Learning Structurally Distinct Factors in Many-Body Approximation for Tensors

Decomposing tensors in an interpretable manner is essential in data analysis and knowledge discovery. Many-body approximation (MBA) for non-negative tensors is a tensor decomposition method that allows each factor to be interpreted as an interaction among a subset of modes of a given tensor. In this paper, we first uncover a structural dependency in MBA: factors containing common modes necessarily share identical slices, regardless of the model parameters. To remove this structural dependency while retaining the convex optimization procedure of MBA, we propose extended many-body approximation (EMBA), which extends each mode of the input tensor by one dimension before applying MBA. We prove that entries selected from different EMBA factors are not identical as functions of the parameters, thereby eliminating the structural dependency present in MBA. Experiments on reconstruction using reduced set of factors and recursive approximation show the improved explanatory capacity and robustness of EMBA over MBA. In factor quantization experiments, EMBA almost perfectly reconstructs synthetic data generated from Boolean factors even under 1-bit quantization. These results demonstrate that EMBA enables structurally distinct factors whose utility extends beyond full-tensor reconstruction, highlighting the importance of factor structure in tensor decomposition.