[Paper Review] Finding a Nonnegative Solution to an M-Tensor Equation

Donghui Li, Hong-Bo Guan+1 more|arXiv (Cornell University)|Nov 28, 2018

Tensor decomposition and applications16 references20 citations

TL;DR

This paper proposes a monotone iterative method, called the Sequential M-matrix Equation Method (S-MEQM), for finding nonnegative solutions to M-tensor equations. The method solves a sequence of linear systems with a fixed M-matrix coefficient, ensuring monotonic and linear convergence to a nonnegative solution when initialized appropriately, offering a robust alternative to Newton’s method when the Jacobian is singular.

ABSTRACT

We are concerned with the tensor equation with an M-tensor or Z-tensor, which we call the M- tensor equation or Z-tensor equation respectively. We derive a necessary and sufficient condition for a Z (or M)-tensor equation to have nonnegative solutions. We then develop a monotone iterative method to find a nonnegative solution to an M-tensor equation. The method can be regarded as an approximation to Newton's method for solving the equation. At each iteration, we solve a system of linear equations. An advantage of the proposed method is that the coefficient matrices of the linear systems are independent of the iteration. We show that if the initial point is appropriately chosen, then the sequence of iterates generated by the method converges to a nonnegative solution of the M- tensor equation monotonically and linearly. At last, we do numerical experiments to test the proposed methods. The results show the efficiency of the proposed methods.

Motivation & Objective

• To develop an efficient iterative method for solving M-tensor equations with nonnegative solutions.
• To address the failure of Newton’s method when the Jacobian is singular, especially at iterates with zero components.
• To ensure monotonic convergence to a nonnegative solution using a fixed M-matrix coefficient in linear subproblems.
• To establish conditions under which the method converges linearly and monotonically.
• To numerically validate the efficiency and robustness of the proposed method across various test problems.

Proposed method

• The method is based on constructing a sequence of linear systems with a fixed M-matrix coefficient derived from the M-tensor structure.
• At each iteration, a system of linear equations is solved using the same coefficient matrix, ensuring computational efficiency.
• The update rule is designed to ensure monotonic decrease in the error while preserving nonnegativity of iterates.
• The method is interpreted as a quasi-Newton approximation, avoiding the need to recompute Jacobians at each step.
• An initial point is chosen such that the iterates remain nonnegative and converge monotonically to a solution.
• An improved variant, the A-Newton method, is introduced with adaptive parameters to enhance convergence speed.

Experimental results

Research questions

• RQ1 Under what conditions does an M-tensor equation admit a nonnegative solution?
• RQ2 Can a monotone iterative method be designed to converge linearly to a nonnegative solution of an M-tensor equation?
• RQ3 How does the performance of the S-MEQM compare to Newton-type methods when the Jacobian is singular?
• RQ4 What is the impact of the parameter α on the convergence behavior and efficiency of the method?
• RQ5 Can the method be extended to handle even-order strong M-tensors with multiple nonnegative solutions?

Key findings

• A necessary and sufficient condition is derived for the existence of nonnegative solutions to M-tensor equations.
• The S-MEQM method converges monotonically and linearly to a nonnegative solution when the initial point is chosen appropriately.
• Numerical experiments show that the method performs better with α ∈ (1, 2) than with α ∈ (0, 1], though convergence theory for this range remains open.
• The A-Newton method outperforms S-MEQM in all tested problems, especially in reducing iteration count and computational time.
• For Problem 1 with n=50, the A-Newton method with α=1.0 achieved 37.8 iterations and 0.3294 seconds, significantly outperforming S-MEQM.
• The method maintains stability and convergence even when Newton’s method fails due to singular Jacobians at zero components.