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This article focuses on the investigation of finite-time dissipative state estimation for Markov jump neural networks. First, in view of the subsistent phenomenon that the state estimator cannot capture the system modes synchronously, the hidden Markov model with partly unknown probabilities is introduced in this article to describe such asynchronization constraint. For the upper limit of network bandwidth and computing resources, a novel dynamic event-triggered transmission mechanism, whose threshold parameter is constructed as an adjustable diagonal matrix, is set between the estimator and the original system to avoid data collision and save energy. Then, with the assistance of Lyapunov techniques, an event-based asynchronous state estimator is designed to ensure that the resulting system is finite-time bounded with a prescribed dissipation performance index. Ultimately, the effectiveness of the proposed estimator design approach combining with a dynamic event-triggered transmission mechanism is demonstrated by a numerical example.

Let G be a connected graph of order n , A ( G ) the adjacency matrix of G and D ( G ) the diagonal matrix of the row-sums of A ( G ) . For every real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . In this paper, we obtain a factorization of the A α -characteristic polynomial of the some families of graphs and determine in these families the conditions about α so that A α ( G ) is positive semidefinite.

This article is devoted to the investigation of reduced-order dissipative filtering for Takagi-Sugeno (T-S) fuzzy Markov jump systems with the event-triggered mechanism. For the proposed event-triggered mechanism, its threshold parameter is constructed as a special diagonal matrix which can improve system performance by flexibly adjusting the matrix elements. Due to the impact of the sampling behaviors and the environmental disturbance, the asynchronization between the filter and the estimated system is considered in this article, which can be characterized by the hidden Markov model. Through handling the linear matrix inequalities (LMIs) with some slack matrices, event-triggered fuzzy filters are designed to guarantee the resulting system is stochastically stable and strictly dissipative. The proposed filter parameters are obtained by solving LMIs. Ultimately, both the effectiveness and advantages of the proposed reduced-order filter with the event-triggered mechanism are verified by a practical example.

The elastic motion estimation is an efficient temporal predictive coding tool for predicting complex motions including zooming, shearing and twisting. However, its optimization solution still suffers from high computational complexity and unstable convergence yet. We thus present a novel optimization framework to improve the elastic motion estimation efficiency. By introducing a diagonal matrix into the Gauss-Newton’s Hessian matrix, we employ the Levenberg-Marquardt optimization to compute the elastic motion vectors. It obtains superior convergence to the conventional Gauss-Newton’s method, especially when the starting point is far away from the globally/locally optimum. Furthermore, we prove that the Gauss-Newton’s Hessian matrix for elastic motion estimation is positive semi-definite by discussing the convexity of the motion-compensated error function. Such a finding allows us to adopt a sign-alternating factor, instead of a positive constant used in the traditional L-M method, to update iteratively the diagonal matrix coefficient. And a bound is determined for the diagonal matrix coefficient to guarantee the non-singularity of the Hessian matrix, with whose constraints we propose an adaptive method to compute the sign-alternating factor to restrict the iterative process into a reasonable motion vector space. Finally, a novel elastic motion estimation algorithm is presented based on the modified Levenberg-Marquardt optimization. Experimental results show that the proposed algorithm gains higher motion-compensated peak signal-to-noise ratio (PSNR) than several benchmark methods. Moreover, our algorithm converges fast which needs only 1-2 iterations before it achieves higher PSNR than conventional elastic motion estimation.

By employing off-diagonal matrix Mittag-Leffler functions and stability theory for line fractional functional differential equations, a new technique is proposed to investigate the existence, uniqueness and global asymptotical stability of S -asymptotically periodic solution for a class of semilinear Caputo fractional functional differential equations. Some better results are derived, which improve and extend the existing research findings in recent years. As an application of the general theory, some decision theorems are established for the asymptotically dynamical behaviors for fractional four-neuron BAM neural networks. The methods used in this paper could be applied to the study of other fractional differential systems in the areas of science and engineering.

