Pittas_BME574_Midterm.ipynb

{
 "cells": [
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   "execution_count": 2,
   "id": "ecf5bb72-c4f6-460f-bb97-8c80eefd0505",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m  Activating\u001b[22m\u001b[39m project at `C:\\Users\\gnpit\\BME574`\n"
     ]
    }
   ],
   "source": [
    "import Pkg\n",
    "Pkg.activate(\".\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "9fba6971-2ed2-4a69-9fbb-354ff54daf95",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m registry at `C:\\Users\\gnpit\\.julia\\registries\\General.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `C:\\Users\\gnpit\\BME574\\Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `C:\\Users\\gnpit\\BME574\\Manifest.toml`\n"
     ]
    }
   ],
   "source": [
    "Pkg.add([\"Random\",\"Images\",\"Colors\",\"MAT\",\"LinearAlgebra\",\"MLDatasets\", \"Plots\", \"Convex\", \"SCS\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "9b3d40e5-e76d-40ed-9505-836c73102594",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Images, Colors, MLDatasets, Plots, Convex, SCS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "abebacf1-30f5-4510-a608-5838317483ec",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dataset MNIST:\n",
       "  metadata  =>    Dict{String, Any} with 3 entries\n",
       "  split     =>    :train\n",
       "  features  =>    28×28×60000 Array{Float32, 3}\n",
       "  targets   =>    60000-element Vector{Int64}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dataset = MNIST()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "3d6a5d01-2ee8-48df-9ae3-5508863793ac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dataset MNIST:\n",
       "  metadata  =>    Dict{String, Any} with 3 entries\n",
       "  split     =>    :test\n",
       "  features  =>    28×28×10000 Array{Float32, 3}\n",
       "  targets   =>    10000-element Vector{Int64}"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "testset = MNIST(:test)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "dab808ec-741d-4b61-82f8-0537f52c14e3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "60000-element Vector{Int64}:\n",
       " 5\n",
       " 0\n",
       " 4\n",
       " 1\n",
       " 9\n",
       " 2\n",
       " 1\n",
       " 3\n",
       " 1\n",
       " 4\n",
       " 3\n",
       " 5\n",
       " 3\n",
       " ⋮\n",
       " 7\n",
       " 8\n",
       " 9\n",
       " 2\n",
       " 9\n",
       " 5\n",
       " 1\n",
       " 8\n",
       " 3\n",
       " 5\n",
       " 6\n",
       " 8"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dataset.targets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "bf64b9e8-451a-49fc-aa68-c612764d48d2",
   "metadata": {},
   "outputs": [
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       "28×28×60000 Array{Float32, 3}:\n",
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       "\n",
       "[:, :, 2] =\n",
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       " ⋮                             ⋮         ⋱                  ⋮         \n",
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       "\n",
       "[:, :, 3] =\n",
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       " 0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0  0.0\n",
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       "\n",
       ";;; … \n",
       "\n",
       "[:, :, 59998] =\n",
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       " 0.0  0.0  0.0  0.0  0.0  0.0       …  0.576471  0.992157   0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0          0.576471  0.992157   0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0          0.619608  0.992157   0.0  0.0  0.0\n",
       " ⋮                        ⋮         ⋱                       ⋮         \n",
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       " 0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0        0.0  0.0  0.0\n",
       "\n",
       "[:, :, 59999] =\n",
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       " 0.0  0.0  0.0       0.0        0.0           0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0       0.0        0.0           0.0  0.0  0.0  0.0  0.0  0.0\n",
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       " 0.0  0.0  0.0       0.0        0.0           0.0  0.0  0.0  0.0  0.0  0.0\n",
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       " 0.0  0.0  0.0       0.0        0.0           0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0       0.0        0.0           0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0       0.0        0.0        …  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0       0.0        0.0           0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0       0.0        0.0           0.0  0.0  0.0  0.0  0.0  0.0\n",
       "\n",
       "[:, :, 60000] =\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.101961  0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.898039  0.286275  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.976471  0.756863  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.690196  0.772549  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.870588  0.52549   0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.956863  0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.172549  0.0       0.0  0.0  0.0\n",
