Lp Regularization Research Articles

Reconstruction of flame permittivity distribution based on ECT and Lp regularized D-bar method

Abstract The measurement of flame permittivity is significant in obtaining the combustion state. A method of Lp regularized D-bar is proposed in this article, which is used to reconstruct flame permittivity distribution by electrical capacitance tomography (ECT). Firstly, the distribution characteristic of flame permittivity is analyzed using the electric probe method. The simulation model of ECT flame measurement is set up based on the result of flame permittivity distribution. Then, the relationship model between the truncation radius of the D-bar algorithm and noise level is fitted based on the simulation reconstruction results of flame permittivity. The truncation radius of the D-bar algorithm for flame permittivity reconstruction is obtained by the relationship model. The Lp regularization is introduced into the scattering transformation solving of the D-bar algorithm, and the simulation results of flame permittivity reconstruction under different p-values are compared. In the experiment, the reconstruction of flame permittivity by the D-bar algorithm with different truncation radii is compared. The experimental results verify the truncation radius strategy based on noise level. Moreover, compared with L 1 and L 2 regularization methods, Lp regularization combined with the D-bar algorithm is more accurate in the experiment results of flame permittivity reconstruction.

Nonnegative iterative reweighted method for sparse linear complementarity problem

Solution of sparse linear complementarity problem (LCP) has been widely discussed in many applications. In this paper, we consider the ℓp regularization problem with nonnegative constraint for sparse LCP, and propose algorithms based on the iterative reweighted method to approach a sparse solution of the LCP, and then show the convergence to the stationary point of ℓp regularization problem. Numerical results on simulated data exhibit an excellent performance of the proposed algorithms on approaching a sparse solution of the LCP. Finally, we apply this method to the frictional and frictionless contact problems. The numerical experiments demonstrate that the contact problems can be efficiently solved by the proposed algorithm.

On sparse regression, Lp‐regularization, and automated model discovery

AbstractSparse regression and feature extraction are the cornerstones of knowledge discovery from massive data. Their goal is to discover interpretable and predictive models that provide simple relationships among scientific variables. While the statistical tools for model discovery are well established in the context of linear regression, their generalization to nonlinear regression in material modeling is highly problem‐specific and insufficiently understood. Here we explore the potential of neural networks for automatic model discovery and induce sparsity by a hybrid approach that combines two strategies: regularization and physical constraints. We integrate the concept of Lp regularization for subset selection with constitutive neural networks that leverage our domain knowledge in kinematics and thermodynamics. We train our networks with both, synthetic and real data, and perform several thousand discovery runs to infer common guidelines and trends: L2 regularization or ridge regression is unsuitable for model discovery; L1 regularization or lasso promotes sparsity, but induces strong bias that may aggressively change the results; only L0 regularization allows us to transparently fine‐tune the trade‐off between interpretability and predictability, simplicity and accuracy, and bias and variance. With these insights, we demonstrate that Lp regularized constitutive neural networks can simultaneously discover both, interpretable models and physically meaningful parameters. We anticipate that our findings will generalize to alternative discovery techniques such as sparse and symbolic regression, and to other domains such as biology, chemistry, or medicine. Our ability to automatically discover material models from data could have tremendous applications in generative material design and open new opportunities to manipulate matter, alter properties of existing materials, and discover new materials with user‐defined properties.

Research on the Grounding Grid Electrical Impedance Imaging Algorithm Based on Improved Tikhonov and Lp Regularization

Research on the Grounding Grid Electrical Impedance Imaging Algorithm Based on Improved Tikhonov and Lp Regularization

Sparse reconstruction of EMT based on compressed sensing and Lp regularization with the split Bregman method

Electromagnetic tomography (EMT) is a promising tomographic method used to reconstruct the conductivity and permeability material owing to its contactless nature. However, the EMT inverse problem is ill-posed due to the soft-field characteristic and limited prior information, which will severely affect the imaging quality. To address the challenge, a novel sparse reconstruction algorithm is proposed in this article on the basis of compressed sensing (CS) and Lp regularization. The EMT inverse problem is converted into the L1 norm regularization problem using the advantages of the compressed sensing theory. To combine the merits of L1 and L2 regularization, Lp regularization is deployed as the regularizer taking the place of L1 norm, which can overcome the over-smoothing and over-sparsity of the reconstruction objects. A modified split Bregman (SB) method is exploited to effectively cope with the Lp regularization problem. An accelerating method is further applied to improve the reconstruction speed. The sensitivity matrix calculation is performed by the perturbation method. Simulation studies are implemented to show the superiority of the presented method in comparison with the classical methods. The effectiveness of the proposed method is further validated by the developed EMT system. Experimental results confirm that the proposed method has better imaging capability, which provides a great candidate for the practical application of EMT.