In this paper, the robust filtering problem for uncertain complex networks with time-varying state delay and stochastic nonlinear coupling based on H ∞ performance criterion is studied. The random connections of coupling nodes are represented by utilizing independent random variables and the multiple fading measurements phenomenon is characterized by introducing diagonal matrices with independent stochastic elements. Moreover, the probabilistic time-varying delays in the measurement outputs are described by white sequences with the Bernoulli distributions. Furthermore, All system’s matrices are supposed to have uncertainty and a quadratic bound is assumed for nonlinear part of the network. This bound can be obtained by solving a sum of squares (SOS) optimization problem. By applying the Lyapunov theory, we design a robust filter for each node of the network so that the filtering error system is asymptomatically stable and the H ∞ performances are met. Then, the parameters of the filters are achieved by solving a linear matrix inequality (LMI) feasibility problem. Finally, the applicability and performance of the proposed H ∞ filtering approach are demonstrated via a practical example.

For a graph G , the signless Laplacian matrix Q ( G ) is defined as Q ( G ) = D ( G ) + A ( G ) , where A ( G ) is the adjacency matrix of G and D ( G ) the diagonal matrix whose main entries are the degrees of the vertices in G . The Q -spectrum of G is that of Q ( G ) . In the present paper, we are interested in the minimum values of the second largest signless Laplacian eigenvalue q 2 ( G ) of a connected graph G . We find the five smallest values of q 2 ( G ) over the set of connected graphs G with given order n . We also characterize the corresponding extremal graphs.

In a previous paper based on the interpolation finite difference method, a calculation system was shown for calculating 1D (one-dimensional) Laplace’s equation and Poisson’s equation using high-order difference schemes. Finite difference schemes, from the usual second-order to tenth-order differences, including odd number order differences, were systematically and instantaneously derived over equally/unequally spaced grid points based on the Lagrange interpolation function. Using the direct method with the band diagonal matrix algorithm, 1D Poisson equations were numerically calculated under double precision floating arithmetic, but it became clear that high accurate calculations could not be secured in high-order differences because “digit-loss errors” caused by the finite precision of computations occurred in the calculations when using the high-order differences. The double precision calculation corresponds to 15 (significant) digit calculation. In this paper, we systematically investigate how the calculation accuracy changes by high precision calculations (30-digit, and 45-digit calculations). Under 45-digit calculation, where the digit-loss error can be almost ignored, the high-order differences enable extremely high-accurate calculations.

For every 2 n × 2 n real positive definite matrix A, there exists a real symplectic matrix M such that M T A M = diag ( D , D ) , where D is the n × n positive diagonal matrix with diagonal entries d 1 ( A ) ≤ ⋯ ≤ d n ( A ) . The numbers d 1 ( A ) , … , d n ( A ) are called the symplectic eigenvalues of A. We derive analogues of Wielandt's extremal principle and multiplicative Lidskii's inequalities for symplectic eigenvalues.

In the radial basis function-based state-dependent autoregressive (RBF-AR) models with regression weights, the local linear models are included between the hidden layers and the output layers of the networks. The parameter estimation for the RBF-AR models with regression weights is studied in this brief. Considering the separable feature of the models, two criterion functions based on the increasing data length are defined to fit the observation data of the whole dynamical process. Two sub-algorithms are proposed by minimizing the criterion functions. Aiming to overcome the existence of the singular matrix during the Newton search and to make the algorithm more stable, a positive definite diagonal matrix is introduced to the algorithm. Based on the hierarchical principle, a hierarchical Newton recursive algorithm is proposed, which can realize the on-line parameter estimation. Simulation results verify the validity.

In this paper, we introduce the class of diagonally dominant with respect to a given LMI region D ⊂ C matrices. They are shown to possess the analogues of well-known properties of (classical) diagonally dominant matrices, e.g. their spectra are localized inside the region D . Moreover, we show that in some cases, diagonal D -dominance implies ( D , D ) -stability (i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix).

An adaptive backstepping control scheme has been put forward to investigate the attitude stabilization problem for a spacecraft with modeling uncertainties, external disturbances, actuator failures, angular velocity limitations and actuator limitations simultaneously. Based on the active disturbance rejection concept (ADRC), a positive definite diagonal matrix is selected by the designer to separate a control item from the system. Then, an extended state observer is constructed to deal with the total uncertain item including actuator failure. In order to deal with the limitation of torques, a saturation coefficient is introduced and its adaptive law is obtained. Meanwhile, a command filter and the compensation signal are constructed to handle the angular velocity limitations. Finally, the stability of the closed-loop system is analyzed. Simulation results are given to verify the effectiveness of the proposed scheme.

