Research on infrared dim and small target detection algorithm based on low-rank tensor recovery

In order to rapidly and accurately detect infrared small and dim targets in the infrared image of complex scene collected by virtual prototyping of space-based downward-looking multiband detection,an improved detection algorithm of infrared small and dim target is proposed in this paper.Firstly,the original infrared images are changed into a new infrared patch tensor mode through data reconstruction.Then,the infrared small and dim target detection problems are converted to low-rank tensor recovery problems based on tensor nuclear norm in accordance with patch tensor characteristics,and inverse variance weighted entropy is defined for self-adaptive adjustment of sparseness.Finally,the low-rank tensor recovery problem with noise is solved by alternating the direction method to obtain the sparse target image,and the final small target is worked out by a simple partitioning algorithm.The test results in various spacebased downward-looking complex scenes show that such method can restrain complex background well by virtue of rapid arithmetic speed with high detection probability and low false alarm rate.It is a kind of infrared small and dim target detection method with good performance.

Higher-order principal component pursuit via tensor approximation and convex optimization

Recovering the low-rank structure of data matrix from sparse errors arises in the principal component pursuit （PCP）. This paper exploits the higher-order generalization of matrix recovery, named higher-order principal component pursuit （HOPCP）, since it is critical in multi-way data analysis. Unlike the convexification （nuclear norm） for matrix rank function, the tensorial nuclear norm is stil an open problem. While existing preliminary works on the tensor completion field provide a viable way to indicate the low complexity estimate of tensor, therefore, the paper focuses on the low multi-linear rank tensor and adopt its convex relaxation to formulate the convex optimization model of HOPCP. The paper further propose two algorithms for HOPCP based on alternative minimization scheme： the augmented Lagrangian alternating direction method （ALADM） and its truncated higher-order singular value decomposition （ALADM-THOSVD） version. The former can obtain a high accuracy solution while the latter is more efficient to handle the computationally intractable problems. Experimental results on both synthetic data and real magnetic resonance imaging data show the applicability of our algorithms in high-dimensional tensor data processing.

Light Propagation in the Second post-Newtonian Approximation of Scalar-Tensor Theory of Gravity

在这份报纸，我们使用公制的系数和在分级张肌的理论的第二 Newtonian 以后近似获得导出秒顺序光繁殖方程和轻偏转角度并且把它与以前的工作作比较的运动的方程。这些结果为象与 micro-arcsecond 和 nano-arcsecond 精确性瞄准天体测量学的 ASTROD， GAIA，达尔文和 SIM 一样的精确天体测量学使命是有用的，并且为第二个 Newtonian 以后框架和位置推算历需要让观察决定星并且太空船位置。

Preservation of Linear Constraints in Approximation of Tensors

For an arbitrary tensor(multi-index array) with linear constraints at each direction,it is proved that the factors of any minimal canonical tensor approximation to this tensor satisfy the same linear constraints for the corresponding directions.

Truncated sparse approximation property and truncated q-norm minimization

This paper considers approximately sparse signal and low-rank matrix’s recovery via truncated norm minimization minx∥xT∥q and minX∥XT∥Sq from noisy measurements.We first introduce truncated sparse approximation property,a more general robust null space property,and establish the stable recovery of signals and matrices under the truncated sparse approximation property.We also explore the relationship between the restricted isometry property and truncated sparse approximation property.And we also prove that if a measurement matrix A or linear map A satisfies truncated sparse approximation property of order k,then the first inequality in restricted isometry property of order k and of order 2k can hold for certain different constantsδk andδ2k,respectively.Last,we show that ifδs(k+|T^c|)<√(s-1)/s for some s≥4/3,then measurement matrix A and linear map A satisfy truncated sparse approximation property of order k.It should be pointed out that when Tc=Ф,our conclusion implies that sparse approximation property of order k is weaker than restricted isometry property of order sk.

Recovery of Corrupted Low-Rank Tensors

This paper studies the problem of recovering low-rank tensors, and the tensors are corrupted by both impulse and Gaussian noise. The problem is well accomplished by integrating the tensor nuclear norm and the l1-norm in a unified convex relaxation framework. The nuclear norm is adopted to explore the low-rank components and the l1-norm is used to exploit the impulse noise. Then, this optimization problem is solved by some augmented-Lagrangian-based algorithms. Some preliminary numerical experiments verify that the proposed method can well recover the corrupted low-rank tensors.