       " ⋮                        ⋮    ⋱                      ⋮         \n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0       0.0       0.0  0.0  0.0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dataset.features"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "136c9576-4e4f-4879-b11e-7de0994de61b",
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/html": [
       "<img src=\"data:image/png;base64,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\">"
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       "28×28 Array{Gray{Float32},2} with eltype Gray{Float32}:\n",
       " Gray{Float32}(0.0)  Gray{Float32}(0.0)  …  Gray{Float32}(0.0)\n",
       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
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       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
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       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
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       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
       " ⋮                                       ⋱  \n",
       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
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       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)\n",
       " Gray{Float32}(0.0)  Gray{Float32}(0.0)     Gray{Float32}(0.0)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gray.(dataset.features[:,:,5]')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b491bcc4-fb79-4b61-ad4d-cf53c770331a",
   "metadata": {},
   "source": [
    "## BME 574 Midterm Exam (take-home)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a3fc123-f991-438e-a7df-662a9ea14bcd",
   "metadata": {},
   "source": [
    "The dataset above is the famous MNIST dataset of handwritten digits (28x28 grayscale images).\n",
    "The challenge for this midterm is to find the most informative pixels for each digit from this dataset\n",
    "using techniques that you have learning previously.  Here is how to start:\n",
    "1. Create a set of output (label) vectors $\\{y_j\\}$ of length 10 that represent the labels (0-9)\n",
    "here are a few examples:\n",
    "$$\n",
    "\\begin{align}\n",
    "^{\"}1^{\"}=\\begin{bmatrix} 1\\\\0\\\\0\\\\ \\vdots \\\\0 \\end{bmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "$$\n",
    "\\begin{align}\n",
    "^{\"}3^{\"}=\\begin{bmatrix} 0\\\\0\\\\1\\\\ \\vdots \\\\0 \\end{bmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "$$\n",
    "\\begin{align}\n",
    "^{\"}0^{\"}=\\begin{bmatrix} 0\\\\0\\\\0\\\\ \\vdots \\\\1 \\end{bmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "2. Now let $B$ be the set of output (label) vectors\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "B = \\begin{bmatrix} y_{1} & y_{2} & y_{3} & \\ldots & y_{n}\\end{bmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "3. Let the matrix $A$ be the corresponding reshaped (vectorized) MNIST images (one in each row)\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "A = \\begin{bmatrix} x_{1} & x_{2} & x_{3} & \\ldots & x_{n}\\end{bmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "4. $AX = B$ maps from the image space to the label space\n",
    "\n",
    "Your task is to use solvers for $AX=B$ that promote sparsity to rank the pixels in the MNIST dataset that are most informative for correctly labeling the digits. You will have to come up with your own heuristics or empirical rules for this. Be sure to visualize the results from X. Apply your most important pixels to the test data set to see how accurate you are with as few pixels as possible. Redo the analysis with each digit individually to ﬁnd the most important pixels for each digit. Think about the interpretation of what you are doing with this $AX = B$ problem.\n",
    "Hint: for testing, it may be a good idea to start with a smaller dataset (e.g. 1000 images)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "71b9670b-4a7d-49d3-9c32-b5655420e5ff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10000-element Vector{Int64}:\n",
       " 7\n",
       " 2\n",
       " 1\n",
       " 0\n",
       " 4\n",
       " 1\n",
       " 4\n",
       " 9\n",
       " 5\n",
       " 9\n",
       " 0\n",
       " 6\n",
       " 9\n",
       " ⋮\n",
       " 5\n",
       " 6\n",
       " 7\n",
       " 8\n",
       " 9\n",
       " 0\n",
       " 1\n",
       " 2\n",
       " 3\n",
       " 4\n",
       " 5\n",
       " 6"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Gather data from MNIST\n",
    "img_train = dataset.features\n",
    "labels_train = dataset.targets\n",
    "img_test = testset.features\n",
    "labels_test = testset.targets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "d6158a10-97c8-4e3a-971f-ccc8566dcead",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Gather/Assign number of images, pixels, and digits in set\n",
    "num_img = size(img_train, 3)\n",
    "num_img_test = 1000\n",
    "num_pixels = size(img_train, 1) * size(img_train, 2)\n",
    "num_digits = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "9f0ff4de-523d-4954-b24b-f2d7cb3a48d8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "28×28×60000 Array{Float64, 3}:\n",
       "[:, :, 1] =\n",
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       "\n",
       "[:, :, 2] =\n",