Let G be a connected graph, we define D Q ( G ) = T r ( G ) + D ( G ) as distance signless Laplacian matrix of G, where T r ( G ) and D ( G ) are diagonal matrix with vertex transmissions of G and distance matrix of G, respectively. In this paper, we characterize the extremal graphs which maximize the D Q -spectral radius among complements of unicyclic graphs and trees, respectively. And we also characterize the unique graph among complements of unicyclic graphs of diameter three which maximize the least D Q -eigenvalues.

Let be a simple graph of order and let be the Seidel matrix of , defined as where if the vertices and are adjacent and if the vertices and are not adjacent and if . Let be the diagonal matrix where denotes the degree of the vertex of . The Seidel Laplacian matrix of a graph is defined as and the Seidel signless Laplacian matrix of a graph is defined as . The zero-divisor graph of a commutative ring , denoted by , is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices are adjacent if and only if . In this paper, we find the Seidel polynomial and Seidel Laplacian polynomial of the join of two regular graphs using the concept of schur complement and coronal of a square matrix. Also we describe the computation of the Seidel Laplacian and Seidel signless Laplacian eigenvalues of the join of more than two regular graphs, using the well known Fiedler's lemma and apply these results to describe these eigenvalues for the zero-divisor graph on . Further we find the Seidel Laplacian and Seidel signless Laplacian spectrum of the zero-divisor graph of for some values of , say , where are distinct primes. We also prove that 0 is a simple Seidel Laplacian eigenvalue of , for any .

For 0 ≤ α ≤ 1 , the A α -matrix of graph G is defined as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , where D ( G ) and A ( G ) are the diagonal matrix of the degrees and the adjacency matrix of G, respectively. On the premise of a fixed number of edges, Zhai et al. (2020) [15] depicted the extremal graph which has the largest A 1 2 -spectral radius, and also characterized the graph maximizing the A 1 2 -spectral radius among all graphs with given clique number (respectively, chromatic number). In this paper, we extend the conclusion of them to A α -spectral radius for α ∈ [ 1 2 , 1 ] .

A derivative-free quasi-Newton-type algorithm in which its search direction is a product of a positive definite diagonal matrix and a residual vector is presented. The algorithm is simple to implement and has the ability to solve large-scale nonlinear systems of equations with separable functions. The diagonal matrix is simply obtained in a quasi-Newton manner at each iteration. Under some suitable conditions, the global and R-linear convergence result of the algorithm are presented. Numerical test on some benchmark separable nonlinear equations problems reveal the robustness and efficiency of the algorithm.

The topic of finite-time stability criterion for neural networks with time-varying delays via a new argument Lyapunov-Krasovskii functional (LKF) was proposed and the time-varying delay of the system is without differentiable. For sufficient conditions of this study, a new (LKF) is combined with improved triple integral terms, namely the functionality of finite-time stability, integral inequality, and a positive diagonal matrix without using a free weighting matrix. The improved finite-time sufficient conditions for the neural network with time varying delay are given in terms of linear matrix inequalities (LMIs) and the results show improvement on previous studies. Numerical examples are given to illustrate the effectiveness of the proposed method.

Let G be a simple connected graph with n vertices. The transmission Tv of a vertex v is defined to be the sum of the distances from v to all other vertices in G, that is, Tv = Σu∈Vduv, where duv denotes the distance between u and v. Let T1,…, Tn be the transmission sequence of G. Let $$\cal{D}=(d_{ij})_{n\times n}$$ be the distance matrix of G, and $$\cal{T}$$ be the transmission diagonal matrix diag(T1,…,Tn). The matrix $$\cal{Q}(G)=\cal{T}+\cal{D}$$ is called the distance signless Laplacian of G. In this paper, we provide the distance signless Laplacian spectrum of complete k-partite graph, and give some sharp lower and upper bounds on the distance signless Laplacian spectral radius $$\mathbb{q}(G)$$ .