Low-Rank Positive Approximants of Symmetric Matrices

Given a symmetric matrix X, we consider the problem of finding a low-rank positive approximant of X. That is, a symmetric positive semidefinite matrix, S, whose rank is smaller than a given positive integer, , which is nearest to X in a certain matrix norm. The problem is first solved with regard to four common norms: The Frobenius norm, the Schatten p-norm, the trace norm, and the spectral norm. Then the solution is extended to any unitarily invariant matrix norm. The proof is based on a subtle combination of Ky Fan dominance theorem, a modified pinching principle, and Mirsky minimum-norm theorem.

Non-intrusive temperature rise fault-identification of distribution cabinet based on tensor block-matching

In this study,a novel non-intrusive temperature rise fault-identification method for a distribution cabinet based on tensor block-matching is proposed.Two-stage data repair is used to reconstruct the temperature-field information to support the demand for temperature rise fault-identification of non-intrusive distribution cabinets.In the coarse-repair stage,this method is based on the outside temperature information of the distribution cabinet,using tensor block-matching technology to search for an appropriate tensor block in the temperature-field tensor dictionary,filling the target space area from the outside to the inside,and realizing the reconstruction of the three-dimensional temperature field inside the distribution cabinet.In the fine-repair stage,tensor super-resolution technology is used to fill the temperature field obtained from coarse repair to realize the smoothing of the temperature-field information inside the distribution cabinet.Non-intrusive temperature rise fault-identification is realized by setting clustering rules and temperature thresholds to compare the location of the heat source with the location of the distribution cabinet components.The simulation results show that the temperature-field reconstruction error is reduced by 82.42%compared with the traditional technology,and the temperature rise fault-identification accuracy is greater than 86%,verifying the feasibility and effectiveness of the temperature-field reconstruction and temperature rise fault-identification.

The Low-Rank Approximation of Fourth-Order Partial-Symmetric and Conjugate Partial-Symmetric Tensor

We present an orthogonal matrix outer product decomposition for the fourth-order conjugate partial-symmetric(CPS)tensor and show that the greedy successive rank-one approximation(SROA)algorithm can recover this decomposition exactly.Based on this matrix decomposition,the CP rank of CPS tensor can be bounded by the matrix rank,which can be applied to low-rank tensor completion.Additionally,we give the rank-one equivalence property for the CPS tensor based on the SVD of matrix,which can be applied to the rank-one approximation for CPS tensors.

APPROXIMATING THE STATIONARY BELLMAN EQUATION BY HIERARCHICAL TENSOR PRODUCTS

We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law.The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations(PDE),which means that the Bellman equation suffers from the curse of dimensionality.Its non linearity is handled by the Policy Iteration algorithm,where the problem is reduced to a sequence of linear equations,which remain the computational bottleneck due to their high dimensions.We reformulate the linearized Bellman equations via the Koopman operator into an operator equation,that is solved using a minimal residual method.Using the Koopman operator we identify a preconditioner for operator equation,which deems essential in our numerical tests.To overcome computational infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats,in particular tensor trains(TT tensors)and multi-polynomials,together with high-dimensional quadrature,e.g.Monte-Carlo.By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidence is given.

基于并行小波算法的多模态数据近似匹配模型构建

针对近似匹配过程易受数据冗余性、异构成分等的影响,提出了基于并行小波算法的多模态数据近似匹配模型。该模型首先采用并行小波算法剔除多模态数据中的噪声,避免噪声对匹配过程产生影响;其次采用张量分解的聚类算法将不同相似度的数据划分到不同类簇中,以消除不同类簇的数据差异度;最后将预处理后的数据输入到基于空间方向近似性的数据匹配模型中,通过计算参考数据与待匹配数据之间的空间方向近似度、编辑距离完成多模态数据的近似匹配。实验结果表明,所提方法的匹配查准率高、查全率高、匹配时间短。

Fast nonnegative tensor ring decomposition based on the modulus method and low-rank approximation

Nonnegative tensor ring(NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on frequent reshaping and permutation operations in the optimization process and use a shrinking step size or projection techniques to ensure core tensor nonnegativity, which leads to a slow convergence rate, especially for large-scale problems. In this paper, we first propose an NTR algorithm based on the modulus method(NTR-MM), which constrains core tensor nonnegativity by modulus transformation. Second, a low-rank approximation(LRA) is introduced to NTR-MM(named LRA-NTR-MM), which not only reduces the computational complexity of NTR-MM significantly but also suppresses the noise. The simulation results demonstrate that the proposed LRA-NTR-MM algorithm achieves higher computational efficiency than the state-of-the-art algorithms while preserving the effectiveness of feature extraction.