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       " 0.0  0.0  0.0  0.0  0.0  0.0     0.764706   0.0  0.0  0.0  0.0  0.0\n",
       " ⋮                        ⋮    ⋱                       ⋮         \n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0470588  0.0  0.0  0.0  0.0  0.0\n",
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       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
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       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       "\n",
       "[:, :, 3] =\n",
       " 0.0  0.0  0.0  0.0       0.0       …  0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0       …  0.152941  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.243137     0.152941  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.494118     0.156863  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.862745     0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.870588     0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.180392  0.960784  …  0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.470588  0.996078     0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.623529  0.996078     0.0       0.0  0.0  0.0  0.0  0.0\n",
       " ⋮                                  ⋱                      ⋮         \n",
       " 0.0  0.0  0.0  0.0       0.466667     0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0       …  0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0       …  0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0       0.0          0.0       0.0  0.0  0.0  0.0  0.0\n",
       "\n",
       ";;; … \n",
       "\n",
       "[:, :, 59998] =\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0        …  0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0        …  1.0       0.47451   0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.572549  0.270588  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0        …  0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " ⋮                        ⋮          ⋱                      ⋮         \n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0        …  0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0862745     0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.768627      0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.45098       0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0        …  0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0           0.0       0.0       0.0  0.0  0.0\n",
       "\n",
       "[:, :, 59999] =\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " ⋮                        ⋮    ⋱                       ⋮         \n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.733333   0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0352941  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  …  0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0     0.0        0.0  0.0  0.0  0.0  0.0\n",
       "\n",
       "[:, :, 60000] =\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0       …  0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0       …  0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.643137  0.733333  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.92549   0.231373  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0       …  0.360784  0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " ⋮                        ⋮              ⋱            ⋮              \n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.258824  …  0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.556863     0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.352941     0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.101961     0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0       …  0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0          0.0       0.0       0.0  0.0"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A=Array{Float64}(undef, 28, 28, num_img) #Initialize array that will become A\n",
    "\n",
    "#Read each image individually and sort into A; otherwise, orientation becomes messed up\n",
    "for i in 1:num_img\n",
    "    A[:, :, i] = img_train[:,:,i]'\n",
    "end\n",
    "return(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "e8d200b3-44fa-4e2f-ae53-679290a66923",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "60000×10 Matrix{Float64}:\n",
       " 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0\n",
       " 1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0\n",
       " 0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " ⋮                        ⋮                   \n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0\n",
       " 0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0\n",
       " 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0\n",
       " 0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0  0.0\n",
       " 0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  1.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0  0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Create the matrix B where each row represents labels 0-9, where the index 1 is placed at the value of the image\n",
    "B = zeros(num_img, num_digits)\n",
    "for i in 1:num_img\n",
    "    B[i, labels_train[i]+1] = 1\n",
    "end\n",
