Chapter 18 - The Singular Value Decomposition.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy as sp\n",
    "import scipy.linalg\n",
    "import sympy as sy\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('ggplot')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.set_printoptions(precision=3)\n",
    "np.set_printoptions(suppress=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> The Singular Values </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have discussed what _Spectral Decomposition_ which can decompose any symmetric matrices unconditionally into three special matrices.However only square matrices have eigenvalues and vectors, however we want to extend a similar concept for any $m \\times n$ matrices.\n",
    "\n",
    "If $A$ is an $m \\times n$ matrix, then $A^TA$ and $AA^T$ are both symmetric and orthogonally diagonalizable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 4, -3],\n",
       "       [-3,  2],\n",
       "       [ 2, -1],\n",
       "       [ 1,  0]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[4, -3], [-3, 2], [2, -1], [1, 0]]); A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute eigenvalues and eigenvectors of $A^TA$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 30, -20],\n",
       "       [-20,  14]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ATA = A.T@A; ATA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "D1, P1 = np.linalg.eig(ATA)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check if $P$ is an orthonormal matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0.],\n",
       "       [0., 1.]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1@P1.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([43.541,  0.459])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The square roots of eigenvalues of $A^TA$ are called <font face=\"gotham\" color=\"red\"> singular values</font>  of $A$, denoted by $\\sigma_1, ..., \\sigma_n$ in decreasing order. \n",
    "\n",
    "We can also show that singular values of $A$ are the lengths of vectors $A\\mathbf{v}_1,..., A\\mathbf{v}_n$, where $\\mathbf{v}_i$ is the eigenvalue of $A^TA$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The length of $A\\mathbf{v}_i$ is $\\|A\\mathbf{v}_i\\|$\n",
    "\n",
    "$$\n",
    "\\|A\\mathbf{v}_i\\| = \\sqrt{(A\\mathbf{v}_i)^TA\\mathbf{v}_i} = \\sqrt{\\mathbf{v}_i^TA^T A\\mathbf{v}_i}=\\sqrt{\\mathbf{v}_i^T(\\lambda_i\\mathbf{v}_i)} = \\sqrt{\\lambda_i}=\\sigma_1\n",
    "$$\n",
    "\n",
    "where $\\sqrt{\\mathbf{v}_i^T\\mathbf{v}_i} = 1$ and $\\lambda_i$'s are eigenvalues of $A^TA$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " # <font face=\"gotham\" color=\"purple\"> Singular Value Decomposition</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Singular Value Decomposition** (SVD) is probably the most important decomposition technique in the history of linear algebra, it combines all the theory we discussed, then culminate at this point."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$A$ is a $m\\times n$ matrix. However $AA^T$ and $A^TA$ are symmetric matrices,then both are orthogonally diagonalizable.\n",
    "\n",
    "$$\n",
    "AA^T = U\\Sigma\\Sigma^T U^T=(U\\Sigma V^T)(V\\Sigma U^T)\\\\\n",
    "A^TA = V\\Sigma^T \\Sigma V^T = (V\\Sigma^T U^T)(U \\Sigma V^T)\n",
    "$$\n",
    "\n",
    "where $\\Sigma\\Sigma^T$ is a diagonal matrix with all eigenvalues of $AA^T$ and $\\Sigma^T \\Sigma$ is a diagonal matrix with all eigenvalues of $VV^T$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because both $AA^T$ and $A^TA$ are symmetric, then $UU^T= U^TU=I_{m\\times m}$ and $VV^T= V^TV=I_{n\\times n}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have implicitly shown the singular value decompositions above, one of the most important concept in linear algebra.\n",
    "\n",
    "<font face=\"gotham\" color=\"red\">\n",
    "$$\n",
    "\\Large\n",
    "SVD:\\quad\n",
    "A_{m\\times n} = U_{m\\times m}\\Sigma_{m \\times n} V^T_{n \\times n}\n",
    "$$\n",
    "</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The SVD theory guarantees that any matrix $A$, no matter its ranks or shapes, can be unconditionally decomposed into three special matrices."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So next question: what is $\\Sigma$? \n",
    "\n",
    "It is an $m\\times n$ main diagonal matrix, with all singular values on the main diagonal. Rewrite"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "A^TA = V\\Sigma^T \\Sigma V^T = V\\Sigma^2 V^T\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Post-multiply both sides by $V$\n",
