Underdetermined Case Research Articles

In defense of a ‘thick’ formal equality principle in healthcare resource distribution

Given resource constraints in healthcare, demands justice and equity require the constant development of material principles for resource distribution. In many cases, such material principles are formulated as mid-level principles, well-adapted to handle healthcare distribution but suffering from aspects outside the healthcare context that affect their application. In healthcare, factors outside the healthcare system will sometimes affect patients’ equal opportunity to receive treatment and achieve health. Examples of such factors might include an individual’s economic means, the cost of drugs, geography, etc. This article explores whether the formal equality principle could help us address such problems. It is argued that the traditional (thin) formal equality principle can aid in priority setting, both as a heuristic tool to aid in consistency and also in handling under-determined cases. However, it appears to be too ‘thin’ to handle problems caused by factors outside the healthcare context. In the article, I explore a more robust version of the formal equality principle and argue that such a principle can be used to address such cases. However, even a more comprehensive formal equality principle will need to consider whether some other actors have a moral responsibility to address the problem or whether the issue could be solved more effectively by others. Furthermore, such a principle needs to consider whether the opportunity cost of achieving ‘thick’ formal equality is acceptable, given the material principles of distribution.

A Separation Method for Electromagnetic Radiation Sources of the Same Frequency

To separate electromagnetic interference sources with an unknown source number, a new separation method is proposed, which includes five key steps: spatial spectrum estimation, source number and direction-of-arrival estimation, mixed matrix estimation, separation matrix estimation, and source signal recovery. A pseudospatial spectrum estimation network based on a convolutional neural network is proposed to estimate the number of electromagnetic radiation sources, their direction of arrival, and the mixing matrix. A new loss function is designed as an optimization criterion for estimating the separation matrix. To ensure generalization, both simulated and measured datasets are used to train the proposed network. Experimental results demonstrate that the proposed separation method outperforms existing source separation techniques in terms of correlation coefficient, root mean square error, and running time. Importantly, it exhibits strong performance in underdetermined cases, as well as in overdetermined or determined cases.

Accelerated generalized minimax-concave sparse regularization for impact force reconstruction and localization

Impact force identification has always been of significance for structure health monitoring especially on the applications involving composite materials. As a typical inverse problem, impact force reconstruction and localization is undoubtedly a challenging task. The well-known ℓ1 sparse regularization has a tendency to underestimate the amplitude of impact forces. To alleviate this limitation, we propose an accelerated generalized minimax-concave (AGMC) for sparse regularization that employs a non-convex generalized minimax-concave (GMC) penalty as the regularizer and incorporates an acceleration technique to expedite the attainment of the global minimum. Compared with the classic ℓ1-norm penalty, the GMC penalty can not only induce sparsity in the estimation, but also maintain the convexity of the cost function, so that the global optimal solution can be obtained through convex optimization algorithms. This method is applied to solve the impact force identification problem with unknown force locations to simultaneously reconstruct and localize impact forces in the under-determined case utilizing a limited number of sensors. Meanwhile, K-sparsity criterion is used to adaptively select regularization parameters by taking advantage of the sparse prior knowledge on impact forces. Simulations and experiments are conducted on a composite plate to verify the computational efficiency and robustness of the AGMC method in terms of impact force reconstruction and localization, particularly in the presence of noise. Results demonstrate that the proposed AGMC method achieves faster convergence and provides more accurate and sparse reconstruction and localization of impact forces compared to other state-of-the-art sparse regularization methods.

A Review of Tags Anti-Collision Identification Methods Used in RFID Technology

With radio frequency identification (RFID) becoming a popular wireless technology, more and more relevant applications are emerging. Therefore, anti-collision algorithms, which determine the time to tag identification and the accuracy of identification, have become very important in RFID systems. This paper presents the algorithms of ALOHA for randomness, the binary tree algorithm for determinism, and a hybrid anti-collision algorithm that combines these two algorithms. To compensate for the low throughput of traditional algorithms, RFID anti-collision algorithms based on blind source separation (BSS) are described, as the tag signals of RFID systems conform to the basic assumptions of the independent component analysis (ICA) algorithm. In the determined case, the ICA algorithm-based RFID anti-collision method is described. In the under-determined case, a combination of tag grouping with a blind separation algorithm and constrained non-negative matrix factorization (NMF) is used to separate the multi-tag mixing problem. Since the estimation of tag or frame length is the main step to solve the RFID anti-collision problem, this paper introduces an anti-collision algorithm based on machine learning to estimate the number of tags.

