t*G(x,t) #S(x,t)\n",
    "            if g(x_new) <= (\n",
    "                    gg - t * torch.sum( x.grad*G(x,t) ) + alpha*t*(\n",
    "                    torch.norm( G(x,t), p=2, dim=0 )\n",
    "                )**2    ):\n",
    "                 # 一旦满足回溯线搜索想要的条件，就接受这个新的参数作为更新，并退出\n",
    "                x_old = x.clone().detach()\n",
    "                x = x_new\n",
    "                break\n",
    "             # 如果不满足（免于退出），就缩减步长，然后又看需不需要再缩减\n",
    "            t=t*beta\n",
    "\n",
    "        x_all.append(x.clone().detach().numpy())\n",
    "        \n",
    "        iter_norm = torch.abs(torch.norm(x) - torch.norm(x_old))\n",
    "        if iter_norm < eps:\n",
    "            return x, f_all, x_all\n",
    "\n",
    "    return x, f_all, x_all"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Nesterov 加速\n",
    "也即 **FISTA**\n",
    "\n",
    "将书上的 notation 改成本题使用的，中文教材 条件 $(8.2.2)$ 可写成  \n",
    "（同时也是参考课件《06-近端梯度下降与加速》第 26 页的写法）\n",
    "$$\n",
    "g(x^+) \\geq g(v) + \\nabla g(v)^T (x^+ -v) + \\dfrac{1}{2t} \\|x^+ -v\\|^2_2\n",
    "$$\n",
    "若转换成代码，其中\n",
    "$x^+$ 写作 `x_new`，$v$ 写作 `x_v`，\n",
    "即转换为\n",
    "\n",
    "```python\n",
    "g(x_new) <= ( gg +  torch.sum( x_v.grad * (x_new - x_v) )\n",
    "            + ( 1/(2*t) ) * ( torch.norm( x_new - x_v,ord=2,dim=0 ) )**2\n",
    "            )\n",
    "```\n",
    "\n",
    "相应地，$x^{(x-1)}$ 写作 `x_old`，\n",
    "\n",
    "$$\n",
    "v=x^{(x-1)} + \\dfrac{k-2}{k+1}\n",
    "\\left(x^{(k-1)} - x^{(k-2)}\\right)\n",
    "$$\n",
    "\n",
    "$$\n",
    "x^{(k)} = \\operatorname{prox}_{t_k}\n",
    "\\left(\n",
    "v - t_k \\nabla g(v)\n",
    "\\right)\n",
    "$$\n",
    "\n",
    "写作\n",
    "\n",
    "```python\n",
    "x_v = ( x + (i-2)/(i+1)*(x-x_old) )\n",
    "x_new = S(x_v-t*x_v.grad,t)\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "def FISTA(x,max_iter=max_iter,beta=beta,eps=epsilon,t=t):\n",
    "    f_all=[]\n",
    "    x_all=[]\n",
    "    x_all.append( x.clone().detach().numpy() )\n",
    "    x_old = x.clone().detach() \n",
    "    x_old.requires_grad = True; x_old.requires_grad_(True)\n",
    "    for i in range(1,int(max_iter)+1):  # 不能从 0 开始\n",
    "        #  FISTA 两阶段法的第一阶段，沿前两步方向找一个新的点\n",
    "        x_v = ( x + (i-2)/(i+1)*(x-x_old) )\n",
    "        x_v = x_v.clone().detach().requires_grad_(True)\n",
    "\n",
    "        gg=g(x_v); ff=f(x)\n",
    "        f_all.append(ff)\n",
    "        gg.backward()\n",
