Minimum Norm Research Articles

Robust low tubal rank tensor completion via factor tensor norm minimization

Recent research has demonstrated that low tubal rank recovery based on tensor has received extensive attention. In this correspondence, we define tensor double nuclear norm and tensor Frobenius/nuclear hybrid norm to induce a surrogate for tensor tubal rank, and prove that they are equivalent to tensor Schatten-p norm for p=1/2 and p=2/3. Based on the definition, we propose two novel tractable tensor completion models called Double Nuclear norm regularized Tensor Completion (DNTC) and Frobenius/Nuclear hybrid norm regularized Tensor Completion (FNTC) by integrating these two norm minimization and factorization methods into a joint learning framework. Furthermore, we adopt invertible linear transforms to obtain low tubal rank tensors, which makes the model more flexible and effective. Two efficient algorithms are designed to solve the proposed tensor completion models by incorporating the convexity of the factor norms. Comprehensive experiments are conducted on synthetic and real datasets to achieve better results in comparison with some state-of-the-art approaches.

Compressive sampling - based extrapolation of free surface displacement data from pressure measurements

This article deals with the problem of extrapolating the time history of the free surface displacement at a given point from wave pressure head records. This problem is relevant to marine engineering applications, where, for instance, the monitoring of marine structures must be arranged without resorting to expensive measurement techniques, or sensor failures must be circumvented through adequate signal processing techniques. Specifically, a Compressive Sensing (CS) based technique is developed by resorting to the concept of sparsity, which is implemented through formulating and solving a L1 – norm minimization problem.Relevant numerical examples show that the proposed CS approach allows extrapolating reliable free surface displacement time histories. Further, the data extrapolated via CS are in better agreement with target free surface data than standard Fourier - based approaches, while possessing the desirable feature of removing the subjective choice of a cut-off frequency requested by the standard techniques. CS limits are highlighted in terms of required optimal pressure sensor position below the mean water level, and of reliability in extrapolating the profiles of steep waves.

Minimal-norm state-feedback globally non-overshooting/ undershooting tracking control of multivariable systems

The non-overshooting/undershooting (NOUS) tracking control objective in the output response of the closed-loop system has several practical applications in “positioning” control problems. Recently, a state-feedback gain matrix based on Moore’s eigenstructure assignment technique to achieve the above control objective with a desired convergence rate in the output error from any initial condition is designed, referred to as globally monotonic tracking control, which is a convex optimization problem. From a practical viewpoint, a feedback gain matrix of a minimal norm is often required to install low-cost actuators in the overall closed-loop system. However, there is always a trade-off between the control objectives of achieving a NOUS tracking response with the desired convergence rate and a minimal norm of the feedback gain matrix. Besides, the latter control requirement renders the globally NOUS control optimization problem nonconvex. In this paper, a convex optimization-based iterative algorithm is developed to synthesize a minimal-norm state-feedback globally NOUS tracking controller. An upper bound on the achievable settling time in all output responses for the closed-loop eigenvalues lying in a prespecified region of the complex plane is also provided. The effectiveness of the proposed algorithm is demonstrated through a numerical example, followed by experimental validation on a multitank system.

Reproducing kernels and minimal solutions of elliptic equations

Abstract Suppose that the set of square-integrable solutions of an elliptic equation which have a value at some given point equal to c is not empty. Then there is exactly one element with minimal L 2 {L^{2}} -norm. Moreover, it is shown that this minimal element depends continuously on a domain of integration, i.e., on the set on which our solutions are defined, and on a weight of integration, i.e., on the deformation of an inner product. The theorems are proved using the theory of reproducing kernels and Hilbert spaces of square-integrable solutions of elliptic equations. We prove the existence of such a reproducing kernel using theory of Sobolev spaces. We generalize the well-known Ramadanov theorem. This is done in three different ways. Two of them are similar to the techniques used by I. Ramadanov and M. Skwarczyński(see [11, 14, 13]) , while the third method using weak convergence is new. Moreover, we show that our reproducing kernel depends continuously on a weight of integration. The idea of using the minimal norm property in such a proof is novel and, which is important, it needs the convergence of weights only almost everywhere.

A gridless DOA estimation algorithm based on unsupervised deep learning

A gridless direction-of-arrival (DOA) estimation algorithm with unsupervised deep learning is proposed to eliminate the influence of the spatial angle discretization on the estimation results and the deep neural network's reliance on labels in the training dataset. This algorithm is inspired by the atomic norm minimization (ANM) algorithm and unsupervised deep learning, and a loss function is designed to solve the ANM problem in the unsupervised deep learning framework. The proposed algorithm can address the grid mismatch issue and be more robust to the training dataset, since it no longer depends on the spatial angle discrete grid and labels in the training dataset. Simulation results show that the proposed algorithm is state of the art when compared with the other advanced algorithms currently in use.