A matrix algebra approach to approximate Hessians

Abstract This work presents a novel matrix-based method for constructing an approximation Hessian using only function evaluations. The method requires less computational power than interpolation-based methods and is easy to implement in matrix-based programming languages such as MATLAB. As only function evaluations are required, the method is suitable for use in derivative-free algorithms. For reasonably structured sample sets, the method is proven to create an order-$1$ accurate approximation of the full Hessian. Under more specialized structures, the method is proved to yield order-$2$ accuracy. The underdetermined case, where the number of sample points is fewer than required for full interpolation, is studied and error bounds are developed for the resulting partial Hessians.

Beamspace-based underdetermined DOA estimator via Khatri-Rao subspace for quasi-stationary signals

The Khatri-Rao subspace methods utilize the characteristics of the Khatri-Rao product of the array response to achieve the extension of the physical aperture and further resolve the quasi-stationary signals in the underdetermined cases. However, the estimation performance of the existing Khatri-Rao subspace methods suffers from the input signal-to-noise (SNR). Especially for a low SNR, the estimation accuracy of the local covariance matrix signals decreases, which will further result in the DOA estimation performance degradation. To address these problems, an enhanced Khatri-Rao subspace method based on beamspace processing for the DOA estimation is proposed. The total array is first split into several identical subarrays, and the beamspace transformation for the data received by subarrays is implemented via beamforming. This results in the improvement of the SNR and further improves the noise suppression ability for the low SNR regions. Then, the Khatri-Rao subspace processing is used for the separated beamspace sources to realize the aperture extension. Finally, the DOAs for the separated sources in the beamspace domain can be further obtained when combined with the subspace-based methods, and the undetermined problem can also be achieved by combining the DOAs obtained from different directions in the beamspace domain.

Impact force identification on composite panels using fully overlapping group sparsity based on Lp-norm regularization

Impact force identification is of great importance for composite structural health monitoring due to the poor impact resistance of composite materials. Convex sparse regularization method based on L1-norm tends to underestimate the amplitude of the impact force. This paper proposes a novel method using fully overlapping group sparsity based on Lp-norm regularization (FOGSLp) for impact force identification, which can localize the impact force and reconstruct its time history simultaneously with limited measurements in under-determined cases. The FOGSLp method takes more sparse prior information into account by combining the non-convex Lp-norm () and fully overlapping group sparsity to promote sparse solutions, thus improving the accuracy of the reconstructed impact force. An accelerated grouped shrinkage-thresholding algorithm is employed to solve the non-convex optimization problem under the majorization-minimization framework. The Nesterov’s acceleration strategy is modified to accommodate the requirements of non-convex optimization. Simulations and experiments are conducted on composite panels to validate the effectiveness of the FOGSLp method. Results demonstrate the efficiency and robustness of the FOGSLp method to localize the impact force and reconstruct its time history simultaneously while 25 potential impact points are monitored using two sensors. Compared with L1-norm, Lp-norm, and L2,1-norm regularization methods, the proposed method performs best in both simulations and experiments.

Linearly-Involved Moreau-Enhanced-Over-Subspace Model: Debiased Sparse Modeling and Stable Outlier-Robust Regression

We present an efficient mathematical framework based on the linearly-involved Moreau-enhanced-over-subspace (LiMES) model. Two concrete applications are considered: sparse modeling and robust regression. The popular minimax concave (MC) penalty for sparse modeling subtracts, from the $\ell_1$ norm, its Moreau envelope, inducing nearly unbiased estimates and thus yielding remarkable performance enhancements. To extend it to underdetermined linear systems, we propose the projective minimax concave penalty using the projection onto the input subspace, where the Moreau-enhancement effect is restricted to the subspace for preserving the overall convexity. We also present a novel concept of stable outlier-robust regression which distinguishes noise and outlier explicitly. The LiMES model encompasses those two specific examples as well as two other applications: stable principal component pursuit and robust classification. The LiMES function involved in the model is an ``additively nonseparable'' weakly convex function but is defined with the Moreau envelope returning the minimum of a ``separable'' convex function. This mixed nature of separability and nonseparability allows an application of the LiMES model to the underdetermined case with an efficient algorithmic implementation. Two linear/affine operators play key roles in the model: one corresponds to the projection mentioned above and the other takes care of robust regression/classification. A necessary and sufficient condition for convexity of the smooth part of the objective function is studied. Numerical examples show the efficacy of LiMES in applications to sparse modeling and robust regression.