    "        x_old.requires_grad = True; x_old.requires_grad_(True)\n",
    "        \n",
    "        while True:\n",
    "            with torch.no_grad():\n",
    "             # 这是个上下文管理器，表示在这个缩进块内不需要计算梯度\n",
    "             # 更新参数不需要梯度\n",
    "                x_new = S(x_v-t*x_v.grad,t)\n",
    "            if g(x_new) <= (\n",
    "                    gg + torch.sum( x_v.grad * (x_new - x_v) )\n",
    "                    + ( 1/(2*t) ) * ( torch.norm( x_new - x_v,p=2,dim=0 ) )**2\n",
    "                ):\n",
    "                #  一旦满足回溯线搜索想要的条件，就接受这个新的参数作为更新，并退出\n",
    "                x_old = x.clone().detach()\n",
    "                x = x_new\n",
    "                break\n",
    "             # 如果不满足（免于退出），就缩减步长，然后又看需不需要再缩减\n",
    "            t=t*beta\n",
    "\n",
    "        x_all.append(x.clone().detach().numpy())\n",
    "\n",
    "        iter_norm = torch.abs(torch.norm(x) - torch.norm(x_old))\n",
    "        if iter_norm < eps:\n",
    "            return x, f_all, x_all\n",
    "\n",
    "    return x, f_all, x_all"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "{{< pagebreak >}}\n",
    "\n",
    "## 交替方向乘子法\n",
    "从 Stephen P. Boyd 等人的 Slides（September 2010）第 17 页得知，在 ADMM 算法迭代中，\n",
    "\n",
    "对于 $\\dfrac{1}{2} \\|Ax-b\\|_2^2 + \\lambda \\|x\\|_1$ 有\n",
    "$$\n",
    "\\begin{aligned}\n",
    "x^{(k+1)}&:=(A^TA+\\rho I)^{-1}(A^Tb+\\rho z^{(k)}-y^{(k)})\n",
    "\\\\\n",
    "z^{(k+1)}&:=S_{\\lambda/\\rho}(x^{(k+1)}+y^{(k)}/\\rho)\n",
    "\\\\\n",
    "y^{(k+1)}&:=y^{(k)}+\\rho(x^{k+1}-z^{(k+1)})\n",
    "\\end{aligned}\n",
    "$$\n",
    "**但是**，这里的二次可微项系数是 $\\frac{1}{2}$，我们求解问题的二次可微项系数是 $1$，不可直接照搬\n",
    "\n",
    "以下从原问题出发重新推导所求问题\n",
    "\n",
    "记 $g(x)=\\|Ax-b\\|^2_2 \\quad h(x)=\\mu \\|x\\|_1 \\quad f(x)=g(x)+h(x)$\n",
    "\n",
    "则原问题写作 $\\enspace \\min_x f(x)$\n",
    "\n",
    "现改写为 $\\enspace \\min_{x,z} g(x)+h(z) \\quad \\text{s. t.} \\enspace x-z=0$\n",
    "\n",
    "若 $x-z=0$ 化为 $A_1 x + A_2 z = b$ 的标准形式，则\n",
    "$\\begin{cases}\n",
    "A_1&=1\n",
    "\\\\\n",
    "A_2&=-1\n",
    "\\\\\n",
    "b&=0\n",
    "\\end{cases}$\n",
    "\n",
    "记 Lagrange 乘子为 $y \\quad$ 罚因子为 $\\rho \\quad$ 固定步长为 $\\tau$\n",
    "\n",
    "对改写形式构造增广 Lagrange 函数\n",
    "\n",
    "$$\n",
    "L_\\rho = g(x)+h(z)+ y^T (x-z) + \\frac{\\rho}{2} \\|x-z\\|^2_2\n",
    "$$\n",
    "\n",
    "对于每次迭代\n",
    "\n",
    "第一步\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "x^{(k+1)}&=\\operatorname{argmin}_x L_\\rho (x,z^{(k)},y^{(k)})\n",
