Communication with the goal of accurately conveying meaning, rather than accurately transmitting symbols, has become an area of growing interest. This paradigm, termed semantic communication, typically leverages modern developments in artificial intelligence and machine learning to improve the efficiency and robustness of communication systems. However, a standard model for capturing and quantifying the details of "meaning" is lacking, with many leading approaches to semantic communication adopting a black-box framework with little understanding of what exactly the model is learning. One solution is to utilize the conceptual spaces framework, which models meaning explicitly in a geometric manner. Though prior work studying semantic communication with conceptual spaces has shown promising results, these previous attempts involve hand-crafting a conceptual space model, severely limiting the scalability and practicality of the approach. In this work, we develop a framework for learning a domain of a conceptual space model using only the raw data with high-level property labels. In experiments using the MNIST and CelebA datasets, we show that the domains learned using the framework maintain semantic similarity relations and possess interpretable dimensions.