Adaptive Low-Rank Tensor Approximation Based on Mixed-Integer Representations

Tensor structures are fundamental in addressing the intricate challenges posed by high-dimensional data across a spectrum of scientific and computational domains. Within this context, low-rank tensor approximation plays a pivotal role in enhancing data processing efficiency. This paper develops a novel adaptive low-rank tensor approximation method by introducing mixed-integer representations to identify an appropriate low-rank approximation for high-dimensional tensors. The approach takes into consideration both tensor rank determination and approximation accuracy, leveraging binary variables to represent tensor ranks that will be optimized as unknowns with the tensor arrays. By integrating the alternating least squares technique with the truncation method, the integrated algorithm effectively achieves a proper low-rank tensor with high approximation accuracy. To substantiate its efficacy and efficiency in solving tensor approximation problems, the paper provides extensive simulation results and analysis.