Diagonal Representation Research Articles

An Unsupervised Feature Extraction Approach Based on Self-Expression.

Feature extraction algorithms lack good interpretability during the projection learning. To solve this problem, an unsupervised feature extraction algorithm, that is, block diagonal projection (BDP), based on self-expression is proposed. Specifically, if the original data are projected into a low-dimensional subspace by a feature extraction algorithm, although the data may be more compact, the new features obtained may not be as explanatory as the original sample features. Therefore, by imposing L2,1 norm constraint on the projection matrix, the projection matrix can be of row sparsity. On one hand, discriminative features can be selected to make the projection matrix to be more interpretable. On the other hand, irrelevant or redundant features can be suppressed. The proposed model integrates feature extraction and selection into one framework. In addition, since self-expression can well excavate the correlation between samples or sample features, the unsupervised feature extraction task can be better guided using this property between them. At the same time, the block diagonal representation regular term is introduced to directly pursue the block diagonal representation. Thus, the accuracy of pattern recognition tasks such as clustering and classification can be improved. Finally, the effectiveness of BDP in linear dimensionality reduction and classification is proved on various reference datasets. The experimental results show that this algorithm is superior to previous feature extraction counterparts.

An enforced block diagonal low-rank representation method for the classification of medical image patterns

Low-rank representation based methods have been used on a variety of medical imaging databases for the segmentation and classification of biomedical images. The subspace segmentation of the data is performed by generating the block diagonal coefficient matrix. Whereas, the data is classified by performing the partitioning of the low-rank representation matrix. There exist several such methods for analysing medical images. The major difference between them lies in the construction of the data dictionary. Most of the time, the input data pattern is used as the dictionary for learning the representation matrix. The direct use of the input data for learning the representation degrades the performance of the model because medical images are subjected to outliers of multiple types, which include environmental lighting, image appearance and varying illumination. These types of errors induce noise in the data. It has been observed that the representation-based model is robust when the training data is clean. If the training data contains corrupted subsamples, the performance of the model drops down. We have addressed the mentioned problem by adopting a class-wise dictionary learning approach. In which the pattern of each class is learnt as the set of tuples in the dictionary. The model has been evaluated on several medical imaging datasets, which includes the Break-his dataset, ALL-IDB, biomedical images, covid CT and chest X-ray. The classification performance of the model is best for the biomedical database (99.16%) followed by the Covid dataset (94%), ALL-IDB database (93.47%) and Break-his dataset (93%).

Multi-view graph embedding clustering network: Joint self-supervision and block diagonal representation

Multi-view clustering has become an active topic in artificial intelligence. Yet, similar investigation for graph-structured data clustering has been absent so far. To fill this gap, we present a Multi-View Graph embedding Clustering network (MVGC). Specifically, unlike traditional multi-view construction methods, which are only suitable to describe Euclidean structure data, we leverage Euler transform to augment the node attribute, as a new view descriptor, for non-Euclidean structure data. Meanwhile, we impose block diagonal representation constraint, which is measured by the ℓ1,2-norm, on self-expression coefficient matrix to well explore the cluster structure. By doing so, the learned view-consensus coefficient matrix well encodes the discriminative information. Moreover, we make use of the learned clustering labels to guide the learnings of node representation and coefficient matrix, where the latter is used in turn to conduct the subsequent clustering. In this way, clustering and representation learning are seamlessly connected, with the aim to achieve better clustering performance. Extensive experimental results indicate that MVGC is superior to 11 state-of-the-art methods on four benchmark datasets. In particular, MVGC achieves an Accuracy of 96.17% (53.31%) on the ACM (IMDB) dataset, which is an up to 2.85% (1.97%) clustering performance improvement compared with the strongest baseline.

Complete/incomplete multi‐view subspace clustering via soft block‐diagonal‐induced regulariser

This study proposes a novel multi-view soft block diagonal representation framework for clustering complete and incomplete multi-view data. First, given that the multi-view self-representation model offers better performance in exploring the intrinsic structure of multi-view data, it can be nicely adopted to individually construct a graph for each view. Second, since an ideal block diagonal graph is beneficial for clustering, a ‘soft’ block diagonal affinity matrix is constructed by fusing multiple previous graphs. The soft diagonal block regulariser encourages a matrix to approximately have (not exactly) K diagonal blocks, where K is the number of clusters. This strategy adds robustness to noise and outliers. Third, to handle incomplete multi-view data, multiple indicator matrices are utilised, which can mark the position of missing elements of each view. Finally, the alternative direction of multipliers algorithm is employed to optimise the proposed model, and the corresponding algorithm complexity and convergence are also analysed. Extensive experimental results on several real-world datasets achieve the best performance among the state-of-the-art complete and incomplete clustering methods, which proves the effectiveness of the proposed methods.

