notebooks/Curse_of_dimensionality.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The curse of dimensionality\n",
    "\n",
    "In general, we don't want to learn with an arbitrarily large number of features.\n",
    "\n",
    "We can still improve on the 2 or 3 variables in a lot of 'standard' modeling tasks (e.g. modeling facies from a few logs, or porosity from facies, or permeability from porosity). But the earth is massively high-dimensional... what's the right number, and why does it matter?\n",
    "\n",
    "- High-dimensional models are hard to explain.\n",
    "- High-dimensional datasets are hard to visualize.\n",
    "- We need a lot of data to properly sample the joint distributions of high-dimensional datasets. \n",
    "- We're more likely to have redundant features in the data.\n",
    "\n",
    "🙌 **This notebook was inspired and informed by Rudy Rucker's _The Fourth Dimension and How to Get There_, Richard Hamming's [ADSE book](https://en.wikipedia.org/wiki/The_Art_of_Doing_Science_and_Engineering), Michael Pyrcz's talk on the curse of dimensionality, and my colleague Floris Elzinga's talk at Equinor's developer conference in 2024. Cheers!**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The unit square\n",
    "\n",
    "Although the unit square (hypercube really) has constant volume ($= 1^d$ where $d$ is the number of dimensions), **its diagonal is unbounded.** In other words, high-dimensional spaces are BIG."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6b5bd08190>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "diag = [np.linalg.norm(np.ones(dim)) for dim in range(1, 10000)]\n",
    "    \n",
    "plt.plot(diag)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sampling hyperspace\n",
    "\n",
    "It's big, so it's hard to sample.\n",
    "\n",
    "Let's say we are training a machine learning model. The features of the data are physical measurements about rocks; maybe one of them is the density of that rock. And imagine that, looking at the distribution of the densities of our rocks, we feel like we can make a decent model if our samples 'cover' 90% of the distribution.\n",
    "\n",
    "So if we have two features, and we have sampled 80% of their respective distributions, then how much of the joint distribution have we sampled? Well, if the properties are independent, then it's $0.8 \\times 0.8 = 0.64$, or 64% of the distribution.\n",
    "\n",
    "How much of the joint distribution have we sampled if we have 100 features?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0370359763344975e-08"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "100 * 0.8 ** 100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Only 0.00000002%, that's not much!\n",
    "\n",
    "What does this mean for the amount of data collection we need to do? Well, let's say it took 10 equally-spaced samples to represent that 80% of the distribution for one feature. Then 2 features will require $10^2 = 100$ samples and **100 features is going to need a googol samples**. I probably cannot afford to collect that much data."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fortunately, our features are usually not independent, or there are structures in the data we can exploit. But the basic point is that you are going to need a lot of samples and/or smart sampling methods for high-dimensional problems. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Volume of unit ball\n",
    "\n",
    "The unit hypercube always has the same volume: 1. But the volume of the unit ball varies quite a bit:\n",
    "\n",
    "- In 1D it is 2. It's the length of the diameter, basically, since the 'ball' is just a line.\n",
    "- In 2D it is $\\pi r^2$, so $\\pi$ since $r = 1$.\n",
    "- In 3D it is $\\tfrac{4}{3} \\pi$\n",
    "- In 4D and above, we'll need a function to compute it...\n",
    "\n",
    "For radius $R$ in $d$ dimensions:\n",
    "\n",
    "$$ V_d(R) = \\frac{\\pi^{d/2}}{\\Gamma\\bigl(\\tfrac d2 + 1\\bigr)}R^d$$\n",
    "\n",
    "Let's use this to compute how the volume varies as a function of dimensionality:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.special import gamma\n",
    "\n",
    "def vol(d, r=1):\n",
    "    return r**d * np.pi**(d/2) / gamma(d/2 + 1)\n",
    "\n",
    "d = np.arange(1, 50)\n",
    "vols = vol(d)\n",
    "maxx = np.argmax(vols) + 1\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(15, 3))\n",
    "ax.plot(d, vols, 'o-')\n",
    "ax.axvline(maxx, c='r', alpha=0.5)\n",
    "ax.set_yscale('log')\n",
    "ax.grid(c='k', alpha=0.1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So the unit ball has maximum volume of 5.26 in 5 dimensions. Above that, ball's volume _decreases_, and with only 100 dimensions its volume is already tiny: $2.4 \\times 10^{-40}$. Amazing."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(2.368202101882829e-40)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vol(100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**So high-dimensional spheres enclose almost no space.** Yet the circumscribed hypercube of side 2 (i.e. the cube that contains the unit ball) grows exponentially in volume:\n",
