Tensor Reconstruction Research Articles

Nonconvex Low-Rank Tensor Completion from Noisy Data

This paper investigates a problem of broad practical interest, namely, the reconstruction of a large-dimensional low-rank tensor from highly incomplete and randomly corrupted observations of its entries. Although a number of papers have been dedicated to this tensor completion problem, prior algorithms either are computationally too expensive for large-scale applications or come with suboptimal statistical performance. Motivated by this, we propose a fast two-stage nonconvex algorithm—a gradient method following a rough initialization—that achieves the best of both worlds: optimal statistical accuracy and computational efficiency. Specifically, the proposed algorithm provably completes the tensor and retrieves all low-rank factors within nearly linear time, while at the same time enjoying near-optimal statistical guarantees (i.e., minimal sample complexity and optimal estimation accuracy). The insights conveyed through our analysis of nonconvex optimization might have implications for a broader family of tensor reconstruction problems beyond tensor completion.

Asymmetric susceptibility tensor imaging.

To investigate the symmetry constraint in susceptibility tensor imaging. The linear relationship between the MRI frequency shift and the magnetic susceptibility tensor is derived without constraining the tensor to be symmetric. In the asymmetric case, the system matrix is shown to be maximally rank 6. Nonetheless, relaxing the symmetry constraint may still improve tensor estimation because noise and image artifacts do not necessarily follow the constraint. Gradient echo phase data are obtained from postmortem mouse brain and kidney samples. Both symmetric and asymmetric tensor reconstructions are applied to the data. The reconstructions are then used for susceptibility tensor imaging fiber tracking. Simulations with ground truth and at various noise levels are also performed. The reconstruction methods are compared qualitatively and quantitatively. Compared to regularized and unregularized symmetric reconstructions, the asymmetric reconstruction shows reduced noise and streaking artifacts, better contrast, and more complete fiber tracking. In simulation, the asymmetric reconstruction achieves better mean squared error and better angular difference in the presence of noise. Decomposing the asymmetric tensor into its symmetric and antisymmetric components confirms that the underlying susceptibility tensor is symmetric and that the main sources of asymmetry are noise and streaking artifacts. Whereas the susceptibility tensor is symmetric, asymmetric reconstruction is more effective in suppressing noise and artifacts, resulting in more accurate estimation of the susceptibility tensor.

Synchronized Entry-Traffic Flow Prediction for Regional Expressway System Based on Multidimensional Tensor

Predicting entry-traffic flows synchronously could enable inferences about the changing trends and spatial structure of dynamic traffic flows in an expressway network. This research develops a synchronized entry-traffic flow prediction method for regional expressway systems. The new method first organizes numerous entry-traffic flows as a three-dimensional (time slots, spatial locations, and vehicle types) tensor, then applies tensor decomposition to extract their temporally changing features. After forecasting the temporally changing features, predicted values of entry-traffic flows can be calculated synchronously by tensor reconstruction. Data from hourly entry-traffic flows involving nine vehicle types and 201 spatial locations in a regional expressway system of China are used to discuss the performance of this new method. The results show that the new method could obtain prediction results with high overall accuracy. Comparative experiments indicate that the new method and existing methods (autoregressive integrated moving average, or ARIMA, and Holt-Winters) could generate prediction results with similar accuracy. However, the proposed method has the advantage of reducing the number of time series that need to be handled in the prediction of numerous entry-traffic flows for regional expressway systems. This method might be helpful for administrators to guide and manage vehicles so that they enter the expressway system effectively.

Polarization-sensitive optical diffraction tomography

Polarization of light has been widely used as a contrast mechanism in two-dimensional (2D) microscopy and also in some three-dimensional (3D) imaging modalities. In this paper, we report the 3D tomographic reconstruction of the refractive index (RI) tensor using 2D scattered fields measured for different illumination angles and polarizations. Conventional optical diffraction tomography (ODT) has been used as a quantitative, label-free 3D imaging method. It is based on the scalar formalism, which limits its application to isotropic samples. We achieve imaging of the birefringence of 3D objects through a reformulation of ODT based on vector diffraction theory. The off-diagonal components of the RI tensor reconstruction convey additional information that is not available in either conventional scalar ODT or 2D polarization microscopy. Finally, we show experimental reconstructions of 3D objects with a polarization-sensitive contrast metric quantitatively displaying the true birefringence of the samples.