Spinc structures on Hantzsche-Wendt manifolds

Using a combinatorial description of Stiefel-Whitney classes of closed flat manifolds with diagonal holonomy representation, we show that no Hantzsche-Wendt manifold of dimension greater than three admits a spinc structure.

Robust multiple kernel subspace clustering with block diagonal representation and low-rank consensus kernel

Multiple kernel learning (MKL) is widely used to deal with nonlinear clustering problem, because MKL can obtain the optimal consensus kernel by designing kernel weighting strategy, which avoids the limitation of single-kernel learning restricted by the selected kernel function. However, existing MKL subspace clustering algorithms ignore noise and the low-rank structure of the data in the feature space. We propose a new robust multiple kernel subspace clustering algorithm (LRMKSC) with block diagonal representation (BDR) and low-rank consensus kernel (LRCK), which is more systematic and comprehensive for data processing. In particular, (1) to learn the optimal consensus kernel, we design an automatic weighting strategy using Mixture Correntropy Induced Metric (MCIM), which not only sets the optimal weight for each kernel, but also improves the robustness of LRMKSC by suppressing noise; (2) to explore low-rank structure of input data in feature space, we propose the Weighted Truncated Schatten p-Norm (WTSN) and apply it to the low-rank constraint of the optimal consensus kernel; (3) considering the block diagonal property of the affinity matrix, we apply block diagonal constraint to the coefficient matrix. LRMKSC combines MKL, LRCK and BDR to solve these problems simultaneously. Through the interaction of three technologies, the results of other technologies are used in the overall optimal solution to iteratively improve the efficiency of each technology, and finally form an overall optimal algorithm for processing nonlinear structural data. Compared with the most advanced MKL subspace clustering algorithms, extensive experiments on image and text datasets verify the competitiveness of LRMKSC.

Adaptive low-rank kernel block diagonal representation subspace clustering

Adaptive low-rank kernel block diagonal representation subspace clustering

Fast numerical approximation for the space-fractional semilinear parabolic equations on surfaces

In this work, a fast numerical method is considered to solve the space-fractional semilinear parabolic equations on closed surfaces. It is a challenge in that how to define the space fractional operator and corresponding semilinear parabolic equation with the energy functional defined on surface. To overcome it, using the local tangential space, we construct the spectral approximation for the space fractional operator on surfaces and apply the matrix transfer technique to avoid the difficulty of fractional nonlocality. The main advantage of the matrix transfer method is the completed diagonal representation of fractional operators from eigenvalue decomposition. Moreover, the time-discrete error estimates are presented as well as the energy stability. Various numerical examples are carried out to verify the theoretical results.

An efficient spectral-Galerkin method for fractional reaction-diffusion equations in unbounded domains

In this work, we apply a fast and accurate numerical method for solving fractional reaction-diffusion equations in unbounded domains. By using the Fourier-like spectral approach in space, this method can effectively handle the fractional Laplace operator, leading to a fully diagonal representation of the fractional Laplacian. To fully discretize the underlying nonlinear reaction-diffusion systems, we propose to use an accurate time marching scheme based on ETDRK4. Numerical examples are presented to illustrate the effectiveness of the proposed method.

Active Block Diagonal Subspace Clustering

Subspace clustering aims to find clusters in the low-dimensional subspaces for high-dimensional data. Subspace clustering with Block Diagonal Representation (BDR) maintains the number of connected components of the graph by Laplacian rank constraint, and the learned affinity matrix shows a block diagonal structure, which will achieve a good segmentation for the dataset by spectral clustering. However, the subspaces of real data may overlap and the learned affinity matrix may be imprecise. In this work, we propose an Active learning framework for BDR(ABDR) to acquire and incorporate prior knowledge to improve the subspace clustering performance. An active selection strategy is designed to acquire labels of the informative data points from both the skeleton of clusters and the boundaries of clusters, and then the labeled data are converted into pairwise constraints, which are incorporated into BDR. The optimization of the new objective function is given and the convergence of ABDR is discussed. Experimental results on three images datasets(MNIST, ORL and COIL-20) and one UCI dataset(ISOLET) demonstrate the effectiveness of ABDR on complex clustering tasks and show that ABDR is superior to multiple state-of-the-art active clustering and learning techniques.

