Identification of unknown micro- and nano-sized mineral phases is commonly achieved by analyzing chemical maps generated from hyperspectral imaging datasets, particularly scanning electron microscope - energy dispersive X-ray spectroscopy (SEM-EDS). However, the accuracy and reliability of mineral identification are often limited by subjective human interpretation, non-ideal sample preparation, and the presence of mixed chemical signals generated within the electron-beam interaction volume. Machine learning has emerged as a powerful tool to overcome these problems. Here, we propose a machine-learning approach to identify unknown phases and unmix their overlapped chemical signals. This approach leverages the guidance of Gaussian mixture modeling clustering fitted on an informative latent space of pixel-wise elemental datapoints modeled using a neural network autoencoder, and unmixes the overlapped chemical signals of phases using non-negative matrix factorization. We evaluate the reliability and the accuracy of the new approach using two SEM-EDS datasets: a synthetic mixture sample and a real-world particulate matter sample. In the former, the proposed approach successfully identifies all major phases and extracts background-subtracted single-phase chemical signals. The unmixed chemical spectra show an average similarity of 83.0% with the ground truth spectra. In the second case, the approach demonstrates the ability to identify potentially magnetic Fe-bearing particles and their background-subtracted chemical signals. We demonstrate a robust approach that brings a significant improvement to mineralogical and chemical analysis in a fully automated manner. The proposed analysis process has been built into a user-friendly Python code with a graphical user interface for ease of use by general users.