    "\\\\\n",
    "&=\\operatorname{argmin}_x \\left\\{\n",
    "    \\|Ax-b\\|^2_2 +\\mu \\|z\\|_1 +(y^{(k)})^T (x-z^{(k)}) + \\frac{\\rho}{2} \\|x-z^{(k)}\\|^2_2\n",
    "    \\right\\}\n",
    "\\\\\n",
    "\\Rightarrow \\quad& \\frac{\\partial L}{\\partial x} = 2 A^T (Ax-b) +y^{(k)} +\\rho(x-z^{(k)})\n",
    "\\\\\n",
    "\\text{令 } \\frac{\\partial L}{\\partial x} &=0 \\text{ 则 }\n",
    "2 A^T Ax - 2A^T b +y^{(k)} + \\rho x - \\rho z^{(k)} =0\n",
    "\\\\\n",
    "\\Rightarrow \\quad& (2 A^T A + \\rho I)x - 2 A^T b + y^{(k)} - \\rho z^{(k)} =0\n",
    "\\\\\n",
    "\\Rightarrow \\quad& (2 A^T A + \\rho I)x = 2 A^T b + \\rho z^{(k)} - y^{(k)}\n",
    "\\\\\n",
    "\\Rightarrow \\quad& x^{(k+1)}= (2 A^T A + \\rho I)^{-1} (2 A^T b + \\rho z^{(k)} - y^{(k)})\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "{{< pagebreak >}}\n",
    "\n",
    "第二步\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "z^{(k+1)}&=\\operatorname{argmin}_z L_\\rho \\left(  x^{(k+1)},z,y^{(k)}  \\right)\n",
    "\\\\\n",
    "&=\\operatorname{argmin}_z\\left\\{\n",
    "    \\|Ax^{(k+1)} -b\\|^2_2 +\\mu \\|z\\|_1 \n",
    "    + (y^{(k)})^T (x^{(k+1)}-z)\n",
    "    + \\frac{\\rho}{2} \\|x^{(k+1)}-z\\|^2_2\n",
    "    \\right\\}\n",
    "\\\\\n",
    "&=\\operatorname{argmin}_z\\left\\{\n",
    "    \\mu \\|z\\|_1\n",
    "    + (y^{(k)})^T (x^{(k+1)}-z)\n",
    "    + \\frac{\\rho}{2} \\|x^{(k+1)}-z\\|^2_2\n",
    "    \\right\\}\n",
    "\\\\ \\quad \\\\\n",
    "&\\text{对要找最值点的目标乘以常数} \\frac{2}{\\rho}\n",
    "\\\\ \\quad \\\\\n",
    "&=\\operatorname{argmin}_z\\left\\{\n",
    "    \\frac{2 \\mu}{\\rho} \\|z\\|_1\n",
    "    + \\frac{2}{\\rho} (y^{(k)})^T (x^{(k+1)}-z)\n",
    "    + \\|x^{(k+1)}-z\\|^2_2\n",
    "    \\right\\}\n",
    "\\\\ \\quad \\\\\n",
    "&=\\operatorname{argmin}_z\\left\\{\n",
    "    \\frac{2 \\mu}{\\rho} \\|z\\|_1\n",
    "    + 2 \\left(\\frac{y^{(k)}}{\\rho}\\right)^T (x^{(k+1)}-z)\n",
    "    + \\|x^{(k+1)}-z\\|^2_2\n",
    "    \\right\\}\n",
    "\\\\ \\quad \\\\\n",
    "&\\text{加上与 } z \\text{ 无关的项 } \\left\\|\\frac{y^{(k)}}{\\rho}\\right\\|^2_2\n",
    "\\\\ \\quad \\\\\n",
    "&=\\operatorname{argmin}_z\\left\\{\n",
    "    \\frac{2 \\mu}{\\rho} \\|z\\|_1\n",
    "    + \\left\\|\\frac{y^{(k)}}{\\rho}\\right\\|^2_2\n",
    "    + 2 \\left(\\frac{y^{(k)}}{\\rho}\\right)^T (x^{(k+1)}-z)\n",
    "    + \\|x^{(k+1)}-z\\|^2_2\n",
    "    \\right\\}\n",
    "\\\\ \\quad \\\\\n",
    "&=\\operatorname{argmin}_z\\left\\{\n",
