Rank Decomposition Research Articles

Energy-Efficient Weighted Observation Fusion Kalman Filtering with Randomly Delayed Measurements

This paper is concerned with the weighted observation fusion Kalman filtering problem for a class of multi-sensor fusion systems with random measurement delays and energy constraints. To reduce the energy consumptions, each sensor intermittently sends information to the fusion center in a random way. By using the full rank decomposition approach, the observation fusion equation is derived. Without resorting to the augmentation technique, optimal weighted observation fusion Kalman filter (WOFKF) is given, and it is proved that the performance of the WOFKF is equivalent to that of the centralized fusion Kalman filter. Simulations show the effectiveness of the proposed fusion methods.

Enhancing sparsity via full rank decomposition for robust face recognition

In this paper, we propose a fast and robust face recognition method named enhancing sparsity via full rank decomposition. The proposed method first represents the test sample as a linear combination of the training data as the same as sparse representation, then make a full rank decomposition of the training data matrix. We obtain the generalized inverse of the training data matrix and then solve the general solution of the linear equation directly. For obtaining the optimum solution to represent the test sample, we use the least square method to solve it. We classify the test sample into the class which has the minimal reconstruction error. Our method can solve the optimum solution of the linear equation, and it is more suitable for face recognition than sparse representation classifier. The extensive experimental results on publicly available face databases demonstrate the effectiveness of the proposed method for face recognition.

Image classification by non-negative sparse coding, correlation constrained low-rank and sparse decomposition

We propose an image classification framework by leveraging the non-negative sparse coding, correlation constrained low rank and sparse matrix decomposition technique (CCLR-Sc+SPM). First, we propose a new non-negative sparse coding along with max pooling and spatial pyramid matching method (Sc+SPM) to extract local feature’s information in order to represent images, where non-negative sparse coding is used to encode local features. Max pooling along with spatial pyramid matching (SPM) is then utilized to get the feature vectors to represent images. Second, we propose to leverage the correlation constrained low-rank and sparse matrix recovery technique to decompose the feature vectors of images into a low-rank matrix and a sparse error matrix by considering the correlations between images. To incorporate the common and specific attributes into the image representation, we still adopt the idea of sparse coding to recode the Sc+SPM representation of each image. In particular, we collect the columns of the both matrixes as the bases and use the coding parameters as the updated image representation by learning them through the locality-constrained linear coding (LLC). Finally, linear SVM classifier is trained for final classification. Experimental results show that the proposed method achieves or outperforms the state-of-the-art results on several benchmarks.

Surveillance video analysis using compressive sensing with low latency

We propose a method for analysis of surveillance video by using low rank and sparse decomposition (LRSD) with low latency combined with compressive sensing to segment the background and extract moving objects in a surveillance video. Video is acquired by compressive measurements, and the measurements are used to analyze the video by a low rank and sparse decomposition of a matrix. The low rank component represents the background, and the sparse component, which is obtained in a tight wavelet frame domain, is used to identify moving objects in the surveillance video. An important feature of the proposed low latency method is that the decomposition can be performed with a small number of video frames, which reduces latency in the reconstruction and makes it possible for real time processing of surveillance video. The low latency method is both justified theoretically and validated experimentally.

An internal model controller design procedure for multivariable system based on the double‐decomposition

A novel design procedure of internal model controller based on the Double‐Decomposition (DD), which includes the Full Rank Decomposition (FRD) and Singular Value Decomposition (SVD), is proposed for the multivariable system with multiple time‐delay plant. In the controller design process, the Reflexive Generalised Inverse Matrix (RGIM) based on FRD is used to replace the generalised inverse matrix to design the internal model controller; improved filter based on SVD is introduced into the design process. In the FRD process for a non‐square system, the Effective Relative Gain Array (ERGA) is applied to select the linear independent column. After model reduction for the internal model controller, the expression of the controller is obtained. Then the square system and the non‐square system are taken as examples. The simulation results show the effectiveness of the proposed approach for the multivariable system with the multiple time‐delay plant.

On Generic Nonexistence of the Schmidt--Eckart--Young Decomposition for Complex Tensors

The Schmidt--Eckart--Young theorem for matrices states that the optimal rank-$r$ approximation of a matrix is obtained by retaining the first $r$ terms from the singular value decomposition of that matrix. This paper considers a generalization of this optimal truncation property to the rank decomposition (Candecomp/Parafac) of tensors and establishes a necessary orthogonality condition. We prove that this condition is not satisfied at least by an open set of positive Lebesgue measure in complex tensor spaces. It is proved, moreover, that for complex tensors of small rank this condition can be satisfied only by a set of tensors of Lebesgue measure zero. Finally, we demonstrate that generic tensors in cubic tensor spaces are not optimally truncatable.

