A Computational Study of Low-Rank Approximation for q-Rung Orthopair Fuzzy Matrices Using Truncated Singular Value Decomposition

Source abstract

In decision support systems, the amount of high-dimensional fuzzy data is increasing at an alarming rate, which is a huge challenge in storing and computing such data. A prevalent method to tackle these issues is to use the truncated singular value decomposition (TSVD) of low-rank matrices, which can help reduce the data size while retaining key information. Based on the view of a q-rung orthopair fuzzy (q-ROF) matrix, this paper is a comprehensive computational study on applying TSVD to a q-ROF matrix, which is a kind of generalization of fuzzy sets that is introduced to flexibly express and manage uncertainty. We systematically study the trade-offs among reconstruction fidelity, compression ratio, and computation cost by separately approximating the membership and non-membership matrices for different values of q: 1.0, 2.0, and 3.0. The rate of singular value decay is very different, as shown in our results; the membership matrix decays rapidly and can be compressed with very little error, while the non-membership matrix decays gently and needs high ranks to be reconstructed accurately. We consider the AIC, BIC, and cumulative energy based rank selection strategies, and find the Pareto optimal configurations for deployment in practice. Based on the synthetic data sets created following the proposed framework, statistical tests show that q = 1.0 is more accurate than higher values of q. The results give a quantitative basis for efficient fuzzy information system design and give clear guidelines to practitioners on when to favor compression or fidelity.