Update README.md

Add examples

Author: sorenhauberg

README.md

@@ -46,17 +46,68 @@ print(J.shape) # jacobian between input and output: torch.size([100, 5, 10])
 
 ### `stochman.manifold`: Interface for working with Riemannian manifolds
 
+A manifold can be constructed simply by specifying its metric. The example below shows a toy example where the metric grows with the distance to the origin.
+
+``` python
+import torch
+from stochman.manifold import Manifold
+
+class MyManifold(Manifold):
+    def metric(self, c, return_deriv=False):
+        N, D = c.shape  # N is number of points where we evaluate the metric; D is the manifold dimension
+        sq_dist_to_origin = torch.sum(c**2, dim=1, keepdim=True)  # Nx1
+        G = (1 + sq_dist_to_origin).unsqueeze(-1) * torch.eye(D).repeat(N, 1, 1)  # NxDxD
+        return G
+        
+model = MyManifold()
+p0, p1 = torch.randn(1, 2), torch.randn(1, 2)
+c, _ = model.connecting_geodesic(p0, p1)  # geodesic between two random points
+```
+
+If you manifold is embedded (e.g. an autoencoder) then you only have to provide a function for realizing the embedding (i.e. a decoder) and StochMan takes care of the rest (you, however, have to learn the autoencoder yourself).
+
+``` python
+import torch
+from stochman.manifold import EmbeddedManifold
+
+class Autoencoder(EmbeddedManifold):
+    def embed(self, c, jacobian = False):
+        return self.decode(c)
+        
+model = Autoencoder()
+p0, p1 = torch.randn(1, 2), torch.randn(1, 2)
+c, _ = model.connecting_geodesic(p0, p1)  # geodesic between two random points
+```
 
 ### `stochman.geodesic`: computing geodesics made easy!
 
+Geodesics are energy-minimizing curves, and StochMan computes them as such. You can use the high-level `Manifold` interface or the more explicit one:
 
-### `stochman.curves`: Simple curve objects
+``` python
+import torch
+from stochman.geodesic import geodesic_minimizing_energy
+from stochman.curves import CubicSpline
 
+model = MyManifold()
+p0, p1 = torch.randn(1, 2), torch.randn(1, 2)
+curve = CubicSpline(p0, p1)
+geodesic_minimizing_energy(curve, model)
+```
 
+### `stochman.curves`: Simple curve objects
 
+We often want to manipulate curves when computing geodesics. StochMan provides an implementation of cubic splines and discrete curves, both with the end-points fixed.
 
+``` python
+import torch
+from stochman.curves import CubicSpline
 
+p0, p1 = torch.randn(1, 2), torch.randn(1, 2)
+curve = CubicSpline(p0, p1)
 
+t = torch.linspace(0, 1, 50)
+ct = curve(t)  # 50x2
+```
 
 ## Licence
 
@@ -73,10 +124,3 @@ If you want to cite the framework feel free to use this (but only if you loved i
   year={2021}
 }
 ```
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