Density operator formulation for magnetic systems: Physical and mathematical aspects

In this work, the density operator diagonal representation in the coherent states basis, known as the Glauber–Sudarshan-P representation, is used to study harmonic oscillator quantum systems and models of spinless electrons moving in a two-dimensional noncommutative space, subject to a magnetic field background coupled with a harmonic oscillator. Relevant statistical properties such as the Q-Husimi distribution and the Wehrl entropy are investigated.

Phase space theory for open quantum systems with local and collective dissipative processes

In this article we investigate driven dissipative quantum dynamics of an ensemble of two-level systems given by a Markovian master equation with collective and local dissipators. Exploiting the permutation symmetry in our model, we employ a phase space approach for the solution of this equation in terms of a diagonal representation with respect to certain generalized spin coherent states. Remarkably, this allows to interpolate between mean field theory and finite system size in a formalism independent of Hilbert-space dimension. Moreover, in certain parameter regimes, the evolution equation for the corresponding quasiprobability distribution resembles a Fokker–Planck equation, which can be efficiently solved by stochastic calculus. Then, the dynamics can be seen as classical in the sense that no entanglement between the two-level systems is generated. Our results expose, utilize and promote techniques pioneered in the context of laser theory, which can be powerful tools to investigate problems of current theoretical and experimental interest.

Cauchy loss induced block diagonal representation for robust multi-view subspace clustering

With the rapid emergence of data that can be described by different feature sets or different “views”, multi-view subspace clustering has attracted considerable research attention. To uncover the common latent structure shared by multiple views, existing models usually impose the sparse or/and low-rank constraint on the coefficients of each view data, and use Frobenius norm or ℓ1-norm based metric to measure the residuals of multi-view data. However, the intuition behind the sparse or low-rank regularization is implicit. Besides, the Frobenius norm or ℓ1-norm based metric is suitable to handle either Gaussian noise or sparse noise, which are very sensitive to larger noise or outliers. When the data is contaminated by large noise or densely corrupted, performance of existing models is degraded dramatically. In this article, we propose a novel multi-view subspace clustering method to provide superior robustness against large noise or outliers embedded in multi-view data. Our method adopts a more direct and intuitive block diagonal regularization to preserve the underlying structure of each view, and meantime introduces the cauchy loss function to deal with large noise. The derived consensus representation matrix can effectively preserve the underlying common structure of multi-view data and be robust to large noise and data corruptions. Experimental results show that our method outperforms the state-of-the-arts on both synthetic and real-world benchmark datasets.

A quantum system with a non-Hermitian Hamiltonian

The relevance in physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics, and quantum field theory. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of H is obtained using pseudo-bosonic operators.

Block diagonal representation learning for robust subspace clustering

Subspace clustering groups a set of data into their underlying subspaces according to the low-dimensional subspace structure of data. The performance of spectral clustering-based approaches heavily depends on the learned block diagonal structure of the affinity matrix. However, this structure is fragile in the presence of noise within data. As such, the clustering performance is degraded significantly. On the other hand, in practice, we often do not have a prior knowledge of error distribution at all, which results in that we cannot model the error with suitable norms. To this end, in this paper, we propose a robust block diagonal representation learning for subspace clustering. Specifically, a non-convex regularizer is directly utilized to constrain the affinity matrix for exploiting the block diagonal structure. Furthermore, we use a penalty matrix to adaptively weight the reconstruction error so that we can handle noise without prior knowledge. We also devise an effective method to compute the parameters related to this matrix, reducing the complexity of the parameter trains. Experimental results show that our method outperformed the state-of-the-art methods on both synthetic data and real-world datasets.

Computation of the Cutoff Wavenumbers of Metallic Waveguides with Symmetries by Using a Nonlinear Eigenproblem Formulation: A Group Theoretical Approach

The evaluation of the cutoff wavenumbers of metallic waveguides can be related to the numerical resolution of a suitable nonlinear eigenproblem defined on the domain C described by the contour of its transverse cross section. In this work, we show that the symmetries of C can be exploited to obtain a block diagonal matrix representation of the nonlinear eigenproblem, which enables a remarkable reduction in the computational effort involved in its resolution.

