## 3 Results
### Tempered EM: Application to Mixtures of Gaussian  
#### 3.2.1 Context and Experimental Protocol  
In this section, we will assess the capacity of tmp-EM to escape from deceptive local maxima. We compare the classical EM to tmp-EM with both a monotonous and an oscillating temperature profile on a very well know toy example: likelihood maximisation within the Gaussian Mixture Model. We confront the algorithms to situations where the true classes have increasingly more ambiguous positions, combined with initialisations designed to be hard to escape from. Although the EM is an optimisation procedure, and the log-likelihood reached is a critical metric, in this example, we put more emphasis on the correct positioning of the cluster centroids, that is to say on the recovery of the \(\mu_{k}\). The other usual metrics are also in favour of tmp-EM, and can be found in Supplementary Materials. For the sake of comparison, the experimental design is similar to the one in [10] on the tmp-SAEM. It is as follows: we have three clusters of similar shape and same weight. One is isolated and easily identifiable. The other two are next to one another, in a more ambiguous configuration. Figure 4 represents the three, gradually more ambiguous configurations. Each configuration is called a \"parameter family\". We use two different initialisation types to reveal the behaviours of the three EMs. The first--which we call _\"barycenter\"_--puts all three initial centroids at the centre of mass of all the observed data points. However, none of the EM procedures would move from this initial state if the three GMM centroids were at the exact same position, hence we actually apply a tiny perturbation to make them all slightly distinct. The blue crosses on Figure 5 represent a typical _barycenter_ initialisation. With this initialisation method, we assess whether the EM procedures are able to correctly estimate the positions of the three clusters, despite the ambiguity, when starting from a fairly neutral position, providing neither direction nor misdirection. On the other hand, the second initialisation type - which we call _\"2v1\"_ - is voluntarily misguiding the algorithm by positioning two centroids on the isolated right cluster and only one centroid on the side of the two ambiguous left clusters. The blue crosses on Figure 6 represent a typical _2v1_ initialisation. This initialisation is intended to assess whether the methods are able to escape the potential well in which they start and make theirs centroids traverse the empty space between the left and right clusters to reach their rightful position. For each of the three parameter families represented on  
Figure 3: (**a**) Visual representation of the number of Riemann intervals over the EM steps for each profile \(\varphi_{i}\). In higher dimension, the computational complexity of the profiles are very different. More precisely, the total number of Riemann squares computed over 100 EM iterations are: 4534 for “square root”, 125,662 for “5 square root”, 348,550 for “low” and 2,318,350 for “medium”. (**b**) For each profile, average evolution of the negative log-likelihood, with standard deviation, over 50 simulations. The “square root” profile performs poorly. However, the other three are comparable despite their different computational complexities. Figure 4, 1000 datasets with 500 observations each are simulated, and the three EMs are ran with both the _barycenter_ and the _2v1_ initialisation. For tmp-EM, we try two profiles. First, a simple _decreasing_ exponential profile as seen in [31]: \(T_{n}=1+(T_{0}-1)\)\(\exp(-r.n)\). Through a grid search, the values \(T_{0}=5\), \(r=2\) for the _barycenter_ initialisation and \(T_{0}=100\,r=1.5\) for the _2v1_ initialisation are picked for this profile. Since Theorem 3 only requires \(T_{n}\longrightarrow 1\), we also propose an _oscillating_ profile inspired from [10]. The exact formula of these oscillations is: \(T_{n}=th(\frac{n}{2r})+(T_{0}-b\frac{2\sqrt{2}}{3\pi})\,a^{n/r}+b\,sinc(\frac{ 3\pi}{4}+\frac{n}{r})\).