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FACTORIZATION OF OPERATORS THROUGH SUBSPACES OF -SPACES
We analyze domination properties and factorization of operators in Banach spaces through subspaces of$L^{1}$-spaces. Using vector measure integration and extending classical arguments based on scalar integral bounds, we provide characterizations of operators factoring through subspaces of$L^{1}$-spaces of finite measures. Some special cases involving positivity and compactness of the operators are considered.
Journal Article
On Unifying Multi-view Self-Representations for Clustering by Tensor Multi-rank Minimization
In this paper, we address the multi-view subspace clustering problem. Our method utilizes the circulant algebra for tensor, which is constructed by stacking the subspace representation matrices of different views and then rotating, to capture the low rank tensor subspace so that the refinement of the view-specific subspaces can be achieved, as well as the high order correlations underlying multi-view data can be explored. By introducing a recently proposed tensor factorization, namely tensor-Singular Value Decomposition (t-SVD) (Kilmer et al. in SIAM J Matrix Anal Appl 34(1):148–172, 2013), we can impose a new type of low-rank tensor constraint on the rotated tensor to ensure the consensus among multiple views. Different from traditional unfolding based tensor norm, this low-rank tensor constraint has optimality properties similar to that of matrix rank derived from SVD, so the complementary information can be explored and propagated among all the views more thoroughly and effectively. The established model, called t-SVD based Multi-view Subspace Clustering (t-SVD-MSC), falls into the applicable scope of augmented Lagrangian method, and its minimization problem can be efficiently solved with theoretical convergence guarantee and relatively low computational complexity. Extensive experimental testing on eight challenging image datasets shows that the proposed method has achieved highly competent objective performance compared to several state-of-the-art multi-view clustering methods.
Journal Article
If K is a Valdivia compact space, then Cp(K) $$C_{\\hspace{-1.111pt}p}(K)$$ is uniformly ψ $$\\psi $$ -separable
We prove that, for any countably compact subspace X of a Σ -product of real lines, the space Cp(X) is uniformly ψ -separable, that is, has a uniformly dense subset of countable pseudocharacter. This result implies that Cp(K) is uniformly ψ -separable whenever K is a Valdivia compact space. We show that the existence of a uniformly dense realcompact subset of Cp(X) need not imply that Cp(X) is realcompact even if the space X is compact. We also establish that Cp(X) can fail to be ω -monolithic if it has a uniformly dense ω -monolithic subspace. Furthermore, an example is given of spaces X and Y such that both Cp(X) and Cp(Y) are Lindelöf but Cp(X×Y) has no uniformly dense Lindelöf subspace.
Journal Article
Deep Multi-Modal Non-Redundant Subspace Clustering By Contrastive Affinity Learning
Most approaches to multi-modal subspace clustering aim to identify modality-specific subspaces and subsequently converge them into a consensus subspace. However, real-world data often presents diverse clustering solutions that focus on alternative underlying structures. In this paper, we propose a deep Multi-Modal Non-Redundant Subspace Clustering (MMNRSC) framework, which considers both cross-modality contrastive subspace preservation and the non-redundancy between the consensus subspace and the given clustering solutions by learning contrastive affinities. Specifically, modality-specific networks are designed to extract intermediate features from all modalities, which are more robust than the raw features. To leverage complementary information from all modalities, we introduce a cross-modality contrastive loss based on modality-specific coefficients. This loss function pulls positive pairs together while pushing negative pairs apart, enhancing the modality coupling effect and reinforcing the subspace-preserving property. Additionally, the proposed MMNRSC framework eliminates the memory-intensive self-expressive layer. Experiments conducted on four publicly available datasets demonstrate the effectiveness of the proposed MMNRSC approach in comparison to competitive non-redundant clustering methods.
Journal Article
A Sequence Convergence of 1 -Dimensional Subspace in a Normed Space
In this paper, the researchers will be introduced the concept of a sequence convergence of 1 -dimensional subspaces (lines) in a normed space and shall discuss some properties of it. Furthermore, it will be proved a continuity property of angles among subspaces in inner product spaces. Finally, the notion of limit of a sequence of 2 -dimensional subspaces (planes) in a normed space is studied. The researchers also obtain a result which describe how the convergent of a sequence of lines is associated to the convergent of a sequence of planes in a normed space.
Journal Article
Multi-view clustering: A survey
In the big data era, the data are generated from different sources or observed from different views. These data are referred to as multi-view data. Unleashing the power of knowledge in multi-view data is very important in big data mining and analysis. This calls for advanced techniques that consider the diversity of different views, while fusing these data. Multi-view Clustering (MvC) has attracted increasing attention in recent years by aiming to exploit complementary and consensus information across multiple views. This paper summarizes a large number of multi-view clustering algorithms, provides a taxonomy according to the mechanisms and principles involved, and classifies these algorithms into five categories, namely, co-training style algorithms, multi-kernel learning, multi-view graph clustering, multi-view subspace clustering, and multi-task multi-view clustering. Therein, multi-view graph clustering is further categorized as graph-based, network-based, and spectral-based methods. Multi-view subspace clustering is further divided into subspace learning-based, and non-negative matrix factorization-based methods. This paper does not only introduce the mechanisms for each category of methods, but also gives a few examples for how these techniques are used. In addition, it lists some publically available multi-view datasets. Overall, this paper serves as an introductory text and survey for multi-view clustering.
Journal Article
Topics in Quaternion Linear Algebra
Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.
eBook
Simple sufficient condition for subspace to be completely or genuinely entangled
We introduce a simple sufficient criterion, which allows one to tell whether a subspace of a bipartite or multipartite Hilbert space is entangled. The main ingredient of our criterion is a bound on the minimal entanglement of a subspace in terms of entanglement of vectors spanning that subspace expressed for geometrical measures of entanglement. The criterion is applicable to both completely and genuinely entangled subspaces. We explore its usefulness in several important scenarios. Further, an entanglement criterion for mixed states following directly from the condition is stated. As an auxiliary result we provide a formula for the generalized geometric measure of entanglement of the d -level Dicke states.
Journal Article
Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces
Many multivariate functions in engineering models vary primarily along a few directions in the space of input parameters. When these directions correspond to coordinate directions, one may apply global sensitivity measures to determine the most influential parameters. However, these methods perform poorly when the directions of variability are not aligned with the natural coordinates of the input space. We present a method to first detect the directions of the strongest variability using evaluations of the gradient and subsequently exploit these directions to construct a response surface on a low-dimensional subspace---i.e., the active subspace ---of the inputs. We develop a theoretical framework with error bounds, and we link the theoretical quantities to the parameters of a kriging response surface on the active subspace. We apply the method to an elliptic PDE model with coefficients parameterized by 100 Gaussian random variables and compare it with a local sensitivity analysis method for dimension reduction.
Journal Article