Closure, Underdetermination and the peculiarity of Sceptical scenarios

AbstractEpistemologists understand radical scepticism as arising from two principles: closure and underdetermination. Both possess intuitive prima facie support for their endorsement. Understanding how they engender scepticism is crucial for any reasonable anti‐sceptical attempt. The contemporary discussion has focused on elucidating the relationship between them to ascertain whether they establish distinct sceptical questions and which of the two constitutes the ultimately fundamental threat. Major contributions to this debate are due to Brueckner, Cohen and Pritchard. This contribution aims at defending Brueckner's contention that underdetermination expresses the fundamental sceptical threat and that the closure‐based argument can ultimately be reduced to it, at least concerning sceptical contexts. This will be achieved by undermining Cohen's objections to Brueckner on both counts. Cohen's argument endorses a picture of evidential underdetermination, which, while apt for non‐sceptical contexts, cannot be applied to radical scepticism. A comparison with a case of scientific underdetermination is developed to argue for this idea. Our argument is then applied to the principle equivalence issue and to objections against it. It is then shown how this analysis possesses a distinctive effect on our understanding of the sceptical threat and the assessment of successful anti‐sceptical strategies.

Acoustic UAV detection method based on blind source separation framework

Many new challenges are encountered in public security because of the rapid development of an unmanned aerial vehicle (UAV) technology. In order to meet these challenges brought by UAV, it is necessary to realize timely and accurate UAV detection. In this paper, an acoustic UAV detection method based on blind source separation (BSS) framework is introduced to solve the UAV sound detection problem with multi-source interference. In the framework proposed in this paper, source number estimation is used to determine the type of BSS problem, and three methods are applied to separate the UAV sound from the mixing signal in different BSS situations. In the experiment, a variety of machine learning algorithms are used to detect UAVs. The robustness of the proposed method is tested on different UAVs and different sound features. Experimental results show that the UAV sound detection method based on the BSS framework realizes the effective detection of UAVs in different tests and achieves excellent robustness. Compared with mixed signal and filtered signal, the detection rate of our scheme is more than 90%, whether in the overdetermined, positive-definite, and underdetermined cases.

Conjugate Augmented Decoupled 3-D Parameters Estimation Method for Near-Field Sources

A near-field source localization method is proposed for 2-D direction-of-arrival (DOA) and range estimation based on a symmetrical cross array. It first employs the conjugate symmetry property of signal autocorrelation for different time delays to construct a conjugate augmented spatial–temporal cross-correlation matrix, then the extended steering vector is decoupled to avoid the usual multidimensional search based on the properties of the Khatri–Rao product, and finally three 1-D MUSIC-type searches are employed to obtain the results. The proposed method can realize automatic pairing of multiple parameters associated with each source and it also works in the underdetermined case. Furthermore, the stochastic Cramer–Rao lower bound with different time delays is derived. Compared with two existing methods, simulation results demonstrate that the proposed method provides satisfactory estimation performance for both the DOA and range parameters at low signal-to-noise ratio and with a small number of snapshots.

Blind modal identification for decentralized sensor network by modified sparse component analysis in frequency-domain subspace

It is practical to develop a modal identification method using a decentralized sensor network for field tests. Sparse component analysis (SCA) is a potential method in the framework of blind source separation to overcome the difficulty of using limited sensors. Nevertheless, it is still challenging for SCA to identify a larger number of modes than the number of measurements using the decentralized sensor network. This is because the partial mode shapes may be similar. It is convenient to perform SCA in frequency-domain subspaces to improve SCA further for decentralized sensor implementation; this is owing to the distinguishability of different partial mode shapes. In this study, a modal identification method applied to a largely underdetermined case caused by a decentralized sensor network is proposed by modifying SCA by automatic segmentation of frequency-domain subspaces. First, a method for determining the segmenting point is developed by identifying the valleys of the energy fluctuation spectrum using the scale-space peak detection method. Second, single-mode-points in the time–frequency plane are detected to facilitate mode shape estimation in the frequency-domain subspaces. Third, density peak clustering with the clustering number determination method is developed to estimate mode shapes, particularly for closely spaced modes. Finally, the modal frequencies and damping ratios are identified through the inverse operator in the frequency-domain subspaces and the curve fitting of the expected instant spectrum. The proposed method is verified by a 10 degree-of-freedom system, a structure with close modes, and an actual cable-stayed bridge. The investigations verified the effectiveness of the proposed method for decentralized sensor networks and practical applications.

