5G mmWave communication systems have promising properties for high precision user localization and environment mapping. Such information is of high value for emerging applications such as connected automated driving (CAD), and it has also potential to be explored to substantially improve efficiency and reliability of mmWave communications itself. However, the acquisition of such information cannot come with too large overhead in the system. Existing studies have so far relied on complete channel measurements, implying a prohibitive channel training overhead. In this paper, we exploit the possibility of 5G positioning using partial channel measurements. We utilize a tensor completion technique to recover the complete channel information from low rank channel measurements. Simulation results demonstrate the trade-off between user positioning accuracy and channel training overhead, and show that sub-meter precision with negligible performance loss is feasible at sample ratio reductions of at least 30%, and meter level precision is achievable with sample ratio reduction of 50%. CCS CONCEPTS • Communication hardware, interfaces and storage → Signal processing systems.