Key Idea

The multi-variate Gaussians are represented in their canonical representation, by precision/information matrix \(\Omega\) and the information vector \(\xi\), where \(\Omega = \Sigma^{-1}\), and \(\xi = \Sigma^{-1} \mu\).

The Gaussian can be redefined as follows:

\begin{equation} p(x) = \eta \text{exp} \left\{ - \frac{1}{2} x^T \Omega x + x^T \xi \right\} \end{equation}

where \(\eta\) has been redefined to subsume a constant. The reason they are called information matrix and vectors is because \(- \log p(x)\) is quadratic in \(\Omega\) and \(\xi\).

For Gaussians, \(\Omega\) is positive semi-definite, so \(- \log p(x)\) is a quadratic distance function with mean \(\mu = \Omega^{-1} \xi\). The matrix \(\Omega\) determines the rate at which the distance function inccreases is different dimensions of the variable \(x\). A quadratic distance that is weighted by a matrix \(\Omega\) is called Mahalanobis distance.

Algorithm

\begin{algorithm} \caption{Information Filter} \label{information_filter} \begin{algorithmic}[1] \Procedure{InformationFilter}{$\xi_{t-1}, \Omega_{t-1}, \mu_t, \z_t$} \State $\overline{\Omega}_t = (A_t \Omega_{t-1}^{-1} A_t^T + R_t)^{-1}$ \State $\overline{\xi}_t = \overline{\Omega_t}\left( A_t \Omega_{t-1}^{-1} \xi_{t-1} + B_t u_t \right)$ \State $\Omega_t = C_t^T Q_t^{-1} C_t + \overline{\Omega}_t$ \State $\xi_t = C_t^T Q_t^{-1}z_t + \overline{\xi}_t$ \State \Return $\xi_t, \Omega_t$ \EndProcedure \end{algorithmic} \end{algorithm}

Pros

  1. Representing global uncertainty is simple: \(\Omega = 0\). With moments, global uncertainty amounts to covariance of infinite magnitude.
  2. More numerically stable for many applications.
  3. Natural fit for multi-robot problems, where sensor data is collected decentrally. Information integration is additive and achieved by summing information from multiple robots. This is because the canonical parameters represent a probability in log form.
  4. Information matrix may be sparse, lending itself to algorithms that are computationally efficient.

Cons

  1. The update step requires the recovery of a state estimate, inverting the information matrix. Matrix inversion is computationally expensive.