code: def main():
      r"""Compute and visualize a geodesic regression on the SE(2).
  
      The generative model of the data is:
      :math:`Z = Exp_{\beta_0}(\beta_1.X)` and :math:`Y = Exp_Z(\epsilon)`
      where:
      - :math:`Exp` denotes the Riemannian exponential,
      - :math:`\beta_0` is called the intercept,
      - :math:`\beta_1` is called the coefficient,
      - :math:`\epsilon \sim N(0, 1)` is a standard Gaussian noise,
      - :math:`X` is the input, :math:`Y` is the target.
      """
      space = SpecialEuclidean(2)
  
      # Generate noise-free data
      n_samples = 20
      X = gs.random.normal(size=(n_samples,))
      X -= gs.mean(X)
  
      intercept = space.random_point()
      coef = space.to_tangent(5.0 * gs.random.rand(3, 3), intercept)
      y = space.metric.exp(X[:, None, None] * coef[None], intercept)
  
      # Generate normal noise in the Lie algebra
      normal_noise = gs.random.normal(size=(n_samples, 3))
      normal_noise = space.lie_algebra.matrix_representation(
          normal_noise
      )
      noise = space.tangent_translation_map(y)(normal_noise) / gs.pi
  
      rss = gs.sum(space.metric.squared_norm(noise, y)) / n_samples
  
      # Add noise
      y = space.metric.exp(noise, y)
  
      # True noise level and R2
      estimator = FrechetMean(space)
      estimator.fit(y)
      variance_ = variance(space, y, estimator.estimate_)
      r2 = 1 - rss / variance_
  
      # Fit geodesic regression
      gr = GeodesicRegression(
          space,
          center_X=False,
          method="riemannian",
          initialization="frechet",
          compute_training_score=True,
      ).set(
          max_iter=100,
          init_step_size=0.1,
          verbose=True,
      )
      gr.fit(X, y)
  
      intercept_hat, beta_hat = gr.intercept_, gr.coef_
  
      # Measure Mean Squared Error
      mse_intercept = space.metric.squared_dist(
          intercept_hat, intercept
      )
      mse_beta = space.metric.squared_norm(
          space.metric.parallel_transport(
              beta_hat,
              intercept_hat,
              space.metric.log(intercept_hat, intercept),
          )
          - coef,
          intercept,
      )
  
      # Measure goodness of fit
      r2_hat = gr.training_score_
  
      print(f"MSE on the intercept: {mse_intercept:.2e}")
      print(f"MSE on the initial velocity beta: {mse_beta:.2e}")
      print(f"Determination coefficient: R^2={r2_hat:.2f}")
      print(f"True R^2: {r2:.2f}")
  
      # Plot
      fitted_data = gr.predict(X)
      fig = plt.figure(figsize=(8, 8))
      ax = fig.add_subplot(111)
      sphere_visu = visualization.SpecialEuclidean2()
      ax = sphere_visu.set_ax(ax=ax)
  
      path = space.metric.geodesic(
          initial_point=intercept_hat, initial_tangent_vec=beta_hat
      )
      regressed_geodesic = path(gs.linspace(min(X), max(X), 100))
  
      sphere_visu.draw_points(ax, y, marker="o", c="black")
      sphere_visu.draw_points(ax, fitted_data, marker="o", c="gray")
      sphere_visu.draw_points(
          ax, gs.array([intercept]), marker="x", c="r"
      )
      sphere_visu.draw_points(
          ax, gs.array([intercept_hat]), marker="o", c="green"
      )
  
      ax.plot(
          regressed_geodesic[:, 0, 2],
          regressed_geodesic[:, 1, 2],
          c="gray",
      )
      plt.show()
  
imports: import matplotlib.pyplot
  import geomstats.backend
  import geomstats.visualization
  from geomstats.geometry.special_euclidean import SpecialEuclidean
  from geomstats.learning.frechet_mean import FrechetMean, variance
  from geomstats.learning.geodesic_regression import GeodesicRegression
label_desc: Use image node_id 1 for calling a global function with example usage: main() without return types