Motion planning for high-DOF multi-arm systems operating in complex environments remains a challenging problem, with many motion planning algorithms requiring evaluation of the minimum collision distance and its derivative. Because of the computational complexity of calculating the collision distance, recent methods have attempted to leverage data-driven machine learning methods to learn the collision distance. Because of the significant training dataset requirements for high-DOF robots, existing kernel-based methods, which require $O(N^2)$ memory and computation resources, where $N$ denotes the number of dataset points, often perform poorly. This paper proposes a new active learning method for learning the collision distance function that overcomes the limitations of existing methods: (i) the size of the training dataset remains fixed, with the dataset containing more points near the collision boundary as learning proceeds, and (ii) calculating collision distances in the higher-dimensional link $SE(3)^n$ configuration space – here $n$ denotes the number of links – leads to more accurate and robust collision distance function learning. Performance evaluations with high-DOF multi-arm robot systems demonstrate the advantages of the proposed active learning-based strategy vis-$\grave{\text{a}}$-vis existing learning-based methods.