    "\n",
    "- 1D: 2\n",
    "- 2D: 2 &times; 2 = 4 square units\n",
    "- 3D: 2 &times; 2 &times; 2 = 8 cubic units\n",
    "- etc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(15, 3))\n",
    "ax.plot(d, d**2, 'o-')\n",
    "ax.set_yscale('log')\n",
    "ax.grid(c='k', alpha=0.1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To put it another way, **the volume of the hypercube is all in the corners**. And a 100-dimensional hypercube has a lot of corners!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1267650600228229401496703205376"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2**100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The volume is in the shell\n",
    "\n",
    "What proportion of the unit 3D sphere's volume is within 0.01 of the surface?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.02970100000000011)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def vol_frac(d, s):\n",
    "    return (vol(d, r=1) - vol(d, r=(1 - s))) / vol(d, r=1)\n",
    "\n",
    "vol_frac(d=3, s=0.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Only about 3% in a 3-dimensional sphere.\n",
    "\n",
    "But in 100 dimensions, 63% of the volume is in this \"1% shell\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.6339676587267707)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vol_frac(d=100, s=0.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Distance between points\n",
    "\n",
    "High dimensional space is BIG (see _The unit square_, above).\n",
    "\n",
    "One consequence of this is that **random points in the unit cube become very far apart**.\n",
    "\n",
    "But what is really strange is that **these random points are approximately equally distant from each other**.\n",
    "\n",
    "So a high dimensional predictive model, i.e. one with a lot of parameters, might struggle to discriminate between samples: everything is far apart, and everything is about the same distance apart. This is one reason why KNNs are less useful for data with a lot of features (like images)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1500x300 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.spatial.distance import pdist, squareform\n",
    "\n",
    "def dists(dim, n=1000, scale=1):\n",
    "    X = np.random.random(size=(n, dim))\n",
    "    return pdist(X)\n",
    "\n",
    "dims = [2, 20, 200, 2000]\n",
    "\n",
    "fig, axs = plt.subplots(ncols=len(dims), figsize=(15, 3), sharey=True)\n",
    "\n",
    "for dim, ax in zip(dims, axs):\n",
    "    d = dists(dim=dim, n=200)\n",
    "    ax.hist(d)\n",
    "    ax.set_xlim(0, 20)\n",
    "    ax.set_title(f'{dim}-D')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The inscribed hypersphere\n",
    "\n",
    "A famous 'experiment' is to inscribe a circle between 4 unit circles:\n",
    "\n",
    "![image](https://media.cheggcdn.com/media/241/241d8144-ba5c-4ca8-9ecc-fbdbbf720152/phpMYRRRJ)\n",
    "\n",
    "Then to imagine how the radius of this inscribed circle depends on the dimensionality of the problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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XsmXL9Ic//EE2my1ZJQMpaf369ZKkK664YtDxRx99VLfeeqsksd6AOGptbdXNN9+s5uZmORwOXXDBBdq4caNWrFghifUGJBLrDYifI0eO6Prrr1dHR4eKi4t18cUX66233orkIxN9vZkMwzCSXQQAAAAAAAAwnpgJBgAAAAAAgLRHCAYAAAAAAIC0RwgGAAAAAACAtEcIBgAAAAAAgLRHCAYAAAAAAIC0RwgGAAAAAACAtEcIBgAAAAAAgLRHCAYAAAAAAIC0RwgGAAAAAACAtEcIBgAAAAAAgLRHCAYAAAAAAIC09/8DEfnqOfuPNqMAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 1500x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def inscribed_r(d):\n",
    "    return np.sqrt(d) - 1\n",
    "\n",
    "radiuses = inscribed_r(d)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(15, 3))\n",
    "ax.plot(d, radiuses, 'o-')\n",
    "ax.axhline(2, c='r', alpha=0.5)\n",
    "ax.set_yscale('log')\n",
    "ax.grid(c='k', alpha=0.1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The bizarre thing is that at about 10 dimensions, **the radius of the inscribed hypersphere extends _beyond_ the 'enclosing' unit hypersheres!** And although it might look asymptotic in this plot, it is unbounded.\n",
    "\n",
    "This seems very strange, until you remember that the space is 'spiky': the volume is in the corners.\n",
    "\n",
    "Richard Hamming says in his book that a consequence of this is that you might need to look quite far from the average values to find optimal solutions to high-dimensional design problems, since the volume of the design space near the mean is basically 0."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "&copy; 2024 Matt Hall, licensed CC BY, please re-use this work."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "py311",
   "language": "python",
   "name": "py311"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}