    "    \\frac{2 \\mu}{\\rho} \\|z\\|_1\n",
    "    + \\left\\|\n",
    "        \\frac{y^{(k)}}{\\rho}\n",
    "        + x^{(k+1)}-z\n",
    "    \\right\\|^2_2\n",
    "    \\right\\}\n",
    "\\\\ \\quad \\\\\n",
    "&=\\operatorname{argmin}_z\\left\\{\n",
    "    \\frac{1}{2} \\left\\|\n",
    "        z-\\left(\n",
    "            x^{(k+1)} + \\frac{y^{(k)}}{\\rho}\n",
    "        \\right)\n",
    "    \\right\\|^2_2\n",
    "    +\\frac{\\mu}{\\rho} \\|z\\|_1\n",
    "    \\right\\}\n",
    "\\\\ \\quad \\\\\n",
    "&=S_{\\left(\\frac{\\mu}{\\rho}\\right)} \\left(\n",
    "    x^{(k+1)}+\\frac{y^{(k)}}{\\rho}\n",
    "\\right)\n",
    "\\qquad \\text{其中 } S_{(\\cdot)} (\\cdot) \\text{ 表示软阈值算子}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "第三步\n",
    "\n",
    "$$\n",
    "y^{(k+1)}=y^{(k)}+\\tau \\rho (x^{(k+1)}-z^{(k+1)})\n",
    "$$\n",
    "\n",
    "---\n",
    "\n",
    "{{< pagebreak >}}\n",
    "\n",
    "总结\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "x^{(k+1)}&= (2 A^T A + \\rho I)^{-1} (2 A^T b + \\rho z^{(k)} - y^{(k)})\n",
    "\\\\ \\quad \\\\\n",
    "z^{(k+1)}&=S_{\\left(\\frac{\\mu}{\\rho}\\right)} \\left(\n",
    "    x^{(k+1)}+\\frac{y^{(k)}}{\\rho}\n",
    "\\right)\n",
    "\\\\ \\quad \\\\\n",
    "y^{(k+1)}&=y^{(k)}+\\tau \\rho (x^{(k+1)}-z^{(k+1)})\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ADMM(x,max_iter=max_iter,eps=epsilon,t=t,rho=rho):\n",
    "    f_all=[]\n",
    "    x_all=[]\n",
    "    x_all.append( x.clone().detach().numpy() )\n",
    "    z = x.clone().detach() \n",
    "    z.requires_grad = True; z.requires_grad_(True)\n",
    "    y = torch.zeros_like(x)  # 初始的 Lagrange 乘子设为零，形状和 x 对齐\n",
    "    for i in range(1,int(max_iter)+1):\n",
    "        #  calculate the loss \n",
    "        ff = g(x) + h(z)\n",
    "        f_all.append(ff)\n",
    "        x_old = x.clone().detach()\n",
    "        x = torch.linalg.inv(2* A.T @ A + rho*torch.eye(n)) @ (2*A.T @ b + rho*z -y)\n",
    "        z = SS(x+y/rho,mu/rho)\n",
    "        y = y + tau * rho * (x-z)\n",
    "        iter_norm = torch.abs(torch.norm(x) - torch.norm(x_old))\n",
    "        if iter_norm < eps:\n",
    "            return x, f_all, x_all\n",
    "        x_all.append(x.data.cpu().numpy())\n",
    "\n",
    "    return x, f_all, x_all"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "{{< pagebreak >}}\n",
    "\n",
    "# 运行＆分开画图\n",
