Structural Health Monitoring (SHM), i.e. the action of monitoring structures in real-time and in an automated manner, is a major challenge in several industrial fields such as aeronautic. SHM is by nature a very high dimensional data-driven problem that possesses several specificities when addressed as a machine learning problem. First of all data in damaged cases are rare and very costly as the generation of damaged data is not always possible and simulations are not reliable especially when dealing with complex structures. SHM is thus by nature an unsupervised problem. Furthermore, any incoming sample should be instantaneously clustered and handcrafted damage indexes are commonly used as a first dimension reduction step due to large datasets to be processed. As a consequence, unsupervised dimensionality reduction (DR) techniques that project very high dimensional data into a two or three-dimensional space (such as t-SNE or UMAP) are very appealing in such a context. However, these methods suffer from one major drawback which is that they are unable to cluster any unknown incoming sample. To solve this we propose to add inductive abilities to these well know methods by associating their projection bases with Deep Neural Networks (DNNs). The resulting DNNs are then able to cluster any incoming unknown samples. Based on those tools, a SHM methodology allowing for unsupervised damage clustering with dimensionality reduction is presented here. To demonstrate the effectiveness of the method, results of damage classification on large experimental data sets coming from complex aeronautical composite structures monitored through Lamb waves are shown. Furthermore, several DR techniques have been benchmarked and recommendations are derived. It is demonstrated that the use of raw Lamb wave signals instead of the associated damage indexes is more effective. This non-intuitive result helps to reduce the gap between laboratory research and the actual start-up of SHM activities in industrial applications.