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__MACOSX/._A5_programming A5_programming/.DS_Store __MACOSX/A5_programming/._.DS_Store __MACOSX/A5_programming/._test __MACOSX/A5_programming/._train __MACOSX/A5_programming/._.ipynb_checkpoints...

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__MACOSX/._A5_programming
A5_programming/.DS_Store
__MACOSX/A5_programming/._.DS_Store
__MACOSX/A5_programming/._test
__MACOSX/A5_programming/._train
__MACOSX/A5_programming/._.ipynb_checkpoints
A5_programming/C3670_ass5.ipyn
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**COMP3670 Assignment 5 - Matrix Decomposition & Dimensionality Reduction**\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Enter Your Student ID:**\n",
"\n",
"**Your Name:** \n",
" \n",
"\n",
"**Submit:** You can write your answers in this file and submit a single Jupyter Notebook file (.ipynb) on Wattle. Rename this file with your student number as 'uXXXXXXX.ipynb'. Otherwise, you can write your programming questions in this file, and submit two files, 'uXXXXXXX.ipynb' for programming and 'uXXXXXXX.pdf' for theory. Please submit them separately instead of a zip file.\n",
" \n",
"**Enter Discussion Partner IDs Below:**\n",
"- \n",
"- \n",
"- \n",
" \n",
"\n",
"**Programming Section**\n",
"- 1 = 10%\n",
"- 2 = 15%\n",
"- 3 = 30%\n",
"- 4 = 10%\n",
"- 5 = 20%\n",
"- 6 = 15%"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"---\n",
"\n",
"\n",
"**PROGRAMMING SECTION**\n",
"---\n",
"\n",
"For all of the following, program the solution yourself. Don't just call a li
ary function that does the whole question for you, or you'll get zero (no, that doesn't mean you can't use any li
ary functions, but it does mean that you have to show you understand how to compute the answer yourself).\n",
"\n",
"**All written answers** should be between 50 and 500 words. If you can describe all the necessary information in 50 words, that's better. However, you'll only be graded on whether you describe the necessary ideas.\n",
"\n",
"\n",
" XXXXXXXXXXn",
"\n",
" **TASK 0.1:** You know the drill. Import Numpy and PyPlot. We're also going to generate a dataset.\n",
"\n",
"\n",
"-----------"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from mpl_toolkits.mplot3d import Axes3D #This is for 3d scatter plots.\n",
"import math\n",
"import random\n",
"from scipy.stats import multivariate_normal\n",
"import os\n",
"from matplotlib.pyplot import imread\n",
"np.random.seed XXXXXXXXXXn"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(77760, 135)\n"
]
},
{
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"text/plain": [
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"width = 320\n",
"height = 243\n",
"images = []\n",
"for file in os.listdir(\"./train\"):\n",
" if file.endswith(\".pgm\"):\n",
" XXXXXXXXXXim = imread(\"./train/\" + file)\n",
" XXXXXXXXXXim = im.flatten('F') # flatten im into a vector\n",
" XXXXXXXXXXimages.append(im) \n",
"A_pp = np.stack(images).T # build a matrix where each column is a flattened image\n",
"print(A_pp.shape)\n",
"plt.imshow(A_pp[:, 134].reshape(width, height).T)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"PROGRAMMING EXERCISE 1 \n",
" XXXXXXXXXXn",
"\n",
"---\n",
" **TASK 1.1:** Complete the function $laplace\\_expansion(A) = |A|$, where $A \\in \\math
{R}^{N \\times N}$, and $|A|$ is the determinant of $A$.\n",
" \n",
"**HINT:**\n",
"- One way to find the determinant is to implement the Laplace Expansion algorithm. This is described in the matrix decomposition lecture slides.\n",
"- $|A| \\neq 0$ iff $A$ is invertible.\n",
"---\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def det(A):\n",
" # WRITE CODE HERE\n",
" return det_A\n",
"\n",
"print(det(np.eye(2)))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"---\n",
"\n",
"**TASK 1.2:** Let $A\\_{pp} \\in \\math
{R}^{D \\times N}$ be a matrix of data. Each column of $A\\_{pp}$ is a sample of data (1 training example for instance). The rows of $A\\_{pp}$ are thus the features (dimensions) of each of these samples. Complete the function $preprocess(A\\_{pp}) = A, Q\\_norms, A\\_means$, for which:\n",
"\n",
"$$Q_{i,:} = A\\_{pp}_{i,:} - \\mu_i$$\n",
"\n",
"...where $\\mu_i = \\frac{1}{m}\\sum_j A\\_{pp}_{ij}$ .\n",
"\n",
"$$A_{i,:} = \\frac{Q_{i,:}}{||Q_{i,:}||_\\infty }$$\n",
"\n",
"\n",
"$A \\in \\math
{R}^{D \\times N}$\n",
"\n",
"$Q_{i,:}$ is the $i^{th}$ row of $Q$.\n",
"\n",
"$A_{i,:}$ is the $i^{th}$ row of $A$.\n",
"\n",
"$||Q_{i,:}||_\\infty$ is the infinity norm of $Q_{i,:}$.\n",
"\n",
"$Q\\_norms \\in \\math
{R}^{D}$ is a vector recording $||Q_{i,:}||_\\infty$ for every feature dimension $i$.\n",
"\n",
"$A\\_means \\in \\math
{R}^{D}$ is a vector recording $\\mu_i$ for every feature dimension $i$.\n",
"\n",
"\n",
"**HINT:** \n",
"- If the norm is 0, divide by 1 instead.\n",
"\n",
"---"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def preprocess(A_pp):\n",
" # WRITE CODE HERE\n",
" return A, Q_norms, mu\n",
"\n",
"A, Q_norms, A_means = preprocess(A_pp)\n",
"print(A)\n",
"print(Q_norms)\n",
"print(A_means)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"---\n",
"\n",
"$A \\in \\math
{R}^{D \\times N}$ as above is a matrix of data where every column is a sample of data, and every row is a feature of that data. In this case, we're going to be working with images. Each column $A_{:,j}$ of $A$ is an image of a face. \n",
"\n",
"\"But an image is a square grid\" you might be thinking. Well, we've simply taken every column of the image and stacked them vertically, converting a $320$ column $ \\times 243$ row pixel image into a $ XXXXXXXXXXtimes 1$ dimensional vector.\n",
"\n",
"Hence $D = 77760$ and we have $N = 135$ images.\n",
"\n",
"According to our lecture, we should first compute the covariance matrix $AA^T$, and then calculate the eigenvalues and eigenvectors of this covariance matrix. However, in our case, $AA^T$ is a $ XXXXXXXXXXtimes 77760$ dimensional matrix. \n",
"\n",
"Luckily, our lecture introduces an efficient method to work with such high-dimensional data. \n",
"\n",
"That is, we have the choice of either
Answered Same Day Oct 24, 2021 Australian National University