    "选分量全是零的向量作初始的 $x$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 3600x900 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=torch.zeros(n).requires_grad_(True)\n",
    "x_star_PGD, f_all_PGD, x_all_PGD = PGD(x)\n",
    "draw(f_all_PGD)\n",
    "plt.savefig('PGD',dpi=500,bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 3600x900 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=torch.zeros(n).requires_grad_(True)\n",
    "x_star_FISTA, f_all_FISTA, x_all_FISTA = FISTA(x)\n",
    "draw(f_all_FISTA)\n",
    "plt.savefig('FISTA',dpi=500,bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "{{< pagebreak >}}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 3600x900 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=torch.zeros(n).requires_grad_(True)\n",
    "x_star_ADMM, f_all_ADMM, x_all_ADMM = ADMM(x)\n",
    "draw(f_all_ADMM)\n",
    "plt.savefig('ADMM',dpi=500,bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 合并画图"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "def trans(flist):\n",
    "    # 将列表中的每个张量转换为NumPy数组\n",
    "    flist_numpy = np.array([t.detach().numpy() for t in flist])\n",
    "    # 对数组中的每个元素取自然对数\n",
    "    log_flist = np.log(flist_numpy)\n",
    "    return log_flist\n",
    "\n",
    "def alldraw(num):\n",
    "    plt.figure()\n",
    "    plt.xlim(0, num)\n",
    "    plt.plot(trans(f_all_PGD), label='PGD')\n",
    "    plt.plot(trans(f_all_FISTA), label='FISTA')\n",
    "    plt.plot(trans(f_all_ADMM), label='ADMM')\n",
    "    plt.xlabel('迭代次数')\n",
    "    plt.ylabel('目标函数值的自然对数')\n",
    "    plt.title('Log Scale 上的收敛速度')\n",
    "    plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "{{< pagebreak >}}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 3600x900 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alldraw(80)\n",
    "plt.savefig('80',dpi=500,bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 3600x900 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alldraw(4)\n",
    "plt.savefig('4',dpi=500,bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "{{< pagebreak >}}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Eu+yyi6Puxx9/jNt248aNjsexEaAHUOwyFaD/+9//7ig39cFfxOmnny6Px2OXFy1a5PhFGACKRW1tra677jq73KdPn0ZfNEfjRlEAxeKggw7Sm2++qTlz5jT6LKC5+NIFAJJL5zlXkmbPnq0xY8ak3L5Lly466aSTHHUff/xx0j7Rn3N4PB5VVlY2b5AAkCPpPudm8nMCrnMB5Lt0n3Ob67bbbtOGDRvs8qRJk+RyuZL24ToXQD7IZOYr0/mDt99+W9XV1Xb5gAMOSOnfiM6dO+uoo45y1LWVzC8BeuTE7NmzVV9fb5cPPPBAtW/fPqW+hx9+uKP8zjvvyOfzpXV8AFCISktLU2qXqaAoAOSrTJwXg8Gg3njjDUfd0UcfnVLfrl27ao899nDUNTW7HAAUovvvv18//