Structural Damage Detection Based on Improved Sensitivity Function of Modal Flexibility and Iterative Reweighted lp Regularization

The [Formula: see text] regularization is usually used to deal with the problems of under-determinacy and measurement noise for the conventional sensitivity-based model updating damage detection methods. However, the [Formula: see text] regularization technique often provides overly smooth solutions and thus cannot exhibit the sparsity of the structural damage due to the promotion of the 2-norm term on smoothness. In the study, a structural damage detection method is proposed based on an improved modal flexibility sensitivity function and an iterative reweighted [Formula: see text] (IR[Formula: see text] regularization. Specifically, the sensitivity function is established by introducing changes in the mode shapes into the derivative of eigenvalue and can be applied to identify the localized damage more accurately. Additionally, IR[Formula: see text] regularization is proposed to deal with the ill-posed problem of damage detection in a noisy environment. The proposed IR[Formula: see text] regularization is compared with the [Formula: see text] and [Formula: see text] regularizations through a numerical and an experimental examples. The numerical and experimental results indicate that the IR[Formula: see text] regularization can more accurately locate and quantify the single and multiple damages under the noise situation. The maximum identification errors are only 5.16% and 5.67%, respectively. Moreover, compared to the basic modal flexibility sensitivity function, the improved function is more sensitive to the damage. The maximum identification error of the improved function is less than 6%, while the relative errors are significantly larger in the basic function.

Convergence Rates for Regularized Optimal Transport via Quantization

We study the convergence of divergence-regularized optimal transport as the regularization parameter vanishes. Sharp rates for general divergences including relative entropy or Lp regularization, general transport costs, and multimarginal problems are obtained. A novel methodology using quantization and martingale couplings is suitable for noncompact marginals and achieves, in particular, the sharp leading-order term of entropically regularized 2-Wasserstein distance for marginals with a finite [Formula: see text]-moment. Funding: This work was supported by the Alfred P. Sloan Foundation [Grant FG-2016-6282] and the Division of Mathematical Sciences [Grants DMS-1812661 and DMS-2106056].

Color Occlusion Face Recognition Method Based on Quaternion Non-Convex Sparse Constraint Mechanism.

As the acquisition and application of color images become more and more extensive, color face recognition technology has also been vigorously developed, especially the recognition methods based on convolutional neural network, which have excellent performance. However, with the increasing depth and complexity of network models, the number of calculated parameters also increases, which means the training of most high-performance models depends on large-scale samples and expensive equipment. Therefore, the key to the current research is to realize a lightweight model while ensuring the recognition accuracy. At present, PCANet, a typical lightweight framework for deep learning, has achieved good results in most of the image recognition tasks, but its recognition accuracy for color face images, especially under occlusion, still needs to be improved. Therefore, a color occlusion face recognition method based on quaternion non-convex sparse constraint mechanism is proposed in this paper. Firstly, a quaternion non-convex sparse principal component analysis network model was constructed based on Lp regularization of strong sparsity. Secondly, the fixed point iteration method and coordinate descent method were established to solve the non-convex optimization problem. Finally, the occlusion recognition performance of the proposed method was verified on Georgia Tech, Color FERET, AR, and LFW-A Color face datasets.

Crosstalk Correction for Color Filter Array Image Sensors Based on Lp-Regularized Multi-Channel Deconvolution

In this paper, we propose a crosstalk correction method for color filter array (CFA) image sensors based on -regularized multi-channel deconvolution. Most imaging systems with CFA exhibit a crosstalk phenomenon caused by the physical limitations of the image sensor. In general, this phenomenon produces both color degradation and spatial degradation, which are respectively called desaturation and blurring. To improve the color fidelity and the spatial resolution in crosstalk correction, the feasible solution of the ill-posed problem is regularized by image priors. First, the crosstalk problem with complex spatial and spectral degradation is formulated as a multi-channel degradation model. An objective function with a hyper-Laplacian prior is then designed for crosstalk correction. This approach enables the simultaneous improvement of the color fidelity and the sharpness restoration of the details without noise amplification. Furthermore, an efficient solver minimizes the objective function for crosstalk correction consisting of regularization terms. The proposed method was verified on synthetic datasets according to various crosstalk and noise levels. Experimental results demonstrated that the proposed method outperforms the conventional methods in terms of the color peak signal-to-noise ratio and structural similarity index measure.

