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Tensor-based subspace clustering algorithms have garnered significant attention for their high efficiency in clustering high-dimensional data. However, when dealing with 2D image data, traditional vectorization operations in most algorithms tend to undermine the correlations of higher-order tensor terms. To tackle this limitation, this paper proposes a non-convex submodule clustering approach (2D-NLRSC) that leverages sparse and low-rank representations for 2D image data. An [script l]r-induced tensor nuclear norm is introduced to approximate the tensor rank precisely. Instead of vectorizing each 2D image, the framework arranges samples as lateral slices of a third-order tensor. It employs the t-product operation to generate an optimal representation tensor with low-rank constraint. The proposed method combines [script l]q-norm induced clustering awareness with laplacian regularization to obtain a representation tensor with a diagonal structure. Additionally, 2D-NLRSC incorporates the [script l]2,p-norm as a regularization term, taking advantage of its excellent invariance, continuity, and differentiability. Experimental results on real image datasets validate the superior performance of the 2D-NLRSC model.

The proximal gradient algorithm for minimizing the sum of a smooth and nonsmooth convex function often converges linearly even without strong convexity. One common reason is that a multiple of the step length at each iteration may linearly bound the “error”—the distance to the solution set. We explain the observed linear convergence intuitively by proving the equivalence of such an error bound to a natural quadratic growth condition. Our approach generalizes to linear and quadratic convergence analysis for proximal methods (of Gauss-Newton type) for minimizing compositions of nonsmooth functions with smooth mappings. We observe incidentally that short step-lengths in the algorithm indicate near-stationarity, suggesting a reliable termination criterion.

In this paper, we introduce and examine the concept of [rho]--statistical boundedness and give some relations between statistical boundedness and [rho]--statistical boundedness. We also introduce the notion of [rho]--statistical upper bound, [rho]--statistical lower bound, [rho]--statistical supremum and [rho]--statistical infimum and investigate their interrelationships. Keywords: [rho]--statistical boundedness, [rho]--statistical convergence, [rho]--statistical supremum, [rho]--statistical infimum.

New estimates for the convergence abscissas of the multiple Laplace–Stieltjes integral are obtained. There is described the relationship between the integrand function, the Lebesgue–Stieltjes measure, and the abscissa of convergence of the multiple Laplace–Stieltjes integral. Since the multiple Laplace–Stieltjes integral is a direct generalization of the Laplace integral and multiple Dirichlet series, known results about convergence domains for the multiple Dirichlet series are obtained as corollaries of the presented more general statements for the multiple Laplace–Stieltjes integral.

In this paper, we introduce the notion of [I.sub.[lambda]]- statistical convergence of sequences as one of the extensions of I-statistical convergence of bi-complex numbers. We investigate some fundamental properties of these notion and its relationship with I-statistical convergence of bi-complex numbers. In the end we introduce and investigate the concept of [I.sub.[lambda]]- statistical limit points, cluster points and establish some implication relations. Keywords: Bi-complex number, ideal, filter, I- convergence, I- statistical convergence, [I.sub.[lambda]]- statistical convergence.

We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number μ, approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a τ preconditioning when the variable coefficient wave number μ is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented.

For the Dirichlet series F(s)=∑n=1∞f[sub.n] expsλ[sub.n] , which is the Hadamard composition of the genus m of similar Dirichlet series F[sub.j] (s) with the same exponents, the growth with respect to the function G(s) given as the Dirichlet series is studied in terms of the Φ-type (the upper limit of M[sub.G] [sup.−1] (M[sub.F] (σ))/Φ(σ) as σ↑A) and convergence Φ-class defined by the condition ∫σ[sub.0] AΦ′(σ)MG−1(MF(σ))/Φ2(σ)dσ<+∞, where M[sub.F] (σ) is the maximum modulus of the function F at an imaginary line and A is the abscissa of the absolute convergence.

We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged mappings to expansive set-valued mappings that characterizes a type of strong calmness of the fixed point mapping. The second component to this analysis is an extension of the well-established notion of metric subregularity—or inverse calmness—of the mapping at fixed points. Convergence of expansive fixed point iterations is proved using these two properties, and quantitative estimates are a natural by-product of the framework. To demonstrate the application of the theory, we prove, for the first time, a number of results showing local linear convergence of nonconvex cyclic projections for inconsistent (and consistent) feasibility problems, local linear convergence of the forward-backward algorithm for structured optimization without convexity, strong or otherwise, and local linear convergence of the Douglas-Rachford algorithm for structured nonconvex minimization. This theory includes earlier approaches for known results, convex and nonconvex, as special cases.

In 1967, I. J. Maddox generalized the spaces [c.sub.0], c, [l.sub.[infinity]] by adding the powers [p.sub.k] (k [member of] N) in the definitions of the spaces to the terms of elements of sequences ([x.sub.k]). Gokhan and Colak in 2004-2006 defined the corresponding double sequence spaces for the Pringsheim and the bounded Pringsheim convergence. We will additionally define the corresponding double sequence spaces for the regular convergence. In 2009, Gokhan, Colak and Mursaleen characterized some classes of matrix transformations involving these double sequence spaces with powers. However, many of their results appeared to be wrong. In this paper, we give corresponding counterexamples and prove the correct results. Moreover, we present the conditions for a wider class of matrices.

We have introduced and studied a new concept of f-lacunary statistical convergence, where f is an unbounded modulus. It is shown that, under certain conditions on a modulus f, the concepts of lacunary strong convergence with respect to a modulus f and f-lacunary statistical convergence are equivalent on bounded sequences. We further characterize those θ for which Sθf=Sf, where Sθf and Sf denote the sets of all f-lacunary statistically convergent sequences and f-statistically convergent sequences, respectively. A general description of inclusion between two arbitrary lacunary methods of f-statistical convergence is given. Finally, we give an Sθf-analog of the Cauchy criterion for convergence and a Tauberian theorem for Sθf-convergence is also proved.