Distance metric learning (DML) aims to learn distance metrics that reflect the interactions between features and labels. Due to the high computational complexity, existing DML models are unsuitable for large-scale datasets. This study proposes a DML approach for large-scale problems by reducing the number of variables, utilizing sparse structures of the optimization problems, and taking advantage of large-scale computation platforms. The proposed approach treats DML as a linear space transformation problem and suggests that a full DML matrix can be approximated by a diagonal matrix in many cases. We solve the diagonal DML problem along with its l 1 and l 2 regularizations via linear and quadratic programming. To facilitate large-scale learning problems, we design a MapReduce framework to build triplets, which are encapsulations of triple data points used for the optimization problem, and develop a weighting mechanism for triplets according to their contributions to the whole distance distortion. Experiments show that the proposed approach is fast in large-scale DML applications with comparable accuracy to much more time-consuming full matrix models. Since the approach is implemented with the Scala language based on the Spark platform, it can be used directly by productive Java applications, which makes it highly practical for large-scale datasets.

This article is concerned with the resilient fault diagnosis (FD) problem for a class of complex networks subject to possible loss of the actuator effectiveness and random variation of the filter gain. An unknown diagonal matrix is employed to characterize the multiplicative loss of actuator effectiveness for each node. The proposed filter utilizes the information from only the local node and the neighboring nodes. Since there is no need to have a center node receiving global information from every node, the developed FD algorithm is truly distributed. In the presence of gain variations, a time-varying filter is constructed to jointly estimate the system state and the loss of actuator effectiveness at each node. An upper bound of the filtering error covariance is calculated and then minimized via appropriately determining the filter gains. The filter is designed by solving two sets of recursive matrix equations, thereby meriting the suitability of online applications. Sufficient conditions are established to guarantee the exponential boundedness in mean square of the filtering error, and the monotonicity of the estimation error covariance with respect to the coupling strength is also investigated. An illustrative example is provided to show the usefulness of our FD strategy.

A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi = (G, \varphi)$ is a graph where the function $\varphi$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation. A $ \mathbb{T} $-gain graph $\Phi$ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is $1$. Signed graphs are special cases of $\mathbb{T}$-gain graphs. The adjacency matrix of $\Phi$, denoted by $ \mathbf{A}(\Phi)$ is defined canonically. The gain Laplacian for $\Phi$ is defined as $\mathbf{L}(\Phi) = \mathbf{D}(\Phi) - \mathbf{A}(\Phi)$, where $\mathbf{D}(\Phi)$ is the diagonal matrix with diagonal entries are the degrees of the vertices of $G$. The minimum number of vertices (resp., edges) to be deleted from $\Phi$ in order to get a balanced gain graph the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of $\mathbf{L}(\Phi)$. We provide several lower and upper bounds for extremal eigenvalues of $\mathbf{L}(\Phi)$ in terms of different graph parameters such as the number of edges, vertex degrees, and average $2$-degrees. The signed graphs are particular cases of the $\mathbb{T}$-gain graphs, all the bounds established in paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erd\H{o}s-Re\'yni graphs. Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds., Comment: Preliminary version. 32 pages

Let G be a connected simple graph with n vertices having chromatic number χ. The distance Laplacian matrix D L ( G ) is defined as D L ( G ) = D i a g ( T r ) − D , where D i a g ( T r ) is the diagonal matrix of vertex transmissions and D is the distance matrix of G. The eigenvalues of D L ( G ) are the distance Laplacian eigenvalues of G and are denoted by ∂ 1 L ( G ) , ∂ 2 L ( G ) , … , ∂ n L ( G ) . The largest eigenvalue ∂ 1 L ( G ) is called the distance Laplacian spectral radius. For a non-complete graph G with n vertices and chromatic number χ, Aouchiche and Hansen (2017) proved that ∂ 1 L ( G ) ≥ n + ⌈ n χ ⌉ . If G is a connected graph with n ≥ 4 vertices and chromatic number χ ≤ n − 2 , we prove that ∂ 2 L ( G ) ≥ n + ⌈ n χ ⌉ and we show the existence of graphs for which the equality holds. Among all graphs with chromatic number χ satisfying n 2 ≤ χ ≤ n − 1 , we show that the graph K 2 , 2 , … , 2 ︸ n − χ , 1 , 1 , … , 1 ︸ 2 χ − n has the minimum distance Laplacian spectral radius. Also, we give the distribution of the distance Laplacian eigenvalues in relation to the chromatic number χ and other graph invariants. We characterize the extremal graphs for some of these results and for others, we illustrate by examples.