PCDXb7hhhuSXvNyoyiAQta/f3/97ne/0yeffKJ58+Zp9OjRadkvX7oAQGOZOudKcjyFOVVDhw51lBNNnBLBU5kA5JNMnnMz+TkB17kA8lEmz7nNsWzZMj344IN2ecSIEY3OjfFwnQugkDWV+cpG/uC1115r0f6lxpnftpJvIECPnPjwww8d5f333z/lvoMGDXI8jtLn8+m7775L29gAoFCtXr3aUU70QSABegBwij4vGoahTp06tXqf33zzjWPmjP79+2v77bdPuf9+++3nKH/11VetHhMA5JPq6mr98Y9/tMs77rijzj///KR9uM4FUMjuu+8+3X333Y2+6GgNvnQBgPgycc5tjVQnTomIvi7mmhhAW5fJc24mPyfgOhdAPmor17m33HKLYyLV2267LaV+XOcCKGRNZb6ykT9oTea3reYbCNAjJxYvXuwoN+fiyzAMDR482FH3zTffpGNYAFCwVq5cqe+//94ut2vXTv3794/blmARADhFnxc7dOggt9vd6n225npYknbbbTdHmethAMXmj3/8ozZu3GiXb7rppibPz1znAkDz8KULAOSHVCdOiSBYBABhmfycgOtcAGiZr7/+Wn/961/t8iGHHKKDDz44pb5c5wIoVKlkvrKRP2jNMX72s585ymvWrNHmzZtTH2CGEKBHTsT+kte3b99m9e/Vq5ejvGTJklaPCQAK2dSpUx3lgw46SKYZ/zIg9gNDHm8GoNhl4pGPXA8DQMutX79e999/v10eOHCgzjzzzCb7EaAHgObhSxcAyA/vvvuuoxx7/o1FsAgAwjL5OQHXuQDQMjfeeKOCwaBdvvXWW1Puy3UugEKVSuYr0/mDH3/8UVVVVXa5Q4cOqqysTHn/JSUl6tq1q6Nu6dKlzRpjJhCgR05E3xEjSX369GlW/969ezvK0X85AQBOy5Yt0z333OOoO+eccxK2j/3A8MADD5RhGDIMQ6WlperRo4d23313XXDBBXriiSe0ZcuWjIwbANqK6PPit99+a58TDcNQZWWl+vXrp4MPPljXXHON/v3vf6e0T66HAaDl7rnnHtXU1Njlq6++Wi6Xq8l+3CgKAM3Dly4A0PYtXbpU8+bNs8sul0sHHnhgwvaWZTme5MQ1MYBilqnPCbjOBYCW+d///qfnn3/eLh944IEaMWJESn25zgVQqFLNfGU6f9Da/adyjFwgQI+sCwaDqqurc9R17NixWfuoqKhwlAlvAkB8VVVVOv744x0Bo912200nn3xywj6xHxhGq6+v15o1a7Rw4UI9/vjjOu+889SrVy/deOON8vl8aR07ALQVyc6Lmzdv1vLly/Wvf/1Lf/jDHzRq1CgNHTpUs2fPTrrP2BmDuB4GgNRUV1froYcessu9e/fWWWedlVJfbhQFgObhSxcAaPt++9vfKhQK2eUjjjhC3bt3T9h+48aNjvaPP/64fU3scrnUpUsX7bTTTjruuON05513xn16CAAUikx9TsB1LgC0zB/+8AdZlmWXr7nmmpT7cp0LoBA1J/OV6fxBa/efyjFygQA9si72P3y3253STHHRysrKku4TACCtXbtWhxxyiBYuXGjXlZSU6Iknnmj0KJ9oyYKi8VRXV+u2227TPvvsox9++KHF4wWAtqq558VFixbpyCOP1O9+97uEbWKvX0tLS5t1DK6HARSrhx56SJs2bbLLEyZMkMfjSakvN4oCQPPwpQsAtG033nijZs6caZcNw9Att9yStE+ya+JQKKQNGzZo6dKleu2113Tttddq4MCBOuqooxo9RQQACkGmPifgOhcAmm/58uV6+umn7fLuu++uMWPGpNyf61wAhaa5ma9M5w9au/9UjpEL7lwPAMUnHX+ZYvvwZTYAOH3yyScaO3asli9fbteZpqlp06Zp2LBhSfuOGjVK/fr103bbbafy8nJ5vV55vV7V1tbqp59+0sKFC/XWW281elzkwoULNXr0aL3zzjvq1KlTRt4XAGRbMBjUFVdcoa5du6pLly4qKSmR1+uVYRjatGmTVq5cqQULFugf//iH4zrXsizdc8898nq9uuOOOxrtt7XXxFwPAyhGW7du1f3332+XO3bsqIsvvjjl/i29UfTVV1/VrFmz1KtXr2b1B4B8x5cuANB23XDDDZo4caKj7sorr9Q+++yTtF9zr4kty9Ibb7yhPfbYQ4899phOO+20Zo8VANqqTH1OwHUuADTfH//4R/n9frt85ZVXNqs/17kACklLMl+Zzh8UauaXAD2yLvqROZIcj99JVfRFkyS1a9euVWMCgEIyefJkXXXVVaqvr7frPB6Ppk6dqjPOOKPJ/qm0kaTZs2frsssu01dffWXXffHFF7r++uv14IMPNn/gANAGuVwu3X777U22q6mp0eTJkzVx4kTV1dXZ9XfddZfGjh2rfffd19G+tdfEXA8DKEZ/+9vftHbtWrt83nnnNev8x42iANA8fOkCAG1PbW2tzj//fD3//POO+mOPPTalzy86deqkm266