Add LaTeX for norm decomposition and plot titles

Atcold · Sep 27, 2021 · 5ec56a3 · 5ec56a3
1 parent bb97aaf
commit 5ec56a3
Showing 1 changed file with 63 additions and 50 deletions.
diff --git a/extra/a-projections.ipynb b/extra/a-projections.ipynb
@@ -10,7 +10,7 @@
     "import random\n",
     "import torch\n",
     "from plot_lib import set_default, show_mat\n",
-    "from matplotlib.pyplot import plot, subplot, axhline, axvline, legend\n",
+    "from matplotlib.pyplot import plot, subplot, axhline, axvline, legend, suptitle, title\n",
     "from numpy import pi as π\n",
     "from numpy import sqrt, cos, arange, arccos"
    ]
@@ -103,7 +103,8 @@
     "ax1.legend([f'{d}D' for d in d_range], ncol=3)\n",
     "ax1.set_ylim(0, 350)\n",
     "ax2.legend([f'{d}D' for d in d_range], ncol=3)\n",
-    "ax2.set_rlim(0, 350)"
+    "ax2.set_rlim(0, 350)\n",
+    "suptitle('Normalised scalar product distribution')"
    ]
   },
   {
@@ -116,36 +117,60 @@
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": null,
+   "cell_type": "markdown",
    "metadata": {},
-   "outputs": [],
    "source": [
-    "# Where are those two blue spikes coming from?\n",
-    "ϕ = arange(0, 1, 1/100) * π\n",
-    "f = cos(ϕ)"
+    "$$\\Vert a - b \\Vert = \\sqrt{\\Vert a \\Vert^2 + \\Vert b \\Vert^2 - 2 a^\\top b}$$"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {},
+   "metadata": {
+    "scrolled": false
+   },
    "outputs": [],
    "source": [
-    "plot(ϕ, f)\n",
-    "axhline()\n",
-    "axvline()\n",
-    "legend(['cosine'])"
+    "# Compute the histogram of points uniformly distributed on a unit sphere in d dimensions\n",
+    "r = 3\n",
+    "bins = 200 + 1\n",
+    "N = 10_000\n",
+    "d_range = (2, 3, 4, 6, 10, 16)\n",
+    "ax1 = subplot(111)\n",
+    "for d in d_range:\n",
+    "    A = torch.randn(N, d) / sqrt(d)\n",
+    "    B = torch.randn(N, d) / sqrt(d)\n",
+    "    h = torch.histc(torch.cdist(A[None,:,:], B[None,:,:]).view(-1), bins, 0, r) / N\n",
+    "    ax1.plot(torch.linspace(0, r, bins).numpy(), h.numpy())\n",
+    "ax1.legend([f'{d}D' for d in d_range], ncol=3)\n",
+    "ax1.set_ylim(ymin=0);\n",
+    "title('Distribution of distances in D dimensions')"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {},
+   "metadata": {
+    "scrolled": false
+   },
    "outputs": [],
    "source": [
-    "c = arange(-1+1/100, 1, 1/100)\n",
-    "g = arccos(c)"
+    "# Compute the histogram of points uniformly distributed on a unit sphere in d dimensions\n",
+    "r = 1\n",
+    "bins = 200 + 1\n",
+    "N = 10_000\n",
+    "d_range = (2, 3, 4, 6, 10, 16)\n",
+    "ax = subplot(111)\n",
+    "for d in d_range:\n",
+    "    A = torch.randn(N, d) / sqrt(d)\n",
+    "    B = torch.randn(N, d) / sqrt(d)\n",
+    "    dist = torch.cdist(A[None,:,:], B[None,:,:])\n",
+    "    h = torch.histc(dist.min(-1)[0].view(-1), bins, 0, r) / N\n",
+    "    ax.plot(torch.linspace(0, r, bins).numpy(), h.numpy())\n",
+    "l = [f'{d}D' for d in d_range]\n",
+    "ax.legend(l, ncol=3)\n",
+    "ax.set_ylim(ymin=0, ymax=.10)\n",
+    "suptitle('Distribution of min distance')"
    ]
   },
   {
@@ -154,11 +179,9 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "plot(c, g)\n",
-    "plot(c[:-1], (g[1:]-g[:-1])*100)\n",
-    "legend(['arccos', 'D[arccos]'])\n",
-    "axhline()\n",
-    "axvline()"
+    "# Where are those two blue spikes coming from?\n",
+    "ϕ = arange(0, 1, 1/100) * π\n",
+    "f = cos(ϕ)"
    ]
   },
   {
@@ -169,19 +192,20 @@
    },
    "outputs": [],
    "source": [
-    "# Compute the histogram of points uniformly distributed on a unit sphere in d dimensions\n",
-    "r = 3\n",
-    "bins = 200 + 1\n",
-    "N = 10_000\n",
-    "d_range = (2, 3, 4, 6, 10, 16)\n",
-    "ax1 = subplot(111)\n",
-    "for d in d_range:\n",
-    "    A = torch.randn(N, d) / sqrt(d)\n",
-    "    B = torch.randn(N, d) / sqrt(d)\n",
-    "    h = torch.histc(torch.cdist(A[None,:,:], B[None,:,:]).view(-1), bins, 0, r) / N\n",
-    "    ax1.plot(torch.linspace(0, r, bins).numpy(), h.numpy())\n",
-    "ax1.legend([f'{d}D' for d in d_range], ncol=3)\n",
-    "ax1.set_ylim(ymin=0);"
+    "plot(ϕ, f)\n",
+    "axhline()\n",
+    "axvline()\n",
+    "legend(['cosine'])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "c = arange(-1+1/100, 1, 1/100)\n",
+    "g = arccos(c)"
    ]
   },
   {
@@ -192,22 +216,11 @@
    },
    "outputs": [],
    "source": [
-    "# Compute the histogram of points uniformly distributed on a unit sphere in d dimensions\n",
-    "r = 1\n",
-    "bins = 200 + 1\n",
-    "N = 10_000\n",
-    "d_range = (2, 3, 4, 6, 10, 16)\n",
-    "ax = subplot(111)\n",
-    "for N in (5_000, 25_000):\n",
-    "    for d in d_range:\n",
-    "        A = torch.randn(N, d) / sqrt(d)\n",
-    "        B = torch.randn(N, d) / sqrt(d)\n",
-    "        dist = torch.cdist(A[None,:,:], B[None,:,:])\n",
-    "        h = torch.histc(dist.min(-1)[0].view(-1), bins, 0, r) / N\n",
-    "        ax.plot(torch.linspace(0, r, bins).numpy(), h.numpy())\n",
-    "l = [f'{d}D' for d in d_range]\n",
-    "ax.legend(l = l, ncol=3)\n",
-    "ax.set_ylim(ymin=0, ymax=.10)"
+    "plot(c, g)\n",
+    "plot(c[:-1], (g[1:]-g[:-1])*100)\n",
+    "legend(['arccos', 'D[arccos]'])\n",
+    "axhline()\n",
+    "axvline()"
    ]
   }
  ],