Resistive crossbars implement parallel vector-matrix multiplication (VMM) in analog fashion, and thus enable fast and energy-efficient neuromorphic systems. However, interconnect resistance and resistive switching devices form a complex resistance network with sneak paths. It could result in severe distortions on the output currents. When implementing neural networks, current distortions also cause significant accuracy loss. This article proposes an accurate and computationally efficient model of VMM in resistive crossbars, called diagonal matrix regression (DMR), and incorporates the model into the topology of neural networks as DMR layer (DMRL). Given an ${m}{\times }{n}$ crossbar, two diagonal matrices are calculated directly according to the resistance network in a time complexity of only ${O}({m}^{2}+{n}^{2})$ . No hyper-parameter needs to be determined manually. Modeling of VMM is implemented in a time complexity of only ${O}({mn})$ . DMRL is developed to replace the weight matrix of neural networks so that the effect of interconnect resistance and the sneak path problem are well handled during ex-situ training. Using this technique, for the task of MNIST and fashion-MNIST classification, the accuracy is dramatically restored.

In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A ( k ) converge to zero. However, this fact alone does not necessarily guarantee that A ( k ) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q ( k ) . We prove that under certain assumptions A ( k ) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q ( k ) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q ( k ) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example.

Most existing super-resolution imaging methods fail to work in low signal-to-noise ratio (SNR) condition due to the ill-posed antenna measurement matrix, but the sparse-truncated singular value decomposition (TSVD) method can effectively suppress noise and improve azimuth resolution in low SNR condition. However, the current sparse-TSVD method encounters large computation cost, resulting in a slow algorithm speed. In this work, a fast sparse-TSVD super-resolution imaging method of real aperture radar is proposed. First, the proposed method is based on the results of TSVD, using the truncated unitary matrix and diagonal matrix to reconstruct the signal convolution model. The dimension of the reconstructed antenna measurement matrix reduces from $N \times N$ to $k \times N$ , and the dimension of the reconstructed echo matrix reduces from $N \times 1$ to $k \times 1$ , where $N$ is azimuth sampling points and $k$ is truncation parameter, $N \gg k$ . Much of the expensive matrix– multiplication computation can then be performed on the smaller matrices, thereby accelerating the algorithm. Second, an objective function is established as the ${l_{1}}$ constraint based on the regularization strategy. Lastly, this article employs iterative reweighted least square (IRLS) method to solve the objective function, and the dimension of the reversed matrix is lessened from $N \times N$ to $k \times k$ , speeding up the algorithm further. The simulation and real data verify that the proposed algorithm not only improves the azimuth resolution in low SNR condition but also increases computational efficiency compared with the sparse-TSVD method.

In this paper we propose the FD-TR: Feature Detector Based on Scale Invariant Feature Transform and Bidirectional Feature Regionalization for digital image watermarking. The Scale Invariant Feature Transform method is applied to extract keypoints and an Edge and Neighbor Filtering method is proposed to generate the candidate feature points. Then the Bidirectional Feature Regionalization method is proposed and applied in order to classify candidate feature points and form candidate feature regions. On this basis, the Candidate Feature Region Filtering method is proposed to select the final feature regions for watermarking. During the watermarking process, the Nonsubsampled Contourlet Transform is employed to the extracted feature regions to extract the low-frequency coefficients. Next, we use the Diagonal Matrix-based Spread Transform Dither Modulation for watermark embedding and extraction. Extensive experiments have been conducted to evaluate the performance of the proposed scheme and the comparison with existing methods demonstrate that the proposed method is superior to the existing methods in terms of robustness and quality.