SV27dlXHjh3ta+JQKKTq6mp98803mj9/vt5++23HZxdbtmzRWWedpYqKCh199NFpf18AkAuZ+pyA61wAaJ7q6mpNmzbNLnfr1k2nnHJKs/bBdS6AQtHSzFem8weFmvklQI+si/0PPzpglKrYPrGPMAOAYvTjjz9q/PjxmjVrlqO+c+fOev7553XooYem9XhHHHGEPvroI40ZM0bvvvuuXf+Xv/xF11xzjfr06ZPW4wFAW1ZRUaFrr71WBx54oMaMGaOamhpJ4V8cb7zxRr355puO9q29JuZ6GEAxeuCBB+x1wzB0ySWXNKs/N4oCQPPwpQsAtC2ffvqpzjjjDMd1qiQdd9xxev755+V2N/2178CBA3XzzTc32W7lypW69tpr9de//tWuCwQCuuCCC7RkyRI+hwBQEDL1OQHXuQDQPNOmTbO/V5Okiy66SF6vt1n74DoXQL5rbeYr0/mDQs38mrkeAIpP+/btHeVQKNTsO6Zra2sd5bbwlwkAcmnOnDkaOnRoowupAw44QJ999lnaw/MRFRUVeu655xzn9kAgoJkzZ2bkeADQ1o0cOVJ33HGHo27u3LnatGmToy72+rW5v2ByPQyg2Lzzzjv69NNP7fLhhx+uXXbZJSPHitwoOnLkSEf9X/7yF61YsSIjxwSAtogvXQCgbbAsS/fee6+GDx/eKDx/+eWX6+WXX1ZJSUlaj9m7d289+eSTuv/++x31P/74o+PGVgAoBs39nIDrXABInWVZjpuRXC6XfvGLX2TseFznAmiL0pH5ynT+oLX7T+UYuUCAHlnncrlUXl7uqFuzZk2z9rF69WpHuUePHq0eFwDkq7vuukujR4/W2rVr7TqPx6Pbb79d8+bNU+/evTN6/J49e+r000931EXPSA8Axebiiy923Fjk9/v14YcfOtp06NDBUeZ6GACSe+ihhxzliy++OKPH40ZRAOBLFwBoC9avX68jjzxSV1xxhePx7Z06ddKMGTN07733yjQz93XvZZddphNPPNFR9/LLL2fseADQVjXncwKucwEgdbNnz9aSJUvs8pgxY7LypHuucwG0FenKfGU6f9Da/adyjFwgQI+cGDBggKO8fPnyZvWPbT948OBWjwkA8k0wGNQFF1yg3//+947HQe6666768MMPde2118rlcmVlLKNGjXKUW3KhBACFoqSkRPvtt5+jLva8yPUwAKSuqqpKr7zyil3u2LGjjjnmmIwflxtFARQ7vnQBgNxasGCBhg0bptmzZzvqDzvsMC1atEgnnHBCVsZxxRVXOMoff/yxI8wPAMUi1c8JuM4FgNRNmzbNUT777LOzdmyucwHkUrozX5nOH+y4446O8sqVKx3jbkogEHBc47Zr1079+vVr1hgzgQA9cmLQoEGO8tdff92s/rHtY/cHAMXgoosu0uOPP+6ou+CCC+wvVrJpu+22c5RramqyenwAaGuaOi+29np48eLFjjIBegCF7LnnntPWrVvt8imnnKKSkpKsHJsbRQEUM750AYDcef3113XggQdq5cqVdl1JSYnuv/9+zZ49W7169craWIYPHy6v12uXQ6GQY2Y8ACgmqXxOwHUuAKSmurpar776ql2urKzUcccdl7Xjc50LIJfSnfnKdP6gc+fOjgyE3+/XsmXLUt7/kiVLFAwG7fKuu+4qwzCaNcZMIECPnBgyZIij/N5776Xc99tvv3X8Ijpw4EDHo9IAoBjce++9jruxvV6vHn/8cU2dOlXl5eVZH091dbWj3LVr16yPAQDakqbOi7HXwx999JF8Pl/K+3///fcd5b333ruZIwSA/PHkk086ymeddVbWjs2NogCKGV+6AEBuzJkzRyeccIJqa2vtuoEDB+o///mPLrvssqyf60zTVJcuXRx1XBcDKFapfE7AdS4ApOb55593TJxy0kknqbS0NGvH5zoXQK5kIvOVjfxBazK/bTXfQIAeOXH44Yc7ynPnzpVlWSn1feuttxzlsWPHpmtYAJAXFi9erKuvvtoue71ezZgxQ+eff37OxvTpp586yr17987NQACgjVi4cKGjHHte7Nevn3beeWe7vHXr1pR/wdy0aZM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      "text/plain": [
       "<Figure size 3600x900 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alldraw(200)\n",
    "plt.savefig('200',dpi=500,bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 3600x900 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alldraw(800)\n",
    "plt.savefig('800',dpi=500,bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "{{< pagebreak >}}\n",