Determination of the right-hand side in elliptic equations

ABSTRACT The problem of determining a term in the right-hand side of elliptic equations from an observation on a part of the boundary is investigated. The inverse problem is formulated as an operator equation and then stabilized by Tikhonov regularization method. The regularized problem is discretized based on Hinze's variational discretization concept and the regularization parameter is chosen guaranteeing that when noise level and the discretization mesh size tend to zero, the solution of the discretized regularized problem converges to the f ∗ -minimum norm solution of the continuous inverse problem. Some numerical examples are presented for illustrating the performance of the proposed method.

A Novel Demixing Algorithm for Joint Target Detection and Impulsive Noise Suppression

This work considers a collocated radar scenario where a probing signal is emitted toward the targets of interest and records the received echoes. Estimating the relative delay-Doppler shifts of the targets allows determining their relative locations and velocities. However, the received radar measurements are often affected by impulsive non-Gaussian noise which makes a few measurements partially corrupted. While demixing radar signal and impulsive noise is challenging in general by traditional subspace-based methods, atomic norm minimization (ANM) has been recently developed to perform this task in a much more efficient manner. Nonetheless, the ANM cannot identify close delay-Doppler pairs and also requires many measurements. Here, we propose a smoothed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula> atomic optimization problem encouraging both the sparse features of the targets and the impulsive noise. We design a majorization-minimization algorithm that converges to the solution of the proposed non-convex problem using alternating direction method of multipliers (ADMM). Simulations results verify the superior accuracy of our proposed algorithm even for very close delay-Doppler pairs in comparison to ANM with around 40 dB improvement.

Reconfigurable Intelligent Surface Enabled Interference Nulling and Signal Power Maximization in mmWave Bands

Reconfigurable intelligent surface (RIS) has emerged as a promising mean to enhance wireless transmission. The effective reflected paths provided by RIS are able to alleviate the susceptibility to blockage effects, especially in high-frequency band communications, where signals experience severe path loss and high directivity. This paper is concerned with an RIS-assisted system over the millimeter wave (mmWave) channel characterized by sparse propagation paths. A base station tries to connect with the desired user through an RIS, while the undesired user can also receive the signal transmitted from BS unavoidably, which is treated as the interference signal. All terminals are assumed to be equipped with a single antenna for the sake of simplicity. The paper aims to propose an appropriate design of the phase shifts of each element at the RIS so as to maximize the received signal power transmitted from the base station (BS) at the desired user, while nulling the received interference signal power at the undesired user. The proposed reflecting design relies on the decomposition of the reflecting beamforming vectors and all channel path vectors into Kronecker product of factors being uni-modulus vectors. By exploiting characteristics of Kronecker mixed products, different factors of the reflecting are designed for either nulling the interference signal at the undesired user, or coherently combining data paths at the desired user. Furthermore, a channel estimation strategy is proposed to enable the proposed reflecting beamforming design. The magnitude, azimuth, and elevation arrival and departure angles of desired and undesired paths are estimated by an efficient 2-dimension (2-D) line spectrum optimization technique based on the atomic norm minimization (ANM) framework. The performance of the reflecting designs and channel estimation scheme is analyzed and demonstrated by simulation results.

Transmission Optimization and Resource Allocation for Wireless Powered Dense Vehicle Area Network With Energy Recycling

The wireless-powered communication paradigm brings self-sustainability to the on-vehicle sensors by harvesting the energy from radiated radio frequency (RF) signals. This paper proposes a novel transmission and resource allocation strategy for the scenario where multiple wireless powered vehicle area networks (VAN) co-existed with high density. The considered multi-VAN system consists of a remote master access point (MAP), multiple on-vehicle hybrid access points (HAPs) and sensors. Unlike previous works, we consider that the sensors can recycle the radiated radio frequency energy from all the HAPs when HAPs communicate with MAP, so the dedicated signals for energy harvesting (EH) are unnecessary. The proposed strategy can achieve simultaneous wireless information and power transfer (SWIPT) without complex receiver architecture requirements. The extra EH and interference caused by the dense distribution of VANs, which are rarely explored, are fully considered. To maximize the sum throughput of all the sensors while guaranteeing the transmission from HAPs to the MAP, we jointly optimize the time allocation, system energy consumption, power allocation, and receive beamforming. Due to the non-convexity of the formulated problem, we address the sub-problems separately through the Rayleigh quotient, Frobenius norm minimization and convex optimization. Then an efficient iterative algorithm to obtain sub-optimal solutions. The simulation results and discussions illustrate the proposed scheme's effectiveness and advantages.