    "display(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "18ff9eb3-0f3c-4fd3-af84-3595dd81bb38",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "lasso (generic function with 1 method)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Lasso Function from Class\n",
    "function lasso(A, y, λ1)\n",
    "    (T, K) = (size(A, 1), size(A, 2))\n",
    "    Q = A'A / T\n",
    "    c = A'y / T                      #c'b = Y'X*b\n",
    "\n",
    "    x = Variable(K)              #define variables to optimize over\n",
    "    L1 = quadform(x, Q; assume_psd=true)            #b'Q*b\n",
    "    L2 = dot(c, x)                 #c'b\n",
    "    L3 = norm(x, 1)                #sum(|b|)\n",
    "\n",
    "    problem = minimize(L1 - 2 * L2 + λ1 * L3)\n",
    "    solve!(problem, SCS.Optimizer; silent = true)\n",
    "    return vec(Convex.evaluate(x)) # \"evaluate\" seems to run into issue with another package, need to specify that Convex is used here\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "d936894f-10cb-4613-9890-3562a5ca10f9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "784×60000 Matrix{Float64}:\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " ⋮                        ⋮              ⋱            ⋮                   \n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Reshape A to become a 2D matrix with the pixels for each image as opposed to the Array\n",
    "A_re = reshape(A, num_pixels, num_img)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "89da86a9-4776-4eb4-b1e7-51e300c1f1f5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "784×10 Matrix{Float64}:\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17  …  -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17     -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17     -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17     -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17     -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17  …  -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17     -8.15998e-18  -1.08033e-16\n",
       "  1.65408e-10  -6.40297e-10   4.70689e-11      2.66052e-11  -5.39054e-11\n",
       "  8.21629e-10  -3.11891e-9    1.88037e-10      1.27392e-10  -5.28147e-10\n",
       "  5.12169e-10  -2.54367e-9    2.08299e-10      3.37645e-11  -5.26821e-10\n",
       "  8.55492e-10  -3.52994e-9    3.23421e-10  …   5.69305e-11  -3.26675e-10\n",
       "  8.68205e-10  -3.52046e-9    3.89579e-10      1.26522e-10  -7.22248e-11\n",
       "  4.8967e-10   -2.68663e-9    3.03522e-10      1.05078e-10  -8.28603e-10\n",
       "  ⋮                                        ⋱                \n",
       " -9.69602e-11  -1.82988e-9    6.50171e-10      6.87896e-11   2.42675e-9\n",
       " -2.89523e-10  -8.65473e-10   5.66285e-11      9.13828e-11   1.73408e-9\n",
       " -3.34704e-10  -1.35395e-9   -7.30359e-10      3.67192e-11  -9.02398e-10\n",
       " -1.57117e-10  -6.0666e-10   -3.76631e-10  …   1.65973e-11  -3.20849e-10\n",
       "  3.34305e-11   6.39968e-10   4.42701e-11     -1.94441e-11   1.80387e-10\n",
       "  7.77475e-11  -8.74616e-10   4.72409e-11      1.85598e-12   6.3315e-10\n",
       "  1.43534e-10  -1.61468e-9    8.72138e-11      3.42655e-12   1.16889e-9\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17     -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17  …  -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17     -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17     -8.15998e-18  -1.08033e-16\n",
       " -1.56678e-17  -6.51916e-17   4.97602e-17     -8.15998e-18  -1.08033e-16"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Starting test value lambda\n",
    "lambda1 = 0.1\n",
    "\n",
    "#Run the lasso where X pulls out the sig parts\n",
    "X = zeros(num_pixels, num_digits)\n",
    "\n",
    "for digit in 1:num_digits\n",
    "    X[:, digit] = lasso(A_re', B[:,digit], lambda1)\n",
    "end\n",
    "\n",
    "display(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "7ffc29ec-573e-4acd-bcdd-bd283762569c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "60000×10 Matrix{Float64}:\n",
       " 0.0444386     0.0568983   …  0.0530068   0.0190314   0.0483887\n",
       " 0.137116     -0.0048454      0.0767031   0.00767274  0.0245812\n",
       " 0.0132683     0.0207781      0.00147695  0.050998    0.0348669\n",
       " 0.0504329     0.183797       0.0536931   0.0666294   0.00211542\n",
       " 0.00757527    0.0760828      0.0660741   0.0621927   0.058607\n",
       " 0.0570565     0.0459131   …  0.0601927   0.0633508   0.0791472\n",
       " 0.000144578   0.194227       0.0349213   0.0675305   0.0128456\n",
       " 0.0551347     0.0986153      0.0232218   0.0851427   0.0840381\n",
       " 9.55981e-5    0.164153       0.0142811   0.0498532   0.00466463\n",
       " 0.0676608     0.0524599      0.0281747   0.054729    0.0600139\n",
       " 0.0707127     0.136897    …  0.0803545   0.0609367   0.0835148\n",
       " 0.0275825     0.0855862      0.0425751   0.0435658   0.000606084\n",
       " 0.0629357     0.133769       0.0624053   0.0543148   0.0795522\n",
       " ⋮                         ⋱                          \n",
       " 0.0471735     0.0333279      0.0792198   4.06348e-5  0.0526676\n",
       " 0.0224587     0.0665099      0.0204307   0.0682661   0.0513396\n",
       " 0.0202818     0.0632658   …  0.036507    0.00269454  0.05413\n",
       " 0.0499806     0.0278239      0.0155019   3.73912e-5  0.0390983\n",