Multistatic MIMO radar target localization via coupled canonical polyadic decomposition

Target localization is considered based on a multistatic MIMO radar that consists of a single transmit array and multiple receive arrays. The output data of the multistatic MIMO radar is formulated into a set of tensors that together admits a coupled canonical polyadic decomposition (C-CPD) model, and a thorough study on the identifiability of the C-CPD model is provided. In addition, a C-CPD based computational framework for target localization is presented, consisting of coupled data compression in the preprocessing step, several (semi-)algebraic and optimization based C-CPD algorithms in different scenarios as the main body of the framework, and a data fusion approach that calculates the optimal position of each target by fusing the C-CPD results in the postprocessing stage. Due to the exploitation of coupling, the proposed C-CPD based framework is free of parameter pairing and can yield more accurate estimates than conventional single-set based methods. In addition, the proposed method does not require prior knowledge of probing waveforms, which can be non-orthogonal, and can work in difficult underdetermined cases. Experiments are provided to illustrate the performance of the proposed framework.

An Underdetermined Convolutional Blind Separation Algorithm for Time–Frequency Overlapped Wireless Communication Signals with Unknown Source Number

It has been challenging to separate the time–frequency (TF) overlapped wireless communication signals with unknown source numbers in underdetermined cases. In order to address this issue, a novel blind separation strategy based on a TF soft mask is proposed in this paper. Based on the clustering property of the signals in the sparse domain, the angular probability density distribution is obtained by the kernel density estimation (KDE) algorithm, and then the number of source signals is identified by detecting the peak points of the distribution. Afterward, the contribution degree function is designed according to the cosine distance to calculate the contribution degrees of the source signals in the mixed signals. The separation of the TF overlapped signals is achieved by constructing a soft mask matrix based on the contribution degrees. The simulations are performed with digital signals of the same modulation and different modulation, respectively. The results show that the proposed algorithm has better anti-aliasing and anti-noise performance than the comparison algorithms.

Polarization Estimation With Vector Sensor Array in the Underdetermined Case

Ship target detection using radar is an important application in military and civilian fields. For the polarization estimation of scattering waves in the underdetermined case, i.e., the number of scattering waves from ships is larger than the number of sensors, this paper proposes two estimation methods with different measurement models. 1) For the single-vector-sensor model, this paper proposes the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">polarization-invariant</i> ESPRIT-based method. This method can estimate the polarization of signals containing target echo, interference, and noise, which can cure the problem that the accuracy of existing method is poor under low interference signal ratio. 2) For the multi-vector-sensor model, this paper proposes an improved ESPRIT method based on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spatial-invariant and time-invariant</i> simultaneously, which can increase the degree of freedom without increasing hardware cost. As for another problem of multi-vector-sensor, i.e., almost all existing methods assume that the number of scattering waves is known, this paper introduces <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">the polarization spectrum</i> for the first time, which can estimate the polarization parameters when the number of scattering waves is unknown. Finally, we analyze the two proposed ESPRIT-based methods comparing with some existing methods through Monte Carlo simulation, which results demonstrate the efficience of the proposed methods.

Underdetermined DOA estimation via multiple time-delay covariance matrices and deep residual network

Higher-order statistics based approaches and signal sparseness based approaches have emerged in recent decades to resolve the underdetermined direction-of-arrival (DOA) estimation problem. These model-based methods face great challenges in practical applications due to high computational complexity and dependence on ideal assumptions. This paper presents an effective DOA estimation approach based on a deep residual network (DRN) for the underdetermined case. We first extract an input feature from a new matrix calculated by stacking several covariance matrices corresponding to different time delays. We then provide the input feature to the trained DRN to construct the super resolution spectrum. The DRN learns the mapping relationship between the input feature and the spatial spectrum by training. The proposed approach is superior to existing model-based estimation methods in terms of calculation efficiency, independence of source sparseness and adaptive capacity to non-ideal conditions (e.g., low signal to noise ratio, short bit sequence). Simulations demonstrate the validity and strong performance of the proposed algorithm on both overdetermined and underdetermined cases.

Bearing fault diagnosis based on kernel independent component analysis and antlion optimization

Fault diagnosis of gearboxes based on vibration signal processing is challenging, as vibration signals collected by acceleration sensors are typically a nonlinear mixture of unknown signals. Furthermore, the number of source signals is usually larger than that of sensors because of the practical limitation on sensor positions. Hence, the fault characterization is actually a nonlinear underdetermined blind source separation (NUBSS) problem. In this paper, a novel NUBSS algorithm based on kernel independent component analysis (KICA) and antlion optimization (ALO) is proposed to address the technical challenge. The mathematical model demonstrates the nonlinear mixing of source signals in the underdetermined cases. Ensemble empirical mode decomposition is used as a preprocessing tool to decompose the observed signals into a set of intrinsic mode functions that suffers from the problem of redundant components. The correlation coefficient is utilized to eliminate the redundant components. An adaptive threshold singular value decomposition method is proposed to estimate the number of source signals. Then a whitening process is carried out to transform the overdetermined blind source separation (BSS) into determined BSS, which can be solved by the KICA method. However, the reasonable selection of parameters in KICA limits its application to some extent. Therefore, ALO and Fisher’s linear discriminant analysis are adopted to further enhance the accuracy of the KICA method. The separation performance of the proposed method is assessed through simulation. The numerical results show that the proposed method can accurately estimate the number of source signals and attains a higher separation quality in tackling nonlinear mixed signals when compared with the existing methods. Finally, the inner ring fault experiment is conducted to preliminarily validate the practicability of the proposed method in bearing fault diagnosis.