Oblique Projection-Based Covariance Matrix Reconstruction and Steering Vector Estimation for Robust Adaptive Beamforming

Adaptive beamforming can efficiently contract interference and noise. Due to high sensitivity of the beamformer to model mismatch, the capability of interference reduction will critically degrade when the signal model mismatch occurs, particularly when the sampling sequence contains the desired signal. For the purpose of enhancing the robustness of beamformers to signal model mismatch, we propose a new robust adaptive beamforming (RAB) method. Firstly, the precise steering vector (SV) associating with the desired signal is estimated by employing the minimum norm of subspace projection (MNSP) approach. Secondly, the nominal interference SVs are estimated via the maximum entropy power spectrum. Subsequently, the corrected interference SVs and powers are obtained by oblique projection. Finally, the interference-plus-noise covariance matrix (INCM) is reconstructed, and the proposed RAB is obtained. Multiple simulations are carried out and demonstrate the robustness of the proposed RAB method.

Energy optimal guidance for proximity approach with obstacle avoidance

This paper presents an energy optimal guidance algorithm for the approach of a damaged spacecraft with the consideration of the arbitrary number of moving obstacles. The proposed optimal guidance laws are derived by projecting the control function in L2 Space to obtain the minimum norm controller. Unlike conventional trajectory optimization, this algorithm constructs an L2 space based on the state transfer matrix and describes the path constraints as inner products in L2 space. Following this, the proximity approach and obstacle avoidance/rendezvous are combined into a single but global optimal step by the Projection Theorem while obtaining the analytical energy consumption. Examples for multiple static and moving obstacles are presented to demonstrate the advantages of the proposed guidance over traditional trajectory optimization methods. The optimization result of the proposed method is slightly better than GPOPS and the computation time is about five-thousandths of GPOPS.

Improved crest factor minimization of multisine excitation signals using nonlinear optimization

In frequency-domain system identification, multisine waveforms are frequently used as an excitation signal. Supplying a low Crest Factor (CF) input signal to the actuator of the system is essential for measuring transfer functions with a high signal-to-noise ratio. In this paper, we present a method for achieving a lower CF compared to state-of-the-art methods. Our two-step method starts with Guillaume’s method, followed by solving a slack reformulation of the ∞-norm minimization problem for additional improvement. The proposed method is compared to the existing ones in terms of the achievable CF. The performance of the CF reduction methods has been analyzed using statistical methods.

Assessing HD-EEG functional connectivity states using a human brain computational model

Objective. Electro/Magnetoencephalography (EEG/MEG) source-space network analysis is increasingly recognized as a powerful tool for tracking fast electrophysiological brain dynamics. However, an objective and quantitative evaluation of pipeline steps is challenging due to the lack of realistic ‘controlled’ data. Here, our aim is two-folded: (a) provide a quantitative assessment of the advantages and limitations of the analyzed techniques and (b) introduce (and share) a complete framework that can be used to optimize the entire pipeline of EEG/MEG source connectivity. Approach. We used a human brain computational model containing both physiologically based cellular GABAergic and Glutamatergic circuits coupled through Diffusion Tensor Imaging, to generate high-density EEG recordings. We designed a scenario of successive gamma-band oscillations in distinct cortical areas to emulate a virtual picture-naming task. We identified fast time-varying network states and quantified the performance of the key steps involved in the pipeline: (a) inverse models to reconstruct cortical-level sources, (b) functional connectivity measures to compute statistical interdependency between regional signals, and (c) dimensionality reduction methods to derive dominant brain network states (BNS). Main results. Using a systematic evaluation of the different decomposition techniques, results show significant variability among tested algorithms in terms of spatial and temporal accuracy. We outlined the spatial precision, the temporal sensitivity, and the global accuracy of the extracted BNS relative to each method. Our findings suggest a good performance of weighted minimum norm estimate/ Phase Locking Value combination to elucidate the appropriate functional networks and ICA techniques to derive relevant dynamic BNS. Significance. We suggest using such brain models to go further in the evaluation of the different steps and parameters involved in the EEG/MEG source-space network analysis. This can reduce the empirical selection of inverse model, connectivity measure, and dimensionality reduction method as some of the methods can have a considerable impact on the results and interpretation.