       " 0.022978      0.070103       0.030797    2.42616e-5  0.0659638\n",
       " 0.0253525     0.00829127     0.0146184   0.0189383   0.0111689\n",
       " 0.00582261    0.116143       0.0169193   0.0647691   0.000396637\n",
       " 0.00462615    0.115298    …  0.0407812   0.0881086   0.0170536\n",
       " 0.0699602     0.110837       0.0221518   0.0409955   0.0643732\n",
       " 0.000424082   0.033328       0.0431678   0.0377534   0.0450287\n",
       " 0.0734493     0.00466086     0.0307147   3.9359e-5   0.0536448\n",
       " 0.0298819     0.0352706      0.019183    0.0355969   0.0586571"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Using the sig parts, predict the image (and its written value from the data set)\n",
    "predictions = A_re' * X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "8e59e3d0-c437-4b45-9c3a-0b8487eb45c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "60000-element Vector{Int64}:\n",
       " 6\n",
       " 0\n",
       " 8\n",
       " 1\n",
       " 1\n",
       " 9\n",
       " 1\n",
       " 1\n",
       " 1\n",
       " 4\n",
       " 1\n",
       " 1\n",
       " 1\n",
       " ⋮\n",
       " 7\n",
       " 8\n",
       " 1\n",
       " 2\n",
       " 1\n",
       " 6\n",
       " 1\n",
       " 1\n",
       " 1\n",
       " 4\n",
       " 4\n",
       " 6"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#To use the predictions to get the written number prediction, find when the value of the row is 1 (subtract one due to index of 1, but first is a zero written number)\n",
    "max_indices = Int[]\n",
    "\n",
    "for row in eachrow(predictions)\n",
    "    predicted_label_idx = argmax(row)[1]\n",
    "    push!(max_indices, predicted_label_idx)\n",
    "end\n",
    "\n",
    "predict = max_indices.-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "b7379448-7e90-46f4-b96a-5c0dce662552",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Accuracy: 43.44166666666667\n"
     ]
    }
   ],
   "source": [
    "#Calculate the accuracy by comparing the labels (predict is the models label) \n",
    "accuracy = sum(predict .== labels_train) / length(labels_train) *100\n",
    "println(\"Accuracy: \", accuracy)\n",
    "\n",
    "#Accuracy is relatively low, thus we need to optimize lambda"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "0c48710f-481a-499b-bf4f-e014607e2377",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best λ: 0.0015625 with an accuracy of: 84.01666666666667\n"
     ]
    }
   ],
   "source": [
    "#To optimize lambda start at a value of 0.1 and keep decreasing the value by half the value until the difference between trials is under a specific threshold (1%)\n",
    "#Note that ideally we could start at 1.0 and keep halving to see the best, but this takes a lot of computational power and time\n",
    "#Instead, from previous testing, increasing from 0.1 made accuracy worse, while decreasing made it better, so the while loop will continously decrease it\n",
    "\n",
    "new_accuracy = 0\n",
    "accuracy = 0 \n",
    "lambda = 0.2\n",
    "best_lambda = 0 \n",
    "per_diff = 100\n",
    "\n",
    "while per_diff >= 1.0\n",
    "    X = zeros(num_pixels, num_digits)\n",
    "    \n",
    "    lambda = lambda / 2\n",
    "\n",
    "    # Repeat the code used in the original test each run\n",
    "    for digit in 1:num_digits\n",
    "        X[:, digit] = lasso(A_re', B[:, digit], lambda)\n",
    "    end\n",
    "\n",
    "    predictions = A_re' * X\n",
    "\n",
    "    max_indices = Int[]\n",
    "    for row in eachrow(predictions)\n",
    "        predicted_label_idx = argmax(row)[1]\n",
    "        push!(max_indices, predicted_label_idx)\n",
    "    end\n",
    "    predict = max_indices .- 1\n",
    "    accuracy = sum(predict .== labels_train) / length(labels_train) * 100\n",
    "\n",
    "    #Compute the percent change to see if the change is significant (greater than 5%)\n",
    "    per_diff = (abs(new_accuracy - accuracy) / accuracy) * 100\n",
    "    \n",
    "    #Compare the accuracy of the runs to see if accuracy improved\n",
    "    if accuracy > new_accuracy\n",
    "        new_accuracy = accuracy\n",
    "        best_lambda = lambda\n",
    "    end\n",
    "    \n",
    "end\n",
    "\n",
    "println(\"Best λ: \", best_lambda, \" with an accuracy of: \", new_accuracy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "10af4521-1ea2-4289-9afe-c2154c51b2bf",
   "metadata": {},
   "outputs": [],
   "source": [
    "#Thus from the above code, if you keep decreasing lambda by a half the accuracy peaks\n",
    "#it might still increase, but the change from the previous to the last is under 1%\n",
    "#Thus, we could keep going to try to increase the accuracy but the computational time and power increases\n",
    "#This code can be easily modified to that extent, however, making this model relatively powerful and a good basis for the stated problem\n",
    "#It can easily collect the data from all 60000 images, collect sig parts, and make predictions based on the pixels to the written number with\n",
    "#an overall accuracy of 84.02%"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b8579cd3-29d0-496e-bcda-1d935d16ee62",
   "metadata": {},
   "outputs": [],
   "source": [
    "84.625"
   ]
  }
 ],
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