Local refinement has emerged as a pivotal topic in the IGA community, driven by its crucial role in enabling efficient and accurate numerical simulations. The introduction of truncated hierarchical B-splines, which maintain the partition of unity with smaller B-spline support, made a significant advancement in the process of local refinement. This talk introduces a novel approach of using low-rank methods combined with the alternating minimal energy solver to efficiently assemble mass and stiffness tensors, when the solution is approximated using THB-splines. We follow an additive heuristic for the level-wise assembly of these system tensors. In order to make it a low-rank method, the presented approach detects and utilizes local tensor product structures and interpolates the weight functions by using the tensor train format to be able to execute univariate quadrature, which addresses the challenge of high memory requirements and efficient computation. Although not universally applicable, our method can be effective in high-dimensional model problems where the refinement areas are less complex. Preliminary numerical tests using MATLAB and the GeoPDEs toolbox validate our approach, highlighting its potential in reducing computational bottlenecks in specific scenarios.