Yesterday we saw that the E.M. G.M.M algorithm approaches the true distribution if allocated a sufficient number of iterations. However, for the test conducted yesterday, the Dirichlet hyperparameter ( $\alpha$ ) — the tuning parameter on the categorical distribution, depicted in Figure 1 and Figure 2 — was fixed at $1e^{-6}$. Today, we would like to see the sensitivity of the estimator to the value of $\alpha$.



Figure 1 :: Infinite Dirichlet Process





Figure 2 :: Alpha Distribution Over Probability Simplex



To conduct this test, a simulated dataset was generated. This simulated dataset contains 10,000 inlier points and 3,000 outlier points. A scatter of the set is depicted in Figure 3, where the black points represent the inliers and the red points represent the outliers.



Figure 3 :: Simulated Distribution



Using the distribution shown above, the E.M. G.M.M algorithm was utilized with a varying $\alpha$ value to test the estimators sensitivity w.r.t $\alpha$. In Figure 4 the error of the estimated covariance w.r.t the $\alpha$ value is depicted. From this figure, it can be seen that the error tends to decreases as the value taken by the $\alpha$ parameter decreases.



Figure 4 :: Estimated Inlier Covariance Error



In addition to the estimator error, we can also look at the complexity of the mixture model selected for each $\alpha$ value. In Figure 5 the number of mixture components is shown w.r.t the value taken by the $\alpha$ parameter. From this figure it can be seen that the mixture model tends to only estimating one inlier and one outlier distribution as the $\alpha$ parameter decreases.



Figure 5 :: Number of Mixture Components



From this simple experiment, we can see that our estimator is not incredibly sensitive to the value taken by the hyperparameter; however, optimal results ( w.r.t. estimator error and complexity ) are achieved by selecting a small $\alpha$ value. To validate this, later we will run a larger Monte-carlo style simulation to show that the trends hold over multiple trials.



Next Steps

Starting testing our mixture model on pose-graphs in G2O.