Structured block diagonal representation for subspace clustering

The aim of the subspace clustering is to segment the high-dimensional data into the corresponding subspace. The structured sparse subspace clustering and the block diagonal representation clustering are quite advanced spectral-type subspace clustering algorithms when handling to the linear subspaces. In this paper, the respective advantages of these two algorithms are fully exploited, and the structured block diagonal representation (SBDR) subspace clustering is proposed. In many classical spectral-type subspace clustering algorithms, the affinity matrix which obeys the block diagonal property can not necessarily bring satisfying clustering results. However, the k-block diagonal regularizer of the SBDR algorithm directly pursues the block diagonal matrix, and this regularizer is obviously more effective. On the other hand, the general procedure of the spectral-type subspace clustering algorithm is to get the affinity matrix firstly and next perform the spectral clustering. The SBDR algorithm considers the intrinsic relationship of the two seemingly separate steps, the subspace structure matrix obtained by the spectral clustering is used iteratively to facilitate a better initialization for the representation matrix. The experimental results on the synthetic dataset and the real dataset have demonstrated the superior performance of the proposed algorithm over other prevalent subspace clustering algorithms.

High-dimensional spatial patterns in fractional reaction-diffusion system arising in biology

Abstract The concept of fractional derivative has been demonstrated to be successful when applied to model a range of physical and real life phenomena, be it in engineering and science related fields. It is a known fact that reaction-diffusion equation permits the use of different numerical methods in space and time. As a result, we introduce the Fourier spectral method for the discretization of space fractional derivative and adapt the modified version of the exponential time-integrator to advance in time in attempt to explore the dynamic richness of fractional reaction-diffusion equations in two and three dimensions. This approach gives a full diagonal representation of the fractional derivative operator and yields a better spectral convergence irrespective of the value of fractional order chosen in the experiment. Recommendations are made based on some amazing results which arise from the computational experiments. We intend to answer the question ’why is wildlife animals going into extinction in Africa?’ to a reasonable extent. We believe that the spatial patterns obtained in the simulation framework to mimic the ones found in wildlife would provide a measure and serves as a good alternative to an act of killing of wildlife animals for ornamental and decorative purposes, also would serve as a guild to textile industries on pattern formations.

Exact diagonal representation of normal mode energy, occupation number, and heat current for phonon-dominated thermal transport

Collective excitations of crystal vibrations or normal modes are customarily described using complex normal mode coordinates. While appropriate for calculating phonon dispersion, the mixed representation involving the complex conjugates does not allow the construction of equivalent phonon occupation number or modal dynamical quantities such as the energy or heat current specific to a wave-vector direction (q). Starting from a canonical solution that includes waves going to the left and right directions, we cast the Hamiltonian, normal mode population, and heat current in an exactly diagonalizable representation using real normal mode amplitudes. We show that the use of real amplitudes obviates the need for a complex modal heat current while making the passage to second quantization more apparent. Using nonequilibrium molecular dynamics simulations, we then compute the net modal energy, heat current, and equivalent phonon population in a linear lattice subjected to a thermal gradient. Our analysis paves a tractable path for probing and computing the direction-dependent thermal-phononic modal properties of dielectric lattices using atomistic simulations.

Quantifying the Unitary Generation of Coherence from Thermal Quantum Systems.

Coherence is associated with transient quantum states; in contrast, equilibrium thermal quantum systems have no coherence. We investigate the quantum control task of generating maximum coherence from an initial thermal state employing an external field. A completely controllable Hamiltonian is assumed allowing the generation of all possible unitary transformations. Optimizing the unitary control to achieve maximum coherence leads to a micro-canonical energy distribution on the diagonal energy representation. We demonstrate such a control scenario starting from a given Hamiltonian applying an external field, reaching the control target. Such an optimization task is found to be trap-less. By constraining the amount of energy invested by the control, maximum coherence leads to a canonical energy population distribution. When the optimization procedure constrains the final energy too tightly, local suboptimal traps are found. The global optimum is obtained when a small Lagrange multiplier is employed to constrain the final energy. Finally, we explore the task of generating coherences restricted to be close to the diagonal of the density matrix in the energy representation.