class_docstr: Class for modeling a linear 1D localization problem.
  
  The state is made of a scalar position and scalar speed, thus a 2D vector.
  A sensor provides acceleration inputs, while another one provides sparse
  measurements of the position.
docstr: Compute the matrix associated to the propagation noise.
  
  The noise is supposed additive and only applies to the speed part.
  The Jacobian is given by :math:`\begin{bmatrix} 0 & \sqrt{dt}
  \end{bmatrix}`.
  
  Parameters
  ----------
  state : unused
  sensor_input : array-like, shape=[2]
      Vector representing the information from the accelerometer.
  
  Returns
  -------
  jacobian : array-like, shape=[dim_noise, dim]
      Jacobian of the propagation w.r.t. the noise.
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code: def noise_jacobian(self, state, sensor_input):
      r"""Compute the matrix associated to the propagation noise.
  
      The noise is supposed additive and only applies to the speed part.
      The Jacobian is given by :math:`\begin{bmatrix} 0 & \sqrt{dt}
      \end{bmatrix}`.
  
      Parameters
      ----------
      state : unused
      sensor_input : array-like, shape=[2]
          Vector representing the information from the accelerometer.
  
      Returns
      -------
      jacobian : array-like, shape=[dim_noise, dim]
          Jacobian of the propagation w.r.t. the noise.
      """
      dt, _ = sensor_input
      dim = self.group.dim
      position_wrt_noise = gs.zeros((dim // 2, dim // 2))
      speed_wrt_noise = gs.sqrt(dt) * gs.eye(dim // 2)
      jac = gs.vstack((position_wrt_noise, speed_wrt_noise))
      return jac
  
imports: import geomstats.backend
  from geomstats.geometry.euclidean import Euclidean
  from geomstats.geometry.matrices import Matrices
  from geomstats.geometry.special_euclidean import SpecialEuclidean
label_desc: Use image node_id 4 for calling the LocalizationLinear obj's underlying member method code with example usage: obj.noise_jacobian(state, sensor_input) and returns: jac