[Paper Review] A nonnegativity preserving algorithm for multilinear systems with nonsingular M-tensors

Xueli Bai, Hongjin He+2 more|arXiv (Cornell University)|Nov 25, 2018

Tensor decomposition and applications23 references4 citations

TL;DR

This paper proposes a Newton-type algorithm, Nonnegativity Preserving Algorithm (NPA), for solving multilinear systems with nonsingular M-tensors and nonnegative right-hand sides. The method ensures all iterates remain nonnegative and converges to a nonnegative solution, outperforming existing solvers when the right-hand side contains zeros, where other methods may fail to preserve nonnegativity or find sparse solutions.

ABSTRACT

This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the multilinear system under consideration involves a nonsingular $\mathcal{M}$-tensor and a nonnegative right-hand side vector, it may have multiple nonnegative solutions. In this paper, we propose an algorithm which can always preserve the nonnegativity of solutions. Theoretically, we show that the sequence generated by the proposed algorithm is a nonnegative decreasing sequence and converges to a nonnegative solution of the system. Numerical results further support the novelty of the proposed method. Particularly, when some elements of the right-hand side vector are zeros, the proposed algorithm works well while existing state-of-the-art solvers may not produce a nonnegative solution.

Motivation & Objective

• To address multilinear systems with nonsingular M-tensors and nonnegative right-hand side vectors, which may have multiple nonnegative solutions.
• To develop an algorithm that preserves nonnegativity throughout the iterative process, ensuring all iterates remain nonnegative.
• To provide a reliable solver for cases where existing methods fail to produce nonnegative solutions, especially when the right-hand side contains zero entries.
• To support the design of algorithms for sparse nonnegative tensor equations arising in data mining and numerical PDEs.

Proposed method

• Proposes a Newton-type algorithm (NPA) that modifies the standard Newton step to enforce nonnegativity of iterates.
• Uses a line search strategy to ensure the step size maintains nonnegativity while reducing the residual.
• Implements a projection mechanism that resets negative components to zero after each update, preserving nonnegativity.
• The algorithm generates a nonnegative, decreasing sequence in the solution norm, which is proven to converge to a nonnegative solution.
• Employs a residual-based stopping criterion with tolerance $10^{-8}$, and defines a component as zero if below $10^{-5}$.
• Applies the method to both small and large-scale problems, including sparse tensors with 60% zero entries and nonnegative right-hand sides generated via `sprand`.

Experimental results

Research questions

• RQ1 Can a Newton-type algorithm be designed to preserve nonnegativity of iterates when solving multilinear systems with nonsingular M-tensors and nonnegative right-hand sides?
• RQ2 Does the proposed algorithm converge to a nonnegative solution even when multiple nonnegative solutions exist?
• RQ3 How does the algorithm perform compared to state-of-the-art solvers like HM, QCA, and NLSQ when the right-hand side vector contains zero entries?
• RQ4 Can the algorithm reliably find sparse nonnegative solutions in cases where other methods produce only positive solutions?
• RQ5 What is the computational efficiency and robustness of the proposed method in large-scale, sparse tensor systems?

Key findings

• The proposed NPA algorithm generates a nonnegative, decreasing sequence that converges to a nonnegative solution of the multilinear system.
• Numerical experiments show that NPA successfully finds nonnegative solutions even when the right-hand side vector contains zeros, while QCA fails in such cases.
• NPA produces sparse nonnegative solutions, as demonstrated in convergence curves and solution sparsity analysis, outperforming HM and NLSQ in sparsity preservation.
• In the $ (m,n) = (3,200) $ case, NPA achieves a higher success rate and lower computing time than HM and NLSQ, despite requiring more iterations.
• The algorithm is robust across 100 random test instances with sparse tensors and nonnegative right-hand sides, consistently returning nonnegative solutions.
• NPA is the only method among those tested that reliably produces nonnegative and sparse solutions when the right-hand side is nonnegative but not strictly positive.