Reconstruction of depth resolved strain tensor in off-axis single crystals: Application to H+ ions implanted LiTaO3

In the context of growing interest in strain engineering, we present a theoretical protocol for the reconstruction of extrinsic and intrinsic strain tensors in single-crystals attached to a template, with an arbitrary oriented coordinate system. Input data for the protocol are extrinsic deformations of lattice planes, i.e., measured with reference to a template. By combining the protocol with elasticity theory, material property modification can be elucidated. Different methods for strain measurements can take advantage of this approach. It has been applied for reconstruction of strain tensor depth distribution in off-axis LiTaO3 crystals implanted with H+ ions, which is the key step for piezoelectric thin film-on-insulator fabrication by the Smart Cut process. Modifications of composition, lattice parameters, and elastic constants are indicated and discussed.

Hyperspectral image super-resolution via subspace-based fast low tensor multi-rank regularization

The demand for acquiring a high-spatial-resolution hyperspectral image (HR-HSI) imposes challenges to existing hyperspectral imaging tasks. Generally, fusing a high-spatial-resolution multispectral image (HR-MSI) with a low-spatial-resolution hyperspectral image (LR-HSI) is likely the most viable path towards HR-HSI. The Low Tensor Multi-Rank (LTMR) method is demonstrated a good fusion performance. However, it takes a long execution time. To rectify this problem, we propose a fast low tensor multi-rank (FLTMR) hyperspectral super-resolution method to improve the performance in three aspects. To speed up the estimation of spectral coefficients, a set of characteristic values representing HR-HSI are extracted from LR-HSI as the truncation value of tensor reconstruction. To further optimize the calculation time of the algorithm, a truncated value weight coefficient is introduced. Furthermore, applying the iterative attenuation coefficient to the iterative solution can accelerate and achieve better coefficient estimation results. Consequently, our FLTMR approach alleviates the excellent fusion results’ requirements and enables rapid processing of the fusion. It can be illustrated that our algorithm consumes only 1/4 to 2/3 of the time while achieves better or similar performance to LTMR. Essentially, owing to our proposed method’s excellent performance in the coefficient tensor reconstruction, our method may have important significance for the study of tensor reconstruction.

Joint DOD and DOA estimation using tensor reconstruction based sparse representation approach for bistatic MIMO radar with unknown noise effect

Achieving a fair balance between accuracy and computational complexity of limited measurement signals by algorithms for the bistatic multiple-input multiple-output (MIMO) radar under unknown noise effect has been a seemingly difficult task for most covariance methods. In this paper, the aim is to present an efficient method to achieve an improved estimation of the joint direction of departure (DOD) and direction of arrival (DOA) for the bistatic MIMO radar with an unknown ‘Toeplitz’ colored noise effect. First, by taking advantage of the static property of the noise effect, a tensor reconstruction-based imaginary Hermitian matrix is developed to eliminate the unknown noise effect. Then the 2D angle estimation problem is then reduced to a 1D sparse recovery problem where the target sparsity is exploited. Further, we formulate a reverse 1D pairwise Simultaneous Orthogonal Matching Pursuit and sparse Bayesian learning algorithms to reconstruct the sparse signal and estimate the joint DOD-DOA of the target. In contrast with the existing tensor-based methods, the proposed approach not only is robust to the influence of Toeplitz colored noise, resolves targets with limited measurement data but also ensures superior performance with lower computational complexity. Numerical simulation conducted under varying conditions verifies the effectiveness of the proposed approach.

Implementation of clinical tractography for pre-surgical planning of space occupying lesions: An investigation of common acquisition and post-processing methods compared to dissection studies.