Theory of network contractor dynamics for exploring thermodynamic properties of two-dimensional quantum lattice models

Based on the tensor network state representation, we develop a nonlinear dynamic theory, coined network contractor dynamics (NCD), to explore the thermodynamic properties of two-dimensional quantum lattice models. By invoking the rank-1 decomposition in the multilinear algebra, the NCD scheme makes the contraction of the tensor network of the partition function be realized through a contraction of a local tensor cluster with vectors on its boundary. An imaginary-time-sweep algorithm for implementation of the NCD method is proposed for practical numerical simulations. We benchmark the NCD scheme on the square Ising model, which shows great accuracy. Also, the results on the spin-1/2 Heisenberg antiferromagnet on a honeycomb lattice are disclosed to be in good agreement with the quantum Monte Carlo calculations. The quasientanglement entropy $S$, Lyapunov exponent ${I}^{\text{Lya}}$, and loop character ${I}^{\text{loop}}$ are introduced within the dynamic scheme, which are found to display ``nonlocality'' near the critical point, and can be applied to determine the thermodynamic phase transitions of both classical and quantum systems.

WE-G-WAB-04: Ultra-Fast Dynamic MRI for Lung Tumor Tracking Based On Compressed Sensing and Low-Rank Decomposition in the Spatial-Temporal Domain

Purpose: Dynamic MRI is an attractive tool for internal organ motion monitoring but its speed is yet to be improved for real time 3D imaging. The purpose of the study is to accelerate the imaging acquisition using a hybrid compressed-sensing and low rank decomposition method (k-t SLR) exploiting data sparsity in the spatial-temporal domain. Methods: Six retrospective lung cancer patients were included in the study. For each subject, 120 continuously dynamic 2D MR images were acquired using a fast steady state precession imaging sequence. To test the feasibility of accelerated MR acquisition, fully sampled k-space data were down-sampled by 5–20 folds before reconstruction. The under-sampled low rank spatial-temporal matrix was recovered by a k-t SLR method computing the matrix of minimum nuclear norm that fits the data. The k-t SLR results were compared against conventional total variation (TV) reconstruction of individual dynamic MR frames. Results: High imaging fidelity and low noise levels were achieved with the k-t SLR method even at the most aggressive down-sampling ratio while TV reconstruction showed substantial loss of details and subsequently greater tracking errors. The k-t SLR method resulted in an average normalized mean square error less than 0.05 as opposed to 0.23 by using the TV reconstruction on individual frames. In addition to the high correlation coefficients (>0.90) in the tumor trajectories automatically tracked from the original images and the k-t SLR reconstructed images, less than 6% points showed tracking errors greater than 1 mm with 20 fold down-sampling, as opposed to the lower correlation coefficient (0.87) and 15% tracking error from using TV alone with 10 fold data down-sampling. Conclusion: We have demonstrated the potential to significantly reduce the amount data samples required for lung MR image reconstruction in the radiotherapy context where long imaging acquisition of a repetitive anatomic process is performed. The study is supported in part by NIH 1R21CA144063 and R21CA161670

Method of visual and infrared fusion for moving object detection

A method based on low rank and sparse decomposition is proposed for moving object detection by the fusion of visual and infrared video. The visual and infrared image sequences are decomposed into the joint low rank background term, the uncorrelated sparse moving nonobject term, and the common sparse moving object term via a joint minimization cost of nuclear norm, F norm, and l(1) norm. This method provides a flexible framework that can easily fuse information from visual and infrared video. The prior fusion strategies are not required. The complementary information on visual and infrared images can be naturally fused in the procedure of object detection. The experimental results show that the proposed algorithm is effective.

Drogue detection for vision-based autonomous aerial refueling via low rank and sparse decomposition with multiple features

The technology of vision-based probe-and-drogue autonomous aerial refueling is an amazing task in modern aviation for both manned and unmanned aircraft. A key issue is to determine the relative orientation and position of the drogue and the probe accurately for relative navigation system during the approach phase, which requires locating the drogue precisely. Drogue detection is a challenging task due to disorderly motion of drogue caused by both the tanker wake vortex and atmospheric turbulence. In this paper, the problem of drogue detection is considered as a problem of moving object detection. A drogue detection algorithm based on low rank and sparse decomposition with local multiple features is proposed. The global and local information of drogue is introduced into the detection model in a unified way. The experimental results on real autonomous aerial refueling videos show that the proposed drogue detection algorithm is effective.