Matrix Completion With Data-Dependent Missingness Probabilities

The problem of completing a large matrix with lots of missing entries has received widespread attention in the last couple of decades. Two popular approaches to the matrix completion problem are based on singular value thresholding and nuclear norm minimization. Most of the past works on this subject assume that there is a single number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> such that each entry of the matrix is available independently with probability <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> and missing otherwise. This assumption may not be realistic for many applications. In this work, we replace it with the assumption that the probability that an entry is available is an unknown function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> of the entry itself. For example, if the entry is the rating given to a movie by a viewer, then it seems plausible that high value entries have greater probability of being available than low value entries. We propose two new estimators, based on singular value thresholding and nuclear norm minimization, to recover the matrix under this assumption. The estimators involve no tuning parameters, and are shown to be consistent under a low rank assumption. We also provide a consistent estimator of the unknown function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> .

Dependent Function Embedding for Distributed Serverless Edge Computing

Edge computing is booming as a promising paradigm to extend service provisioning from the centralized cloud to the network edge. Benefit from the development of serverless computing, an edge server can be configured as a carrier of limited serverless functions, in the way of deploying Docker runtime and Kubernetes engine. Meanwhile, an application generally takes the form of directed acyclic graphs (DAGs), where vertices represent dependent functions and edges represent data traffic. The status quo of minimizing the completion time (a.k.a. makespan) of the application motivates the study on optimal function placement. However, current approaches lose sight of proactively splitting and mapping the traffic to the logical data paths between the heterogeneous edge servers, which could affect the makespan significantly. To remedy that, we propose an algorithm, termed as Dependent Function Embedding (DPE), to get the optimal edge server for each function to execute and the moment it starts executing. DPE finds the best segmentation of each data traffic by exquisitely solving several infinity norm minimization problems. DPE is theoretically verified to achieve the global optimality. Extensive experiments on Alibaba cluster trace show that DPE significantly outperforms two baseline algorithms in makespan by 43.19% and 40.71%, respectively.

A physics-based crystal plasticity model for the prediction of the dislocation densities in micropillar compression

• Generalized constitutive relations that incorporate the thermally activated and drag mechanisms. • Continuum description of crystal plasticity framework combined with the geometrically necessary dislocation densities and the mobile/immobile statistically stored dislocation densities. • Computation of the geometrically necessary dislocation densities based on the L 1 -norm minimization scheme to provide physical interpretation of the lower bound on overall GND population. • Dislocation mechanisms depending on generation, annihilation, interactions, trapping, and recovery. • Evolution laws of dislocation densities with mechanism-based material parameters. A new nonlocal crystal plasticity model is developed to study the mechanical behavior of face-centered cubic single crystals under heterogeneous inelastic deformation through a crystal plasticity finite element method. The main feature of this work is generalized constitutive relations that incorporate the thermally activated and drag mechanisms to cover different kinetics of viscoplastic flow in metals at a variety of ranges of stresses and strain rates. The constitutive laws are founded upon integrating continuum description of crystal plasticity framework with dislocation densities which is relevant to the geometrically necessary dislocation densities and the statistically stored dislocation densities. The model describes the plastic flow and the yielding of face-centered cubic single-crystal employing evolution laws of dislocation densities with mechanism-based material parameters passed from experiments or small-scale computational models. The geometrically necessary dislocation densities evolve on account of the curl of the plastic deformation gradient where its associated closure failure of the Burgers circuit exists. A minimization scheme termed L 1 -norm is utilized to secure the physical values of the geometrically necessary dislocation densities on slip systems. The evolution equations of statistically stored dislocation densities describe the complex interactions between two distinct dislocation populations, mobile, and immobile statistically stored dislocation dislocations, relying on generation, annihilation, interactions, trapping, and recovery. The experiments of a micropillar compression for the copper single crystal are compared to the computational results obtained using the formulation. The physics-based model clarifies the complex microstructural evolution of dislocation densities in metals and alloys, allowing for more accurate prediction. .

Sparsity-Based Two-Dimensional DOA Estimation for Co-Prime Planar Array via Enhanced Matrix Completion

In this paper, the two-dimensional (2-D) direction-of-arrival (DOA) estimation problem is explored for the sum-difference co-array (SDCA) generated by the virtual aperture expansion of co-prime planar arrays (CPPA). Since the SDCA has holes, this usually causes the maximum virtual aperture of CPPA to be unavailable. To address this issue, we propose a complex-valued, sparse matrix recovery-based 2-D DOA estimation algorithm for CPPA via enhanced matrix completion. First, we extract the difference co-arrays (DCA) from SDCA and construct the co-array interpolation model via nuclear norm minimization to initialize the virtual uniform rectangular array (URA) that does not contain the entire rows and columns of holes. Then, we utilize the shift-invariance structure of the virtual URA to construct the enhanced matrix with a two-fold Hankel structure to fill the remaining empty elements. More importantly, we apply the alternating direction method of the multipliers (ADMM) framework to solve the enhanced matrix completion model. To reduce the computational complexity of the traditional vector-form, sparse recovery algorithm caused by the Kronecker product operation between dictionary matrices, we derive a complex-valued sparse matrix-recovery model based on the fast iterative shrinkage-thresholding (FISTA) method. Finally, simulation results demonstrate the effectiveness of the proposed algorithm.