Background and purposeThere is limited standardization of acquisition and processing methods in diffusion tractography for pre-surgical planning, leading to a range of approaches. In this study, a number of representative acquisition variants and post processing methods are considered, to assess their importance when implementing a clinical tractography program.MethodsDiffusion MRI was undertaken in ten healthy volunteers, using protocols typical of clinical and research scanning: a 32-direction diffusion acquisition with and without peripheral gating, and a non-gated 64 diffusion direction acquisition. All datasets were post-processed using diffusion tensor reconstruction with streamline tractography, and with constrained spherical deconvolution (CSD) with both streamline and probabilistic tractography, to delineate the cortico-spinal tract (CST) and optic radiation (OR). The accuracy of tractography results was assessed against a histological atlas using a novel probabilistic Dice overlap technique, together with direct comparison to tract volumes and distance of Meyer’s loop to temporal pole (ML-TP) from dissections studies. Three clinical case studies of patients with space occupying lesions were also investigated.ResultsTracts produced by CSD with probabilistic tractography provided the greatest overlap with the histological atlas (overlap scores of 44% and 52% for the CST and OR, respectively) and best matched tract volume and ML-TP distance from dissection studies. The acquisition protocols investigated had limited impact on the accuracy of the tractography. In all patients, the CSD based probabilistic tractography created tracts with greatest anatomical plausibility, although in one case anatomically plausible pathways could not be reconstructed without reducing the probabilistic threshold, leading to an increase in false positive tracts.ConclusionsAdvanced post processing techniques such as CSD with probabilistic tractography are vital for pre-surgical planning. However, overall accuracy relative to dissection studies remains limited.

Nearest Neighbor Gaussian Process Emulation for Multi-Dimensional Array Responses in Freeze Nano 3D Printing of Energy Devices

Abstract Energy 3D printing processes have enabled the manufacturing of energy storage devices with complex structures, high energy density, and high power density. Among these processes, freeze nano printing (FNP) has risen as a promising process. However, quality problems are among the biggest barriers for FNP and other 3D printing processes. Particularly, the droplet solidification time in FNP governs the thermal distribution and subsequently determines the product solidification, formation, and quality. To describe the solidification time, a physical-based heat transfer model is built. But, it is computationally intensive. The objective of this work is to build an efficient emulator for the physical model. There are several challenges unaddressed before: (1) the solidification time at various locations, which is a multi-dimensional array response, needs to be modeled and (2) the construction and evaluation of the emulator at new process settings need to be quick and accurate. Here, we integrate joint tensor decomposition and nearest neighbor Gaussian process (NNGP) to construct an efficient multi-dimensional array response emulator with process settings as inputs. Specifically, structured joint tensor decomposition decomposes the multi-dimensional array responses at various process settings into the setting-specific core tensors and shared low-dimensional factorization matrices. Then, each independent entry of the core tensor is modeled with an NNGP, which addresses the computationally intensive model estimation problem by sampling the nearest neighborhood samples. Finally, tensor reconstruction is performed to make predictions of the solidification time for new process settings. The proposed framework is demonstrated by emulating the physical model of FNP and compared with alternative tensor (multi-dimensional array) regression models.

A protocol for the analysis of DTI data collected from young children

Analysis of scalar maps obtained by diffusion tensor imaging (DTI) produce valuable information about the microstructure of the brain white matter. The DTI scanning of child populations, compared with adult groups, requires specifically designed data acquisition protocols that take into consideration the trade-off between the scanning time, diffusion strength, number of diffusion directions, and the applied analysis techniques. Furthermore, inadequate normalization of DTI images and non-robust tensor reconstruction have profound effects on data analyses and may produce biased statistical results. Here, we present an acquisition sequence that was specifically designed for pediatric populations, and describe the analysis steps of the DTI data collected from extremely preterm-born young school-aged children and their age- and gender-matched controls. The protocol utilizes multiple software packages to address the effects of artifacts and to produce robust tensor estimation. The computation of a population-specific template and the nonlinear registration of tensorial images with this template were implemented to improve alignment of brain images from the children.