Projected Tikhonov Regularization of Large-Scale Discrete Ill-Posed Problems

The solution of linear discrete ill-posed problems is very sensitive to perturbations in the data. Confidence intervals for solution coordinates provide insight into the sensitivity. This paper presents an efficient method for computing confidence intervals for large-scale linear discrete ill-posed problems. The method is based on approximating the matrix in these problems by a partial singular value decomposition of low rank. We investigate how to choose the rank. Our analysis also yields novel approaches to the solution of linear discrete ill-posed problems with solution norm or residual norm constraints.

Localizing the Elliott conjecture at strongly self-absorbing C* -algebras

Abstract. We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C* -algebra 𝒟 ${\mathcal {D}}$ ; we also explain how the known classification theorems for nuclear C* -algebras fit into this concept. As a new result in this direction, we employ recent results of H. Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C* -algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang–Su algebra 𝒵 $\mathcal {Z}$ . Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the K-theory restriction entirely. When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense. The strategy to attack the general case aims at combining these two approaches. Our classification theorem covers simple ASH algebras for which projections separate traces (and the K-groups of which have finitely generated torsion part); it does, however, not at all depend on an inductive limit structure. Also, in the monotracial case it does not rely on the existence or absence of projections in any way. In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear and stably finite C* -algebras of real rank zero (the K-groups of which have finitely generated torsion part) as well as the (projectionless) Jiang–Su algebra itself.

A fast direct matrix solver for surface integral equation methods for electromagnetic wave scattering from non‐penetrable targets

The implementation details of a fast direct solver is described herein for solving dense matrix equations from the application of surface integral equation methods for electromagnetic field scatterings from non‐penetrable targets. The proposed algorithm exploits the smoothness of the far field and computes a low rank decomposition of the off‐diagonal coupling blocks of the matrices through a set of skeletonization processes. Moreover, an artificial surface (the Huygens' surface) is introduced for each clustering group to efficiently account for the couplings between well‐separated groups. Furthermore, a recursive multilevel version of the algorithm is presented. Although asymptotically the algorithm would not alter the bleak outlook of the complexity of the worst case scenario,O(N3) for required CPU time where N denotes the number of unknowns, for electrically large electromagnetic (EM) problems; through numerical examples, we found that the proposed multilevel direct solver can scale as good as O(N1.3) in memory consumption and O(N1.8) in CPU time for moderate‐sized EM problems. Note that our conclusions are drawn based on a few sample examples that we have conducted and should not be taken as a true complexity analysis for general electrodynamic applications. However, for the fixed frequency (h‐refinement) scenario, where the discretization size decreases, the computational complexities observed agree well with the theoretical predictions. Namely, the algorithm exhibits O(N) and O(N1.5) complexities for memory consumption and CPU time, respectively.

Biologically realistic excitatory and inhibitory cell properties emerge from a sparse coding network

Neurons in the primary visual cortex exhibit a baffling array of tuning properties, often unaccountable by the classical linear feedforward model. Specifically, excitatory neurons display a number of nonlinear effects collectively known as non-classical receptive field (nCRF) effects [1], and inhibitory neurons have diverse orientation tuning properties [2]. Furthermore, excitatory cells outnumber inhibitory cells by a ratio of 9:1 [3], yet the excitatory and inhibitory drives are balanced. Efficient coding models of early vision have been shown to be able to explain key features of linear filtering properties [4] and some single cell nonlinear effects [5]. However, population statistics of nonlinear properties have not been studied in these models. In addition, inhibitory cells were not typically modeled. Here we demonstrate that many of the aforementioned excitatory cell and inhibitory cell properties emerge naturally from a network that implements sparse coding. To be specific, several nCRF effects including surround suppression, contrast invariant orientation tuning, and cross orientation suppression emerge in the excitatory cell population as a result of sparse coding strategy; the excitatory to inhibitory cell ratio could be understood largely as a result of the overcompletness of representation; moreover, a subpopulation of inhibitory interneurons exhibit orientation tuning due to sparse recurrent connections with the principal cells; another subpopulation displays untuned properties due to low rank connectivity patterns. We also demonstrate that the network exhibits balanced excitation and inhibition, as a result of the receptive field structure. We simulated a population of 2048 excitatory neurons with graded response described by the dynamics of locally competitive algorithm (LCA; [6]), which converges to the sparse coding representation at steady state. Inhibitory interneurons were described by linear units. The low rank and sparse recurrent connectivity pattern was a result of low rank plus sparse decomposition [7] of the LCA connectivity matrix. Non-classical receptive field effects were studied by presenting bar and drifting grating stimuli to the simulated network. Receptive fields of the inhibitory cells were mapped by sparse dots patterns.