Solution

Sandeep Kumar answered on Nov 24 2021
151 Votes
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**COMP3670 Assignment 5 - Matrix Decomposition & Dimensionality Reduction**\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Enter Your Student ID:**\n",
"\n",
"**Your Name:** \n",
" \n",
"\n",
"**Submit:** You can write your answers in this file and submit a single Jupyter Notebook file (.ipynb) on Wattle. Rename this file with your student number as 'uXXXXXXX.ipynb'. Otherwise, you can write your programming questions in this file, and submit two files, 'uXXXXXXX.ipynb' for programming and 'uXXXXXXX.pdf' for theory. Please submit them separately instead of a zip file.\n",
" \n",
"**Enter Discussion Partner IDs Below:**\n",
"- \n",
"- \n",
"- \n",
" \n",
"\n",
"**Programming Section**\n",
"- 1 = 10%\n",
"- 2 = 15%\n",
"- 3 = 30%\n",
"- 4 = 10%\n",
"- 5 = 20%\n",
"- 6 = 15%"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"---\n",
"\n",
"\n",
"**PROGRAMMING SECTION**\n",
"---\n",
"\n",
"For all of the following, program the solution yourself. Don't just call a li
ary function that does the whole question for you, or you'll get zero (no, that doesn't mean you can't use any li
ary functions, but it does mean that you have to show you understand how to compute the answer yourself).\n",
"\n",
"**All written answers** should be between 50 and 500 words. If you can describe all the necessary information in 50 words, that's better. However, you'll only be graded on whether you describe the necessary ideas.\n",
"\n",
"\n",
"-----------\n",
"\n",
" **TASK 0.1:** You know the drill. Import Numpy and PyPlot. We're also going to generate a dataset.\n",
"\n",
"\n",
"-----------"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from mpl_toolkits.mplot3d import Axes3D #This is for 3d scatter plots.\n",
"import math\n",
"import random\n",
"from scipy.stats import multivariate_normal\n",
"import os\n",
"from matplotlib.pyplot import imread\n",
"np.random.seed(13579201)\n"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(77760, 135)\n"
]
},
{
"data": {
"image/png":...
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