Nonconvex Robust Low-Rank Tensor Reconstruction via an Empirical Bayes Method

Low-rank tensor reconstruction has attracted a great deal of research interest in signal processing, image processing and machine learning. To deal with outliers that are now ubiquitous in many modern applications, a representative approach is to decompose and reconstruct the data into a low-rank component and a sparse outlier component. As surrogate functions of the matrix rank and the sparsity that are non-convex and discontinuous, the nuclear norm and the $\ell _1$ norm are often exploited for recovering higher order tensors and removing outliers, respectively. However, the nuclear norm and the $\ell _1$ norm are loose approximations of the matrix rank and the sparsity, respectively. Furthermore, the tensor nuclear norm is not guaranteed to be the tightest convex envelope of a multilinear rank. In this paper, we develop a new method for robust tensor reconstruction. By capitalizing on the empirical Bayes method, whose performance has been shown to act as state-of-the-art to the performance of standard sparse and low-rank matrix recovery, we formulate the objective function of the empirical Bayes method for robust tensor reconstruction, and then adjust the objective function to facilitate algorithm derivation, while keeping desirable attributes including concavity and symmetry of the objective function of the empirical Bayes method. We demonstrate the superior performance of the proposed algorithm in comparison with state-of-the-art alternatives by conducting experiments on both synthetic data and real data.

Robust approximations of low-rank minimization for tensor completion

Motivated by the nuclear norm of tensors and nonconvex approximations of matrix rank, we propose three robust approximations of multi-linear rank for tensor completion. For each method, we develop an efficient algorithm to solve the corresponding optimization problem. Besides, we prove that every cluster point of the sequence, generated by the respective algorithm, is a stationary point. To obtain a more robust reconstruction, we design an updating rule of parameters for each method. Our empirical experiments on real-world data show that the proposed methods deliver state-of-the-art performance in the reconstruction of low-rank tensors.

Coherent and dissipative quantum process tensor reconstructions in two-dimensional electronic spectroscopy.

A major goal of time-resolved spectroscopy is to resolve the dynamical processes that follow photoexcitation. This amounts to identifying all the quantum states involved and the rates of population transfer between them. Unfortunately, such quantum state and kinetic reconstructions are ambiguous using one-dimensional methods such as transient absorption even when all the states of the system are fully resolved. Higher-dimensionality methods like two-dimensional spectroscopy lift some of the ambiguity, but unless the spectral features are well-separated, current inversion methods generally fail. Here, we show that, using both coherence and population signals of the nonlinear response, it is indeed possible to accurately extract both static and dynamic information from the 2D spectrum even when features are highly congested. Coherences report on the positions of the vibronic states of the system, providing a useful constraint for extracting the full kinetic scheme. We model time-resolved 2D photon echo spectra using a sum-over-states approach and show in which regimes the Hamiltonian and kinetic schemes may be recovered. Furthermore, we discuss how such algorithms may be applied to experimental data and where some of the underlying assumptions may fail. The ability to systematically extract the maximal information content of multidimensional spectroscopic data is an important step toward utilizing the full power of these techniques and elucidating the structure and dynamics of increasingly complex molecular systems.

L1-Norm Tucker Tensor Decomposition

Tucker decomposition is a standard multi-way generalization of Principal-Component Analysis (PCA), appropriate for processing tensor data. Similar to PCA, Tucker decomposition has been shown to be sensitive against faulty data, due to its L2-norm-based formulation which places squared emphasis to peripheral/outlying entries. In this work, we explore L1-Tucker, an L1-norm based reformulation of Tucker decomposition, and present two algorithms for its solution, namely L1-norm Higher-Order Singular Value Decomposition (L1-HOSVD) and L1-norm Higher-Order Orthogonal Iterations (L1-HOOI). The proposed algorithms are accompanied by complexity and convergence analysis. Our numerical studies on tensor reconstruction and classification corroborate that L1-Tucker decomposition, implemented by means of the proposed algorithms, attains similar performance to standard Tucker when the processed data are corruption-free, while it exhibits sturdy resistance against heavily corrupted entries.

PARAFAC Tensor Reconstruction for Recommender System based on Apache Spark

PARAFAC Tensor Reconstruction for Recommender System based on Apache Spark

The use of diffusion tractography to characterize a corpus callosum malformation in a dog.