Low-CP-Rank Tensor Completion via Practical Regularization

Dimension reduction is analytical methods for reconstructing high-order tensors that the intrinsic rank of these tensor data is relatively much smaller than the dimension of the ambient measurement space. Typically, this is the case for most real world datasets in signals, images and machine learning. The CANDECOMP/PARAFAC (CP, aka Canonical Polyadic) tensor completion is a widely used approach to find a low-rank approximation for a given tensor. In the tensor model (Sanogo and Navasca in 2018 52nd Asilomar conference on signals, systems, and computers, pp 845–849, https://doi.org/10.1109/ACSSC.2018.8645405, 2018), a sparse regularization minimization problem via ell _1 norm was formulated with an appropriate choice of the regularization parameter. The choice of the regularization parameter is important in the approximation accuracy. Due to the emergence of the massive data, one is faced with an onerous computational burden for computing the regularization parameter via classical approaches (Gazzola and Sabaté Landman in GAMM-Mitteilungen 43:e202000017, 2020) such as the weighted generalized cross validation (WGCV) (Chung et al. in Electr Trans Numer Anal 28:2008, 2008), the unbiased predictive risk estimator (Stein in Ann Stat 9:1135–1151, 1981; Vogel in Computational methods for inverse problems, 2002), and the discrepancy principle (Morozov in Doklady Akademii Nauk, Russian Academy of Sciences, pp 510–512, 1966). In order to improve the efficiency of choosing the regularization parameter and leverage the accuracy of the CP tensor, we propose a new algorithm for tensor completion by embedding the flexible hybrid method (Gazzola in Flexible krylov methods for lp regularization) into the framework of the CP tensor. The main benefits of this method include incorporating the regularization automatically and efficiently as well as improving accuracy in the reconstruction and algorithmic robustness. Numerical examples from image reconstruction and model order reduction demonstrate the efficacy of the proposed algorithm.

Unsupervised discovery of interpretable hyperelastic constitutive laws

We propose a new approach for data-driven automated discovery of isotropic hyperelastic constitutive laws. The approach is unsupervised, i.e., it requires no stress data but only displacement and global force data, which are realistically available through mechanical testing and digital image correlation techniques; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a large catalogue of candidate functions; it is one-shot, i.e., discovery only needs one experiment — but can use more if available. The problem of unsupervised discovery is solved by enforcing equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity of the solution is achieved by ℓp regularization combined with thresholding, which calls for a non-linear optimization scheme. The ensuing fully automated algorithm leverages physics-based constraints for the automatic determination of the penalty parameter in the regularization term. Using numerically generated data including artificial noise, we demonstrate the ability of the approach to accurately discover five hyperelastic models of different complexity. We also show that, if a “true” feature is missing in the function library, the proposed approach is able to surrogate it in such a way that the actual response is still accurately predicted.

Early diagnosis model of Alzheimer’s disease based on sparse logistic regression with the generalized elastic net

Accurate prediction of high-risk group who may convert to Alzheimer’s disease (AD) patients is critical for the future treatment of patients. Recently, logistic regression is used for the early diagnosis of AD. However, due to the high-dimensional small sample characteristics of AD data, this brings difficulties to logistic regression-aided diagnosis. To solve the problem, in this paper, we propose sparse logistic regression with the generalized elastic net for the early diagnosis of AD. The generalized elastic net is composed of Lp regularization and L2 regularization. The Lp regularization can produce sparse solutions. L2 regularization ensures that the correlated brain regions are in solution. We evaluate our proposed method on 197 subjects from the baseline MRI data of ADNI database. Our proposed method achieves classification accuracy of 96.10, 84.67, and 75.87 %, for AD vs. HC, MCI vs. HC, and cMCI vs. sMCI, respectively. Experimental results show that, compared with previous methods, our proposed method captures distinct brain regions that are significantly related to AD conversion and provides a significant enhancement in AD classification.