We address the structure identification and the uniform approximation of two fully nonlinear layer neural networks of the type$$f(x)=1^T h(B^T g(A^T x))$$f(x)=1Th(BTg(ATx))on$$\mathbb R^d$$Rd, where$$g=(g_1,\dots , g_{m_0})$$g=(g1,⋯,gm0),$$h=(h_1,\dots , h_{m_1})$$h=(h1,⋯,hm1),$$A=(a_1|\dots |a_{m_0}) \in \mathbb R^{d \times m_0}$$A=(a1|⋯|am0)∈Rd×m0and$$B=(b_1|\dots |b_{m_1}) \in \mathbb R^{m_0 \times m_1}$$B=(b1|⋯|bm1)∈Rm0×m1, from a small number of query samples. The solution of the case of two hidden layers presented in this paper is crucial as it can be further generalized to deeper neural networks. We approach the problem by sampling actively finite difference approximations to Hessians of the network. Gathering several approximate Hessians allows reliably to approximate the matrix subspace$$\mathcal W$$Wspanned by symmetric tensors$$a_1 \otimes a_1,\dots ,a_{m_0}\otimes a_{m_0}$$a1⊗a1,⋯,am0⊗am0formed by weights of the first layer together with the entangled symmetric tensors$$v_1 \otimes v_1 ,\dots ,v_{m_1}\otimes v_{m_1}$$v1⊗v1,⋯,vm1⊗vm1, formed by suitable combinations of the weights of the first and second layer as$$v_\ell =A G_0 b_\ell /\Vert A G_0 b_\ell \Vert _2$$vℓ=AG0bℓ/‖AG0bℓ‖2,$$\ell \in [m_1]$$ℓ∈[m1], for a diagonal matrix$$G_0$$G0depending on the activation functions of the first layer. The identification of the 1-rank symmetric tensors within$$\mathcal W$$Wis then performed by the solution of a robust nonlinear program, maximizing the spectral norm of the competitors constrained over the unit Frobenius sphere. We provide guarantees of stable recovery under a posteriori verifiable conditions. Once the 1-rank symmetric tensors$$\{a_i \otimes a_i, i\in [m_0]\}\cup \{v_\ell \otimes v_\ell , \ell \in [m_1] \}$${ai⊗ai,i∈[m0]}∪{vℓ⊗vℓ,ℓ∈[m1]}are computed, we address their correct attribution to the first or second layer ($$a_i$$ai’s are attributed to the first layer). The attribution to the layers is currently based on a semi-heuristic reasoning, but it shows clear potential of reliable execution. Having the correct attribution of the$$a_i,v_\ell $$ai,vℓto the respective layers and the consequent de-parametrization of the network, by using a suitably adapted gradient descent iteration, it is possible to estimate, up to intrinsic symmetries, the shifts of the activations functions of the first layer and compute exactly the matrix$$G_0$$G0. Eventually, from the vectors$$v_\ell =A G_0 b_\ell /\Vert A G_0 b_\ell \Vert _2$$vℓ=AG0bℓ/‖AG0bℓ‖2’s and$$a_i$$ai’s one can disentangle the weights$$b_\ell $$bℓ’s, by simple algebraic manipulations. Our method of identification of the weights of the network is fully constructive, with quantifiable sample complexity and therefore contributes to dwindle the black box nature of the network training phase. We corroborate our theoretical results by extensive numerical experiments, which confirm the effectiveness and feasibility of the proposed algorithmic pipeline.

The kernel subspace clustering algorithm aims to tackle the nonlinear subspace model. The block diagonal representation subspace clustering has a more promising capability in pursuing the k-block diagonal matrix. Therefore, the low-rankness and the adaptivity of the kernel subspace clustering can boost the clustering performance, so an adaptive low-rank kernel block diagonal representation (ALKBDR) subspace clustering algorithm is put forward in this work. On the one hand, for the nonlinear nature of the practical visual data, a kernel block diagonal representation (KBDR) subspace clustering algorithm is put forward. The proposed KBDR algorithm first maps the original input space into the kernel Hilbert space which is linearly separable, and next applies the spectral clustering on the feature space. On the other hand, the ALKBDR algorithm uses the adaptive kernel matrix and makes the feature space low-rank to further promote the clustering performance. The experimental results on the Extended Yale B database and the ORL dataset have proved the excellent quality of the proposed KBDR and ALKBDR algorithm in comparison with other advanced subspace clustering algorithms that also are tested in this paper.