Operational modal identification based on parallel factor decomposition with the presence of harmonic excitation

One of the main difficulties of the operational modal analysis is to deal with underdetermined problems inwhich the number of sensors is less than the number of active modes. In the last decade, methods based on the PARAllel FACtor (PARAFAC) decomposition have attracted a lot of attention in the field of modal analysis because it has been proven that these methods can deal with underdetermined cases, as well as the presence of harmonic excitations. Moreover, in combination with kurtosis value as a harmonic indicator, this makes them more efficient in distinguishing between harmonic and structural components. However, it can lead to distorted results as it does not take into account the variation in the length of the covariance functions of the modal coordinates. Since the kurtosis values are estimated from these covariance functions, the length of the latter directly affects the kurtosis. To overcome this limit, the present study proposes to introduce the choice of the length of these functions based on their frequency and damping coefficient. This change improves the existing method by more efficient separating between harmonics and modal components. The proposed procedure is validated using numerical simulations, followed by ambient vibration measurements. © 2021 Academie des sciences, Paris and the authors.

Quantum-classical algorithms for skewed linear systems with an optimized Hadamard test

The solving of linear systems provides a rich area to investigate the use of nearer-term, noisy, intermediate-scale quantum computers. In this work, we discuss hybrid quantum-classical algorithms for skewed linear systems for over-determined and under-determined cases. Our input model is such that the columns or rows of the matrix defining the linear system are given via quantum circuits of poly-logarithmic depth and the number of circuits is much smaller than their Hilbert space dimension. Our algorithms have poly-logarithmic dependence on the dimension and polynomial dependence in other natural quantities. In addition, we present an algorithm for the special case of a factorized linear system with run time poly-logarithmic in the respective dimensions. At the core of these algorithms is the Hadamard test and in the second part of this paper we consider the optimization of the circuit depth of this test. Given an $n$-qubit and $d$-depth quantum circuit $\mathcal{C}$, we can approximate $\langle 0|\mathcal{C}|0\rangle$ using $(n + s)$ qubits and $O\left(\log s + d\log (n/s) + d\right)$-depth quantum circuits, where $s\leq n$. In comparison, the standard implementation requires $n+1$ qubits and $O(dn)$ depth. Lattice geometries underlie recent quantum supremacy experiments with superconducting devices. We also optimize the Hadamard test for an $(l_1\times l_2)$ lattice with $l_1 \times l_2 = n$, and can approximate $\langle 0|\mathcal{C} |0\rangle$ with $(n + 1)$ qubits and $O\left(d \left(l_1 + l_2\right)\right)$-depth circuits. In comparison, the standard depth is $O\left(d n^2\right)$ in this setting. Both of our optimization methods are asymptotically tight in the case of one-depth quantum circuits $\mathcal{C}$.

Tensor-based approach for underdetermined operational modal identification

During these years, blind source separation (BSS) techniques have been demonstrated as a promising tool for operational modal identification of large-scale engineering structures only from output responses. However, plenty of BSS identification approaches are based on the assumption that the sources are sparse or statistically independent, limiting their application scopes. Furthermore, it has been challenging to perform operational modal identification in underdetermined cases where the number of observed sensors is less than the number of active modes. In allusion to the problems above, a novel tensor-based approach for operational modal identification with limited sensors is proposed, in which the low-rank characteristics of vibration measurements is utilized. This paper reveals the intrinsic connection between tensor decomposition and modal expansion. Firstly, a third-order tensor is constructed through a set of generated matrices, in which each observed signal is reshaped into a matrix by segmentation operation. Then, the tensorial observed signals are decomposed into multilinear rank-Lr,Lr,1 terms by block-termed decomposition (BTD). And a collection of sub-tensors that correspond to the mode shapes matrix and modal responses can be obtained, from which the modal parameters are estimated. Finally, the effectiveness of the proposed method is validated with a series of numerical studies and experimental investigations, even in closely-spaced modes. The simulation and experimental results indicate that the proposed method can identify the modal parameters accurately and robustly in both determined and underdetermined situations.