Proximal methods for reweighted l-regularization of sparse signal recovery

Abstract To recover a sparse signal from a noised linear measurement system A x = b + e , convex lp regularization methods (i.e., 1 ≤ p p = 1 ) are commonly used under certain conditions. Recently, however, more attentions have been paid to nonconvex lq regularization methods (i.e., 0 q = 1 / 2 ) for recovering a sparse signal. In this paper, we use proximal methods to study both convex and nonconvex reweighted lQ regularization for recovering a sparse signal. Convex lQ regularization is introduced by S. Voronin and I. Daubechies [19]. We extend it to the nonconvex case and our results therefore supplement those of Voronin and Daubechies [19] . We also study Nesterov’s acceleration method for the nonconvex case. Our numerical experiments show that nonconvex lQ regularization can more effectively recover sparse signals.

ℓ1/2-based penalized clustering with half thresholding algorithm

Clustering is a widely applied method in data analysis. As a novel framework of clustering analysis, penalized clustering bases itself on the sparsity of solution, which contributes to its ability of determining the best number of clusters automatically rather than specified in advance. Moreover, ℓ1/2 regularization has been recognized extensively in recent studies. Compared with other ℓp (0 < p < 1) regularization, it can always obtain sparser solution. Motivated by these two points, we propose a ℓ1/2-based penalized clustering model, and further transform it into a more general form by introducing the vector comprising pairwise differences between centroids and an auxiliary transfer matrix composed of identity matrixes and null matrixes. Finally, the transformed model is solved by the efficient half thresholding algorithm, which can not only obtain an exact analytic expression of ℓ1/2 solutions, but also provide an effective parameter selection strategy. We also prove the convergence of the proposed half thresholding algorithm solving ℓ1/2-based penalized clustering model. Lastly but not least importantly, benchmark experiments on both synthesis and real data sets from UCI have been conducted to prove the superiority of the proposed method.

Projection Data Smoothing for Low-Dose CT Based on $$\ell _p$$ℓp Regularization

Projection data smoothing is a traditional technique for low-dose computed tomography. The projection data can be modeled as a piecewise smooth function. It’s well known that $$\ell _1$$ regularization of the image gradients tries to recover piecewise constant functions, while $$\ell _2$$ regularization recovers smooth functions. This motivates us to propose the $$\ell _p$$ regularization with $$1<p<2$$ for low-dose projection data smoothing. Besides, the non-stationary Gaussian noise model for the projection data is built into the regularization term. The resulting model is then linearized such that the fast split-Bregman algorithm can be applied. Experiments on simulated projection data as well as real data show that $$\ell _p$$ regularization with $$1<p<2$$ could achieve better reconstruction compared to $$\ell _{1}$$ regularization.

Multiple-prespecified-dictionary sparse representation for compressive sensing image reconstruction with nonconvex regularization

Multiple-prespecified-dictionary sparse representation (MSR) has shown powerful potential in compressive sensing (CS) image reconstruction, which can exploit more sparse structure and prior knowledge of images for minimization. Due to the popular L1 regularization can only achieve the suboptimal solution of L0 regularization, using the nonconvex regularization can often obtain better results in CS reconstruction. This paper proposes a nonconvex adaptive weighted Lp regularization CS framework via MSR strategy. We first proposed a nonconvex MSR based Lp regularization model, then we propose two algorithms for minimizing the resulting nonconvex Lp optimization problem. According to the fact that the sparsity levels of each regularizers are varying with these prespecified-dictionaries, an adaptive scheme is proposed to weight each regularizer for optimization by exploiting the difference of sparsity levels as prior knowledge. Simulated results show that the proposed nonconvex framework can make a significant improvement in CS reconstruction than convex L1 regularization, and the proposed MSR strategy can also outperforms the traditional nonconvex Lp regularization methodology.

