Fix rendering

docs/tutorials/MetaOT.ipynb

@@ -406,7 +406,7 @@
     "## Creating the initializer\n",
     "\n",
     "We can create the initializer by providing it with the shared\n",
-    "geometry of the problem instances. First, we create an MLP, which takes in the concatenated :math:`\\alpha` :math:`\\beta` probability measures and outputs the potential."
+    "geometry of the problem instances. First, we create an MLP, which takes in the concatenated $\\alpha$ and $\\beta$ probability measures and outputs the potential."
    ]
   },
   {
@@ -671,7 +671,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.10"
+   "version": "3.10.6"
   }
  },
  "nbformat": 4,

docs/tutorials/Monge_Gap.ipynb

@@ -459,14 +459,11 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We can now examine how these maps vary, depending on whether the Monge gap was used or not."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## No Monge Gap\n",
+    "## Experiments\n",
+    "\n",
+    "We can now examine how these maps vary, depending on whether the Monge gap was used or not.\n",
+    "\n",
+    "### No Monge Gap\n",
     "When fitting without a Monge gap, one recovers an arbitrary push-forward, which has no reason to be optimal for any cost."
    ]
   },
@@ -535,6 +532,8 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
+    "### Monge gap with $c = \\ell_2$\n",
+    "\n",
     "For the Monge gap with $c = \\ell_2$, we obtain a \"needle\" alignment (without crossing lines) because $c$ is a distance: this is known as the Monge Mather shortening principle (see e.g. {cite}`villani:09`)."
    ]
   },
@@ -605,6 +604,8 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
+    "### Monge gap with $c = \\ell_2^2$\n",
+    "\n",
     "For $c = \\ell_2^2$, we observe crossings when the sum of the squared diagonals of the quadrilateral induced by 4 points $(x_1, x_2, T(x_1), T(x_2))$ is lower than the sum of the squared sides. "
    ]
   },
@@ -688,7 +689,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.10"
+   "version": "3.10.6"
   }
  },
  "nbformat": 4,