Factorization of completely positive matrices using iterative projected gradient steps

Abstract

We aim to factorize a completely positive matrix by using an optimization approach which consists in the minimization of a nonconvex smooth function over a convex and compact set. To solve this problem we propose a projected gradient algorithm with parameters that take into account the effects of relaxation and inertia. Both projection and gradient steps are simple in the sense that they have explicit formulas and do not require inner loops. Furthermore, no expensive procedure to find an appropriate starting point is needed. The convergence analysis shows that the whole sequence of generated iterates converges to a critical point of the objective function and it makes use of the Łojasiewicz inequality. Its rate of convergence expressed in terms of the Łojasiewicz exponent of a regularization of the objective function is also provided. Numerical experiments demonstrate the efficiency of the proposed method, in particular in comparison to other factorization algorithms, and emphasize the role of the relaxation and inertial parameters.

Computing the completely positive factorization via alternating minimization

“…In this section, we conduct three groups of experiments to demonstrate the numerical efficiency of our proposals. In particular, we present comparisons with the modified version of the method of alternating projections (MoMAP) developed in Reference 3, the difference‐of‐convex approach with variable step‐size (SpFeas) introduced in Reference 22 (see Algorithm 1 in Reference 22) and with the FISTA‐type method (IPG‐FISTA) described in Example 7 of Reference 23. Both our procedures and the methods of other authors were coded in Matlab.…”

“…In the first experiment, as in Reference 23, we examine the effectiveness and efficiency of the algorithms in factorizing a $$$ 40\times 40 $$$ completely positive matrix when the parameter $$$ r $$$ varies. In particular, we consider the general matrix presented below $$$$ {A}_n={\left[\begin{array}{cc}0& {\mathbf{1}}_{n-1}^{\top}\\ {}{\mathbf{1}}_{n-1}& {I}_{n-1}\end{array}\right]}^{\top}\left[\begin{array}{cc}0& {\mathbf{1}}_{n-1}^{\top}\\ {}{\mathbf{1}}_{n-1}& {I}_{n-1}\end{array}\right]\in {\mathbb{R}}^{n\times n}, $$$$ where $$$ {\mathbf{1}}_n\in {\mathbb{R}}^n $$$ denotes the all‐ones‐vector and $$$ {I}_n $$$ denotes the $$$ n $$$‐by‐$$$ n $$$ identity matrix.…”

Completely positive factorization by a Riemannian smoothing method

“…Groetzner and Dür [18] and Chen et al [19] proposed different methods to solve (FeasCP). Boţ and Nguyen [20] tried to solve another model (2). However, the methods they used do not belong to the Riemannian optimization techniques, but are rather Euclidean ones, since they treated the set O(r) ∶= {X ∈ ℝ r×r ∶ X ⊤ X = I} as a usual constraint in Euclidean space.…”

“…In fact, we will solve this equivalent feasibility problem (FeasCP) by other means in this paper. In 2021, Boţ and Nguyen [20] proposed a projected gradient method with relaxation and inertia parameters for the CP factorization problem, aimed at solving where B(0, ) ∶= {X ∈ ℝ n×r | ‖X‖ ≤ } is the closed ball centered at 0. The authors argued that its optimal value is zero if and only if A ∈ CP n .…”

“…The implementation details regarding the parameters we used are the same as in the numerical experiments reported in [19, Section 6.1]. • RIPG_mod [20]: This is a projected gradient method with relaxation and inertia parameters for solving (2). As shown in [20, Section 4.2], RIPG_mod is the best among the many strategies of choosing parameters.…”