IntroductionThe pathogenesis of corpus callosum malformations (CCM) is not well defined in the dog because of inherent limitations of structural magnetic resonance imaging (MRI) to evaluate the white matter. We used the advanced neuroimaging technique of tractography to virtually dissect the white matter projections in a dog with a CCM and in a normal control dog.MethodsA 9‐month‐old male Coonhound that had a previous structural MRI diagnosis of CCM and a normal control dog underwent anesthesia and 3‐Tesla MRI. Diffusion‐tensor imaging and 3D T1‐weighted and 2D T2‐weighted sequences were acquired. Diffusion data were processed before tensor reconstruction and fiber tracking. Virtual dissections were performed to dissect out the major white matter projections in each dog.ResultsIn the dog with CCM, the corpus callosum exhibited interhemispheric crossing fibers at the level of the splenium and formed longitudinal callosal fasciculi (Probst bundles). In addition, the fornix was small and the cingula enlarged and exhibited increased dorsal connectivity relative to the normal control.Conclusions and Clinical ImportanceWe used tractography to describe a white matter malformation in a dog. The results suggest that, embryologically, formed axons fail to cross midline and instead create Probst bundles.

High performance hardware architecture for singular spectrum analysis of Hankel tensors

This paper presents a hardware architecture for singular spectrum analysis of Hankel tensors, including computation of tucker decomposition, tensor reconstruction and final Hankelization. In the proposed design, we explore two level of optimization. First, in algorithm level, we optimize the calculation process by exploiting the Hankel property to reduce the computation complexity and on-chip BRAM resource usage. Secondly, in hardware level, parallelism is explored for acceleration. Resource sharing is applied to reduce look-up tables (LUTs) usage. To enable flexibility, the number of processing elements (PEs) can be changed through parameter setting. Our proposed design is implemented on Field-Programmable Gate Arrays (FPGAs) to process third order tensors. Experiment results show that our design achieve a speed-up from 172 to 1004 compared with CPU implementation via Intel MKL and 5 to 40 compared with GPU implementation.

Non-intrusive Tensor Reconstruction for High-Dimensional Random PDEs

Abstract This paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank solutions of high-dimensional parametric random PDEs, which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel black-box rank-adapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to (Quasi-)Monte Carlo sampling.

Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

Abstract We consider hyperelastic problems and their numerical solution using a conforming finite element discretization and iterative linearization algorithms. For these problems, we present equilibrated, weakly symmetric, H ⁢ ( div ) {H(\mathrm{div)}} -conforming stress tensor reconstructions, obtained from local problems on patches around vertices using the Arnold–Falk–Winther finite element spaces. We distinguish two stress reconstructions: one for the discrete stress and one representing the linearization error. The reconstructions are independent of the mechanical behavior law. Based on these stress tensor reconstructions, we derive an a posteriori error estimate distinguishing the discretization, linearization, and quadrature error estimates, and propose an adaptive algorithm balancing these different error sources. We prove the efficiency of the estimate, and confirm it on a numerical test with an analytical solution. We then apply the adaptive algorithm to a more application-oriented test, considering the Hencky–Mises and an isotropic damage model.

Discriminative tracking via supervised tensor learning

Abstract Discriminative tracking algorithms have witnessed continued progress for distinguishing the target from background in unconstrained environments. The learning and detection task in existing visual tracking methods often convert a multidimensional data array into a vector-based observation. By altering the 2-D spatial structure of the image, transformation variants and global noises influence the discriminative ability of target representation, often result in degradation of performance. Different from vector representations, this paper presents a tensor-based large margin discriminative framework for visual tracking that utilizes the supervised tensor learning. In our method, an online structured support tensor classifier is designed which produces the multi-linear decision function, incorporating the nonlinearity of tensor-based feature over the target. In order to provide better spatial cues of target representation against noises and facilitate online tracking, we further introduce truncated tucker decomposition in structured multi-linear learning. The proposed algorithm poses an effective parameter tensor reconstruction in the classifier updating procedure and has a robust discriminative ability against several video background variants. Furthermore, a tensor block coordinate descent optimization is presented to achieve a closed form solution specific to the proposed truncated structured Tucker machine (TSTM). Experiment results on a recent comprehensive tracking benchmark demonstrate a promising performance of the proposed method subjectively and objectively compared with several state-of-the-art algorithms.