    "\n",
    "$$\n",
    "A^TAV = V\\Sigma^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the matrix version of $A\\mathbf{v}_i = \\lambda_i \\mathbf{v}_i$, but here the matrix of interest is $A^TA$ rather than $A$. Similarly it can be written with singular values\n",
    "\n",
    "$$\n",
    "A^TA\\mathbf{v}_i = \\sigma_i^2\\mathbf{v}_i\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because $U$ and $V$ are not unique, we tend to standardize the solution by arranging $\\sigma_1 \\geq \\sigma_2 \\geq \\sigma_3\\geq ... \\geq\\sigma_r$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Why we only arrange $r$ singular values? Because it is the rank of $A$, so is the rank of $A^TA$. Explicitly $\\Sigma$ looks like\n",
    "\n",
    "\n",
    "$$\\Sigma =\\left[\\begin{array}{cccccc}\n",
    "\\sqrt{\\lambda_{1}} & & & & &\\\\\n",
    "& \\sqrt{\\lambda_{2}} & & & &\\\\\n",
    "& & \\ddots & & &\\\\\n",
    "& & & \\sqrt{\\lambda}_{\\mathrm{r}} & &\\\\\n",
    "& & & & \\ddots &\\\\\n",
    "& & & & & 0 \n",
    "\\end{array}\\right]\n",
    "=\\left[\\begin{array}{cccccc}\n",
    "\\sigma_1 & & & & &\\\\\n",
    "& \\sigma_2 & & & &\\\\\n",
    "& & \\ddots & & &\\\\\n",
    "& & & \\sigma_r & &\\\\\n",
    "& & & & \\ddots &\\\\\n",
    "& & & & & 0 \n",
    "\\end{array}\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can do the same for $AA^T$ and get\n",
    "\n",
    "$$\n",
    "AA^TU = U \\Sigma^2\n",
    "$$\n",
    "\n",
    "or \n",
    "\n",
    "$$\n",
    "AA^T\\mathbf{u}_i = \\sigma_i^2\\mathbf{u}_i\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have shown why $A_{m\\times n} = U_{m\\times m}\\Sigma_{m \\times n} V^T_{n \\times n}$ holds."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To perfomr a SVD on $A$, we just need two equations and this is also a mannual procedure to decompose any matrix.\n",
    "\n",
    "$$\n",
    "A^TA = V\\Sigma^T \\Sigma V^T\\\\\n",
    "AV = U\\Sigma\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's an example, let's say we have we have a data set $A$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.random.rand(10, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Give it a $\\text{SVD}$ decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "U, S, VT = np.linalg.svd(A, full_matrices=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's say we want to reduce $A_{10\\times 3}$ in to $A_{10\\times 2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Keep only the first dimension\n",
    "A_reduced = np.dot(U[:, :1], np.dot(np.diag(S[:1]), VT[:1, :]))\n",
    "\n",
    "# Plot original data\n",
    "plt.scatter(A[:, 0], A[:, 1])\n",
    "plt.title(\"Original $A$\")\n",
    "plt.show()\n",
    "\n",
    "# Plot reduced data\n",
    "plt.scatter(A_reduced[:, 0], A_reduced[:, 1])\n",
    "plt.title(\"$A$ with Reduced Dimensionality\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Generate some random data\n",
    "np.random.seed(0)\n",
    "data = np.random.randn(200, 2)\n",
    "\n",
    "# Perform SVD on the data\n",
    "U, s, VT = np.linalg.svd(data)\n",
    "\n",
    "# Perform PCA on the data\n",
    "mean = np.mean(data, axis=0)\n",
    "data_pca = data - mean\n",
    "cov = np.cov(data_pca.T)\n",
    "eigenvalues, eigenvectors = np.linalg.eig(cov)\n",
    "\n",
    "# Plot the data\n",
    "plt.scatter(data[:, 0], data[:, 1])\n",
    "plt.title(\"Original Data\")\n",
    "plt.show()\n",
    "\n",
    "# Plot the SVD components\n",
    "plt.scatter(U[:, 0], U[:, 1])\n",
    "plt.title(\"SVD Components\")\n",
    "plt.show()\n",
    "\n",
    "# Plot the PCA components\n",
    "plt.scatter(eigenvectors[:, 0], eigenvectors[:, 1])\n",
    "plt.title(\"PCA Components\")\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "## <font face=\"gotham\" color=\"purple\"> Reformulate SVD</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rewrite $SVD$\n",
    "\n",
    "$$\n",
    "AV = U\\Sigma\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "vector version is \n",
    "\n",
    "$$\n",
    "A\\mathbf{v}_i = \\sigma_i \\mathbf{u}_i\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There two implications from the equation above: $(a)$ $A$ can be decomposed into\n",
    "\n",
    "$$\n",
    "A = \\sum_{i}^r\\sigma_i\\mathbf{u}_i \\mathbf{v}_i^T\n",
    "$$\n",
    "\n",
    "$(b)$ We can compute $\\mathbf{u}_i$ by using \n",
    "\n",
    "$$\\mathbf{u}_i = \\frac{A\\mathbf{v}_i}{\\sigma_i}$$"
   ]
  }
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