Relativistic symmetries with the trigonometric Pschl-Teller potential plus Coulomb-like tensor interaction

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Poeschl-Teller （tPT） potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ（κ± i 1）r^-2. In view of spin and pseudo-spin （p-spin） symmetries, the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method （AIM）. We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ. The non-relativistic limit is also obtained.

Construction and Application of 3-Point Tensor Product Scheme

In this paper, we propose and analyze a tensor product subdivision scheme which is the extension of three point scheme for curve modeling. The usefulness of the scheme is illustrated by considering different examples along with its application in surface modeling.

Generalized Nonconvex Low-Rank Algorithm for Magnetic Resonance Imaging (MRI) Reconstruction

In recent years,utilizing the low-rank prior information to construct a signal from a small amount of measures has attracted much attention.In this paper,a generalized nonconvex low-rank(GNLR) algorithm for magnetic resonance imaging(MRI)reconstruction is proposed,which reconstructs the image from highly under-sampled k-space data.In the algorithm,the nonconvex surrogate function replacing the conventional nuclear norm is utilized to enhance the low-rank property inherent in the reconstructed image.An alternative direction multiplier method(ADMM) is applied to solving the resulting non-convex model.Extensive experimental results have demonstrated that the proposed method can consistently recover MRIs efficiently,and outperforms the current state-of-the-art approaches in terms of higher peak signal-to-noise ratio(PSNR) and lower high-frequency error norm(HFEN) values.

Parallel Active Subspace Decomposition for Tensor Robust Principal Component Analysis

Tensor robust principal component analysis has received a substantial amount of attention in various fields.Most existing methods,normally relying on tensor nuclear norm minimization,need to pay an expensive computational cost due to multiple singular value decompositions at each iteration.To overcome the drawback,we propose a scalable and efficient method,named parallel active subspace decomposition,which divides the unfolding along each mode of the tensor into a columnwise orthonormal matrix(active subspace)and another small-size matrix in parallel.Such a transformation leads to a nonconvex optimization problem in which the scale of nuclear norm minimization is generally much smaller than that in the original problem.We solve the optimization problem by an alternating direction method of multipliers and show that the iterates can be convergent within the given stopping criterion and the convergent solution is close to the global optimum solution within the prescribed bound.Experimental results are given to demonstrate that the performance of the proposed model is better than the state-of-the-art methods.

E-characteristic Polynomials of Real Rectangular Tensor

By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.

Robust least squares projection twin SVM and its sparse solution

Least squares projection twin support vector machine(LSPTSVM)has faster computing speed than classical least squares support vector machine(LSSVM).However,LSPTSVM is sensitive to outliers and its solution lacks sparsity.Therefore,it is difficult for LSPTSVM to process large-scale datasets with outliers.In this paper,we propose a robust LSPTSVM model(called R-LSPTSVM)by applying truncated least squares loss function.The robustness of R-LSPTSVM is proved from a weighted perspective.Furthermore,we obtain the sparse solution of R-LSPTSVM by using the pivoting Cholesky factorization method in primal space.Finally,the sparse R-LSPTSVM algorithm(SR-LSPTSVM)is proposed.Experimental results show that SR-LSPTSVM is insensitive to outliers and can deal with large-scale datasets fastly.

Low-rank spectral estimation algorithm of learning Markov model

This paper proposes a low-rank spectral estimation algorithm of learning Markov model.First,an approximate projection algorithm for the rank-constrained frequency matrix set is proposed,and thereafter its local Lipschitzian error bound established.Then,we propose a low-rank spectral estimation algorithm for estimating the state transition frequency matrix and the probability matrix of Markov model by applying the approximate projection algorithm to correct the maximum likelihood estimation of the frequency matrix,and prove that there is only a multiplying constant difference in estimation errors between the low-rank spectral estimation and the maximum likelihood estimation under appropriate conditions.Finally,numerical comparisons with the prevailing maximum likelihood estimation,spectral estimation,and rank-constrained maxi-mum likelihood estimation show that the low-rank spectral estimation algorithm is effective.

Best rank one approximation of real symmetric tensors can be chosen symmetric

We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.