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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
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
Point-of-interest recommendation based on non-adjacent trajectory interaction model
Next Point-of-Interest(POl)recommendation aims to predict users'future behaviors based on their historical trajectories, providing significant value toboth users and service providers. Most models fail to capture users'non-adjacent trajectory features, leading to insufficient modeling of users'long-term preferences. Therefore, this paper proposes a Non-Adjacent Trajectory Interaction(NATI) model. The NATI model first uses a multi-dimensional embedding layer to represent user trajectories, then employs multi-head self-attention to capture non-adjacent spatio-temporal features across different subspaces, updating users'long-term preferences.Finally, matching attention is used to match potential locations and predict users'possible POls. Validation on two public datasets demonstrates that the proposed model outperforms baseline models by 8%-16%.
Journal Article
Subspace Discrimination for Multiway Data
Sampled values of volumetric data are expressed as third-order tensors. Object-oriented data analysis requires us to process and analyse volumetric data without embedding into a higher-dimensional vector space. Multiway forms of volumetric data require quantitative methods for the discrimination of multiway forms. Tensor principal component analysis is an extension of image singular value decomposition for planar images to higher-dimensional images. It is an efficient discrimination analysis method when used with the multilinear subspace method. The multilinear subspace method enables us to analyse spatial textures of volumetric images and spatiotemporal variations of volumetric video sequences. We define a distance metric for subspaces of multiway data arrays using the transport between two probability measures on the Stiefel manifold. Numerical examples show that the Stiefel distance is superior to the Euclidean distance, Grassmann distance and projection-based similarity for the longitudinal analysis of volumetric sequences.
Journal Article
A Compressive Sensing Based Computationally Efficient High-Resolution DOA Estimation of Wideband Signals Using Generalized Coprime Arrays
Most recent state-of-the-art wideband direction of arrival (DOA) estimation techniques achieve reasonable accuracy at the expenses of high computational complexity. In this paper, a new computationally efficient approach based on compressive sensing (CS) is introduced for high resolution wideband DOA estimation. The low software complexity is achieved utilizing CS with deterministic chaotic Chebyshev sensing matrices that allow reducing the measurement vector dimension, while the high-resolution DOA estimation is acquired utilizing an efficient generalized coprime array configuration. The effectiveness of the introduced approach in enhancing the DOA estimation precision and reducing the computational complexity is studied along with a detailed comparison between three state-of-the-art wideband DOA estimation techniques, namely incoherent signal subspace method (ISSM), focusing signal subspace (FSS), and modified test of orthogonality of projected subspaces (mTOPS) with and without applying the proposed CS technique. The performance is examined utilizing various assessment metrics such as the spatial spectrum, the computational time, and the root mean square error between estimated and actual DOAs when varying the signal-to-noise ratio and number of elements. Results reveal that applying the proposed CS technique to the three algorithms (ISSM, FSS, and mTOPS) provides significant reduction in the execution time needed for the DOA estimation without affecting the resolution accuracy under various set of parameters. This reveals the importance of the proposed approach in wideband wireless communication systems.
Journal Article