Sparse non-convex Lp regularization for cone-beam X-ray luminescence computed tomography

ABSTRACTCone-beam X-ray luminescence computed tomography (CB-XLCT) is an attractive hybrid imaging modality, and it has the potential of monitoring the metabolic processes of nanophosphors-based drugs in vivo. However, the XLCT imaging suffers from a severe ill-posed problem. In this work, a sparse nonconvex Lp (0 < p < 1) regularization was utilized for the efficient reconstruction for early detection of small tumour in CB-XLCT imaging. Specifically, we transformed the non-convex optimization problem into an iteratively reweighted scheme based on the L1 regularization. Further, an iteratively reweighted split augmented lagrangian shrinkage algorithm (IRW_SALSA-Lp) was proposed to efficiently solve the non-convex Lp (0 < p < 1) model. We studied eight different non-convex p-values (1/16, 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8) in both 3D digital mouse experiments and in vivo experiments. The results demonstrate that the proposed non-convex methods outperform L2 and L1 regularization in accurately recovering sparse targets in CB-XLCT. And among all the non-convex p-values, our Lp(1/4 < p < 1/2) methods give the best performance.

Formula omitted] optimization for multigrid fluorescence tomography of small animals using simplified spherical harmonics

We present the first Lq−Lp optimization scheme for fluorescence tomographic imaging. This is then applied to small animal imaging. Fluorescence tomography is an ill-posed, and in full generality, a nonlinear problem that seeks to image the 3D concentration distribution of a fluorescent agent inside a biological tissue. Standard candidates for regularization to deal with the ill-posedness of the image reconstruction problem include L1 and L2 regularization. In this work, a general Lq−Lp regularization framework (Lq discrepancy function - Lp regularization term) is introduced for fluorescence tomographic imaging. A method to calculate the gradient for this general framework is developed which allows evaluating the performance of different cost functions/regularization schemes in solving the fluorescence tomographic problem. The simplified spherical harmonics approximation is used to accurately model light propagation inside the tissue. Furthermore, a multigrid mesh is utilized to decrease the dimension of the inverse problem and reduce the computational cost of the solution. The inverse problem is solved iteratively using an lm-BFGS quasi-Newton optimization method. The simulations are performed under different scenarios of noisy measurements. These are carried out on the Digimouse numerical mouse model with the kidney being the target organ. The evaluation of the reconstructed images is performed both qualitatively and quantitatively using several metrics including QR, RMSE, CNR, and TVE under rigorous conditions. The best reconstruction results under different scenarios are obtained with an L1.5−L1 scheme with premature termination of the optimization process. This is in contrast to approaches commonly found in the literature relying on L2−L2 schemes.

A novel Scan SAR imaging method for maritime surveillance via Lp regularization

ABSTRACTIn this article, a novel Scan mode synthetic aperture radar (SAR) imaging method for maritime surveillance is presented. Conventional Scan SAR is generally operated with severe azimuth resolution loss in order to cover a large area. The proposed imaging method changes the way Scan SAR illuminates sub-scenes and presents a new radar illuminating strategy based on ships’ spatial distribution in each sub-scene. To gain ships’ spatial distribution, a scene sensing algorithm based on radar range profiles together with a peak-seeking and clustering algorithm is introduced. After that, a Markov transfer-probability matrix is generated to make sure that radar illuminates each sub-scene randomly under the probability we calculated before. Finally, an imaging algorithm within the Lp (0 < p ≤ 1) regularization framework is utilized to reconstruct each sub-scene; the regularization problem is solved by an improved iterative thresholding algorithm. The whole wide swath image is joined by putting all the sub-scenes together. Experimental results support that the proposed imaging method can perform high-resolution wide swath SAR imaging effectively and efficiently without reducing the image resolution.

Online gradient method with smoothing ℓ0 regularization for feedforward neural networks

ℓp regularization has been a popular pruning method for neural networks. The parameter p was usually set as 0<p≤2 in the literature, and practical training algorithms with ℓ0 regularization are lacking due to the NP-hard nature of the ℓ0 regularization problem; however, the ℓ0 regularization tends to produce the sparsest solution, corresponding to the most parsimonious network structure which is desirable in view of the generalization ability. To this end, this paper considers an online gradient training algorithm with smoothing ℓ0 regularization (OGTSL0) for feedforward neural networks, where the ℓ0 regularizer is approximated by a series of smoothing functions. The underlying principle for the sparsity of OGTSL0 is provided, and the convergence of the algorithm is also theoretically analyzed. Simulation examples support the theoretical analysis and illustrate the superiority of the proposed algorithm.