This paper aims to discuss the evolution of the notion of "metric" from the retrospective point of view. Since the standard metric allows us to build the entire foundation of analysis, enabling us to define continuity, derivatives, and integrals, various distinct metric notions are needed to develop the analysis. Thus, we aim to have the opportunity to conduct better analyses of real-world problems. This article will examine the historical evolution of the metric concept and introduce the Phi-augmented metric concept, which has a highly comprehensive structure. We shall support the comprehensiveness of this concept with examples and a generalization of the Banach contraction principle.
To enhance the numerical performance of projection-type methods in solving variational inequality problems, we propose two Bregman-based extragradient methods, together with a novel step size strategy. Under the mild condition that the underlying mapping is Lipschitz continuous and pseudomonotone, we prove that the sequences generated by the proposed methods globally converge to the solution. We compare the performance of the two proposed methods with different distance function settings and methodologies, demonstrating their effectiveness.
Subspace clustering has been widely applied in various fields, including computer vision, bioinformatics, and signal processing. However, real-world data is often intrinsically high-dimensional and contaminated with substantial redundancy. Existing substantial literature indicates that low-rank representation can effectively suppress noise interference and reveal the underlying essence of data. To more sufficiently capture this low-rank structure, we introduce a novel non-convex schatten-p norm model and analyze its Karush-Kuhn-Tucker (KKT) conditions. For addressing the significant computational burden inherent in high-dimensional data, we propose an efficient splitting tri-factorization method (STFM). Under mild assumptions, we rigorously prove that our algorithm achieves a convergence rate of O(1/epsilon(2)). Extensive experiments on synthetic datasets, three real-world clustering benchmarks, and challenging foreground extraction tasks demonstrate the superior efficiency and robustness of our approach. The proposed algorithm exhibits competitive performance against state-of-theart methods.
This paper initiates the study of a novel class of equations, referred to as the greatest integer value equation, given by Ax - Lxi = b, where A is an element of R-nxn ,b is an element of R-n and [x] denotes the componentwise greatest integer value of x is an element of R-n. This formulation represents a significant extension of the classical absolute value equation to an integer-type structure, and to the best of our knowledge constitutes the first such investigation in the literature. To solve this new equation, we propose an innovative approach by reformulating it as a combinatorial problem: counting integer lattice points within a parallelotope. This perspective allows us to establish a sufficient condition for the existence of solutions and to derive an explicit upper bound on the number of such solutions in R-2.
In this paper, we study a class of DC composite optimization problems where the objective function is formulated as the difference of two convex functions, with one involving a convex outer function composed with a differentiable inner mapping. To address the challenges arising from the composite structure, we first employ a linearization strategy to approximate the inner mapping. We then utilize the Fenchel-Young inequality to construct a surrogate function for the second DC component, which enables us to apply the Moreau decomposition theorem to uniquely determine an approximate subgradient. Based on this approach, we propose two inertial proximal DC algorithms that incorporate inertial steps to accelerate convergence. Theoretically, we establish the global convergence of the sequences generated by our algorithms to a critical point under suitable assumptions. Numerical experiments on a toy example demonstrate that the proposed algorithms consistently converge to a global minimizer from various initial points, unlike existing methods that may fail for certain starting points. These results partially confirm the effectiveness and robustness of our approaches for DC composite optimization.