    "\n",
    "# 检查找到的自变量有多优\n",
    "\n",
    "## 用 CVXPY 的结果比较\n",
    "\n",
    "### 找到 CVXPY 结果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "最终找到的 x\n",
      " [-2.03295226e-10 -2.35416950e-10  5.29574432e-10 ...  7.08918339e-10\n",
      "  8.83934837e-11 -4.95711989e-11]\n",
      "最终向下找到的目标函数自然对数值 -0.19062034860889124\n"
     ]
    }
   ],
   "source": [
    "import cvxpy as cp\n",
    "# 将数据转换为 NumPy 数组，因为 CVXPY 使用 NumPy\n",
    "A_np = A.numpy(); b_np = b.numpy()\n",
    "x = cp.Variable(n) # 定义变量\n",
    "# 定义目标函数\n",
    "obj = cp.Minimize(\n",
    "    cp.norm(A_np@x - b_np, 2)**2 + mu*cp.norm(x, 1) \n",
    "    )\n",
    "problem = cp.Problem(obj) # 定义问题\n",
    "problem.solve() # 解决问题\n",
    "print(\"最终找到的 x\\n\", x.value)\n",
    "print(\"最终向下找到的目标函数自然对数值\", np.log(problem.value))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "{{< pagebreak >}}\n",
    "\n",
    "### 比较找到的 x 的方向和位置\n",
    "\n",
    "用余弦相似度比较方向，用欧几里得距离比较位置\n",
    "\n",
    "#### ISTA 和 CVXPY"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "余弦相似度: 0.6960036213820899\n",
      "欧几里得距离: 8.263830775789481\n"
     ]
    }
   ],
   "source": [
    "x_star_PGD = x_star_PGD.cpu()\n",
    "similarity = cosine_similarity(x.value, x_star_PGD.detach().numpy())\n",
    "print(\"余弦相似度:\", similarity)\n",
    "distance = np.linalg.norm(x.value - x_star_PGD.detach().numpy())\n",
    "print(\"欧几里得距离:\", distance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### FISTA 和 CVXPY"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "余弦相似度: 0.6981356313569587\n",
      "欧几里得距离: 8.240015587019371\n"
     ]
    }
   ],
   "source": [
    "x_star_FISTA = x_star_FISTA.cpu()\n",
    "similarity = cosine_similarity(x.value, x_star_FISTA.detach().numpy())\n",
    "print(\"余弦相似度:\", similarity)\n",
    "distance = np.linalg.norm(x.value - x_star_FISTA.detach().numpy())\n",
    "print(\"欧几里得距离:\", distance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ADMM 和 CVXPY"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "余弦相似度: 0.9999999198404822\n",
      "欧几里得距离: 0.005486678363559426\n"
     ]
    }
   ],
   "source": [
    "x_star_ADMM = x_star_ADMM.cpu()\n",
    "similarity = cosine_similarity(x.value, x_star_ADMM.detach().numpy())\n",
    "print(\"余弦相似度:\", similarity)\n",
    "distance = np.linalg.norm(x.value - x_star_ADMM.detach().numpy())\n",
    "print(\"欧几里得距离:\", distance)"
   ]
  }
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