Part1.BasicRecAlgorithms/SVD.ipynb

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    "import numpy as np"
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    "### 1. SVD矩阵分解\n",
    "\n",
    "在 Python 中，`numpy` 提供了简单的方法来进行 SVD（奇异值分解），可以使用 `numpy.linalg.svd` 函数。SVD 将一个矩阵分解为三个矩阵的乘积：\r\n",
    "\r\n",
    "$$ \r\n",
    "A = U \\Sigma V^T \r\n",
    "$$\r\n",
    "\r\n",
    "其中：\r\n",
    "- $ A $ 是原始矩阵，\r\n",
    "- $ U $ 是一个正交矩阵，包含了左奇异向量，\r\n",
    "- $ \\Sigma $ 是一个对角矩阵，包含了奇异值，\r\n",
    "- $ V^T $ 是 $ V $ 的转置，$ V $ 是另一个正交矩阵，包含了右奇异向量。\r\n",
    "\r\n",
    "\r\n"
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      "U矩阵：\n",
      "[[-0.21483724  0.88723069  0.40824829]\n",
      " [-0.52058739  0.24964395 -0.81649658]\n",
      " [-0.82633754 -0.38794278  0.40824829]]\n",
      "奇异值（S）：\n",
      "[1.68481034e+01 1.06836951e+00 1.40266540e-16]\n",
      "V^T 矩阵：\n",
      "[[-0.47967118 -0.57236779 -0.66506441]\n",
      " [-0.77669099 -0.07568647  0.62531805]\n",
      " [ 0.40824829 -0.81649658  0.40824829]]\n"
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    "# 创建示例矩阵\n",
    "A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])\n",
    "\n",
    "# 进行 SVD 分解\n",
    "U, S, VT = np.linalg.svd(A)\n",
    "\n",
    "print(\"U矩阵：\")\n",
    "print(U)\n",
    "print(\"奇异值（S）：\")\n",
    "print(S)\n",
    "print(\"V^T 矩阵：\")\n",
    "print(VT)"
   ]
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   "source": [
    "def svd(data, k):\n",
    "    U, S, VT = np.linalg.svd(A)\n",
    "    Sigma_k = np.diag(S[:k])\n",
    "    U_k = U[:, :k]\n",
    "    VT_k = VT[:k, :]\n",
    "    return U, S, VT"
   ]
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    "### 2. 将SVD用作推荐\n",
    "\n",
    "\n",
    "\n",
    "$$ \n",
    "r = U_{u} \\Sigma V_{v}^{\\top}\n",
    "$$\n",
    "\n",
    "其中：\n",
    "- $ r $ 是用户 $ u $ 对物品 $ v $ 的评分预测\n",
    "\n",
    "\n",
    "\n"
   ]
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