Food security in Ukraine: state and current issues of nowadays

The problem of food security in Ukraine and in the world in general has become especially relevant due to the growing population on the planet, depletion of natural resources, declining soil productivity, climate change and military conflicts. The number of countries with existing food security problems has increased in recent years. Different views on food security, state of food security in Ukraine are analyzed and comparisons with the countries of Europe and the world are made. According to statistics, in 2020 the caloric content of the Ukrainian diet was only 7% higher than the threshold level of 2.500 kcal/day, but less than the minimum WHO physiological standard (3.000 kcal/day) while the caloric content of the Ukrainian diet in 1990 was 3597 kcal/day. In addition, the diet of the population of Ukraine remains unbalanced with a predominance of plant products. The most critical is the state of consumption of complete protein foods (meat, milk, fish) and vitamin products (fruits). Ukrainians consume even less than the minimum norm for fish and fish products, as well as fruit. The indicator of import dependence is calculated, according to which as of 2020 it was the highest for fish and fish products — 82%, which is 52% above the threshold level of 30%. The predominant share of imported supplies falls on fish species that are caught exclusively in the waters of maritime economic zones of other countries, while the consumption of fish and fish products by the population of Ukraine remains below the recommended dietary norms. It is established that due to domestic production more than 90% of our country meets the necessary consumer needs of the population for most foods. The analysis and comparative assessment of the structure of total consumer expenditures of households in 2016 and 2020, as well as comparisons with EU countries. According to the State Statistics Service, consumer spending of Ukrainian households is 91.4% of income. The largest share of their income (48.1%) Ukrainians spent on food, housing and utilities — 14.4%. In the EU, housing, utilities and fuel rank first in the structure of household consumption expenditures — 25.7% of total expenditures. Citizens of Slovakia and Finland spend the most on this item — 30.7% and 30.5% of all expenses, respectively. The lowest one is 15.5% in Lithuania. Europeans spent on food and non-alcoholic beverages on average 14.8% of the total consumer spending structure, second only to utilities.

A reweighted alternating direction method of multipliers for joint phase adjustment and ISAR imaging

ABSTRACT In view of inverse synthetic aperture radar (ISAR) imaging, sparse aperture not only affects the imaging quality of traditional range Doppler (RD) algorithm but also increases the difficulty of translational motion compensation (TMC). For the purpose of reducing the influence of motion errors on the target imaging, reweighted alternating direction method of multipliers (RADMM) is proposed to realize the joint processing of phase adjustment and ISAR imaging. Combined with pattern-coupled sparse structure (PCSS) information of the target, the reweighted norm minimization problem is established first. Then, the minimum entropy method is adopted to estimate the phase error, and the iterative process of RADMM is deduced. Finally, the experimental results based on two sets of measured data demonstrate the validity of the proposed algorithm under the conditions of noise and sparse aperture.

Spatio-Temporal Changes in Air Quality of the Urban Area of Chongqing from 2015 to 2021 Based on a Missing-Data-Filled Dataset

Air pollution is one of the severe environmental issues in Chongqing. Many measures made by the government for improving air quality have been put into use these past few years, while the influence of these measures remains unknown. This study analyzed the changes in the air quality of the urban area of Chongqing between 2015 and 2021 using a complete in situ observation dataset that all missing data were filled by the interpolation of a low-rank tensor completion model with truncate nuclear norm minimization (LRTC-TNN). The results include: (1) the LRTC-TNN model robustly performs to reconstruct missing data of pollutant concentrations with an R2 of 0.93 and an RMSE of 7.78; (2) the air quality index (AQI) decreases by 15.96%, and the total polluted days decrease by 21.05% from 2015 to 2021, showing an obvious promotion in air quality; and (3) the changing air quality is attributed to decreasing concentrations of PM2.5 (34.10%), PM10 (25.03%), and NO2 (5.53%) from 2015 to 2021, whereas an increasing concentration of O3 (10.49%) is observed. The processing method for missing data, intact AQI datasets, and analysis of changes are beneficial to policy-making for environmental improvement and fill the gap in the field of data interpolation for air quality datasets in mountainous areas.