Surveillance video processing using compressive sensing

A compressive sensing method combined with decomposition of a matrix formed with image frames of a surveillance video into low rank and sparse matrices is proposed to segment the background and extract moving objects in a surveillance video. The video is acquired by compressive measurements, and the measurements are used to reconstruct the video by a low rank and sparse decomposition of matrix. The low rank component represents the background, and the sparse component is used to identify moving objects in the surveillance video. The decomposition is performed by an augmented Lagrangian alternating direction method. Experiments are carried out to demonstrate that moving objects can be reliably extracted with a small amount of measurements.

Classes of graphs with small rank decompositions are [formula omitted]-bounded

A class of graphs G is χ-bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is χ-bounded, then every class of graphs admitting a decomposition along cuts of small rank to graphs from G is χ-bounded. As a corollary, we obtain that every class of graphs with bounded rank-width (or equivalently, clique-width) is χ-bounded.

Tensor Learning for Regression

In this paper, we exploit the advantages of tensorial representations and propose several tensor learning models for regression. The model is based on the canonical/parallel-factor decomposition of tensors of multiple modes and allows the simultaneous projections of an input tensor to more than one direction along each mode. Two empirical risk functions are studied, namely, the square loss and ε -insensitive loss functions. The former leads to higher rank tensor ridge regression (TRR), and the latter leads to higher rank support tensor regression (STR), both formulated using the Frobenius norm for regularization. We also use the group-sparsity norm for regularization, favoring in that way the low rank decomposition of the tensorial weight. In that way, we achieve the automatic selection of the rank during the learning process and obtain the optimal-rank TRR and STR. Experiments conducted for the problems of head-pose, human-age, and 3-D body-pose estimations using real data from publicly available databases, verified not only the superiority of tensors over their vector counterparts but also the efficiency of the proposed algorithms.

The Corona Factorization Property, Stability, and the Cuntz Semigroup of a C*-algebra

The Corona Factorization Property, originally invented to study exten- sions of C � -algebras, conveys essential information about the intrinsic structure of the C � -algebra. We show that the Corona Factorization Property of a �-unital C � -algebra is completely captured by its Cuntz semigroup (of equivalence classes of positive ele- ments in the stabilization of A). The corresponding condition in the Cuntz semigroup is a very weak comparability property termed the Corona Factorization Property for semigroups. Using this result one can for example show that all unital C � -algebras with finite decomposition rank have the Corona Factorization Property. Applying similar techniques we study the related question of when C � -algebras are stable. We give an intrinsic characterization, that we term property (S), of C � - algebras that have no non-zero unital quotients and no non-zero bounded 2-quasitraces. We then show that property (S) is equivalent to stability provided that the Cuntz semigroup of the C � -algebra satisfies another (also very weak) comparability property, that we call the !-comparison property.

Tensor decomposition in post-Hartree–Fock methods. I. Two-electron integrals and MP2

A new approximation for post-Hartree-Fock (HF) methods is presented applying tensor decomposition techniques in the canonical product tensor format. In this ansatz, multidimensional tensors like integrals or wavefunction parameters are processed as an expansion in one-dimensional representing vectors. This approach has the potential to decrease the computational effort and the storage requirements of conventional algorithms drastically while allowing for rigorous truncation and error estimation. For post-HF ab initio methods, for example, storage is reduced to O(d·R·n) with d being the number of dimensions of the full tensor, R being the expansion length (rank) of the tensor decomposition, and n being the number of entries in each dimension (i.e., the orbital index). If all tensors are expressed in the canonical format, the computational effort for any subsequent tensor contraction can be reduced to O(R(2)·n). We discuss details of the implementation, especially the decomposition of the two-electron integrals, the AO-MO transformation, the Møller-Plesset perturbation theory (MP2) energy expression and the perspective for coupled cluster methods. An algorithm for rank reduction is presented that parallelizes trivially. For a set of representative examples, the scaling of the decomposition rank with system and basis set size is found to be O(N(1.8)) for the AO integrals, O(N(1.4)) for the MO integrals, and O(N(1.2)) for the MP2 t(2)-amplitudes (N denotes a measure of system size) if the upper bound of the error in the l(2)-norm is chosen as ε = 10(-2). This leads to an error in the MP2 energy in the order of mHartree.

Ranks of operators in simple [formula omitted]-algebras

Let A be a unital simple separable C ∗ -algebra with strict comparison of positive elements. We prove that the Cuntz semigroup of A is recovered functorially from the Murray–von Neumann semigroup and the tracial state space T ( A ) whenever the extreme boundary of T ( A ) is compact and of finite covering dimension. Combined with a result of Winter, we obtain Z ⊗ A ≅ A whenever A moreover has locally finite decomposition rank. As a corollary, we confirm Elliott's classification conjecture under reasonably general hypotheses which, notably, do not require any inductive limit structure. These results all stem from our investigation of a basic question: what are the possible ranks of operators in a unital simple C ∗ -algebra?