Lasso, fractional norm and structured sparse estimation using a Hadamard product parametrization

Hoff, Peter D.

doi:10.1016/j.csda.2017.06.007

Cited by 32 publications

(23 citation statements)

References 19 publications

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“…(2.12) is generally difficult to be solved because of its nonconvex property. However, if the L p (0< p<1) regularization is transformed into a series of simple L 2 regularizations [29], to which the existing L 2 regularization algorithms, such as the SVD method, can be efficiently applied, the non-convex problem will be solved easily. To introduce the algorithm, the cost function (2.12) is rewritten as…”

Section: Appendix A: Determination Of Parameters In the Com Methodsmentioning

confidence: 99%

“…where J u,v is differentiable and biconvex and its local minimum can be found by using a very simple alternating Tikhonov regularization (L 2 regularization) method. Besides, there is a correspondence between the minimums of J u,v and J r [29]. Namely, any local…”

Section: Appendix A: Determination Of Parameters In the Com Methodsmentioning

confidence: 99%

“…(2.12) is generally difficult to be solved because of its nonconvex property. To address this issue, the HPP algorithm proposed by [29] uses the Hadmard product parametrization to transform the L p (0 < p < 1) regularization into a series of simple L 2 regularizations, to which the existing L 2 regularization algorithms can be efficiently applied, so that the non-convex problem will be solved easily. Here, the SVD algorithm is applied to solve each Tikhonov regularization for high inversion accuracy, and the regularization parameter is determined adaptively by the L-curve method (Fig.…”

Section: Super-resolution Inversion Based On Thementioning

confidence: 99%

“…Most L p (0< p<1) regularization algorithms in the literatures are somewhat opaque to researchers who are not well-versed in the theory of optimization, so that it's difficult to program. Therefore, to easily solve this inversion problem by developing the work of [29], we propose a stable and simple algorithm, called SVD-HPP [27], to solve the L p (0 < p ≤ 1) regularization, which can turn L p (0 < p ≤ 1) regularization into a series of L 2 regularizations. Then the existing algorithms can be applied to solve each L 2 regularization [30].…”

Section: Introductionmentioning

confidence: 99%

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Super-Resolution Inversion of Non-Stationary Seismic Traces

Gao¹

2021

CSIAM-AM

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In reflection seismology, the inversion of subsurface reflectivity from the observed seismic traces (super-resolution inversion) plays a crucial role in target detection. Since the seismic wavelet in reflection seismic data varies with the travel time, the reflection seismic trace is non-stationary. In this case, a relative amplitude-preserving super-resolution inversion has been a challenging problem. In this paper, we propose a super-resolution inversion method for the non-stationary reflection seismic traces. We assume that the amplitude spectrum of seismic wavelet is a smooth and unimodal function, and the reflection coefficient is an arbitrary random sequence with sparsity. The proposed method can obtain not only the relative amplitude-preserving reflectivity but also the seismic wavelet. In addition, as a by-product, a special Q field can be obtained. The proposed method consists of two steps. The first step devotes to making an approximate stabilization of non-stationary seismic traces. The key points include: firstly, dividing non-stationary seismic traces into several stationary segments, then extracting wavelet amplitude spectrum from each segment and calculating Q value by the wavelet amplitude spectrum between adjacent segments; secondly, using the estimated Q field to compensate for the attenuation of seismic signals in sparse domain to obtain approximate stationary seismic traces. The second step is the super-resolution inversion of stationary seismic traces. The key points include: firstly, constructing the objective function, where the approximation error is measured in L 2 space, and adding some constraints into reflectivity and seismic wavelet to solve ill-conditioned problems; secondly, applying a Hadamard product parametrization (HPP) to transform the non-convex problem based on the L p (0 < p < 1) constraint into a series of convex optimization problems in L 2 space, where the convex optimization problems are solved by the singular value decomposition (SVD) method and the regularization parameters are determined by the L-curve method in the case of single-variable inversion. In this paper, the effectiveness of the proposed method is demonstrated by both synthetic data and field data.

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Section: Appendix A: Determination Of Parameters In the Com Methodsmentioning

confidence: 99%

Section: Appendix A: Determination Of Parameters In the Com Methodsmentioning

confidence: 99%

Section: Super-resolution Inversion Based On Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Super-Resolution Inversion of Non-Stationary Seismic Traces

Gao¹

2021

CSIAM-AM

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“…The inverse problem belongs to the group of so-called ill-posed problems. According to Hadamard, well-conditioned problems must meet three criteria: the solution exists, the solution is unambiguous and the solution is stable [5]. A well-posed problem is likely to be solved on a computer using a stable algorithm [6,7,8].…”

Section: Introductionmentioning

confidence: 99%

Logistic Regression for Machine Learning in Process Tomography

Rymarczyk

Kozłowski

Kłosowski

et al. 2019

Sensors

118

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The main goal of the research presented in this paper was to develop a refined machine learning algorithm for industrial tomography applications. The article presents algorithms based on logistic regression in relation to image reconstruction using electrical impedance tomography (EIT) and ultrasound transmission tomography (UST). The test object was a tank filled with water in which reconstructed objects were placed. For both EIT and UST, a novel approach was used in which each pixel of the output image was reconstructed by a separately trained prediction system. Therefore, it was necessary to use many predictive systems whose number corresponds to the number of pixels of the output image. Thanks to this approach the under-completed problem was changed to an over-completed one. To reduce the number of predictors in logistic regression by removing irrelevant and mutually correlated entries, the elastic net method was used. The developed algorithm that reconstructs images pixel-by-pixel is insensitive to the shape, number and position of the reconstructed objects. In order to assess the quality of mappings obtained thanks to the new algorithm, appropriate metrics were used: compatibility ratio (CR) and relative error (RE). The obtained results enabled the assessment of the usefulness of logistic regression in the reconstruction of EIT and UST images.

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Denoising matrix factorization for high-dimensional time series forecasting

Chen,

Fang,

2023

Neural Comput & Applic

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Lasso, fractional norm and structured sparse estimation using a Hadamard product parametrization

Cited by 32 publications

References 19 publications

Super-Resolution Inversion of Non-Stationary Seismic Traces

Super-Resolution Inversion of Non-Stationary Seismic Traces

Logistic Regression for Machine Learning in Process Tomography

Denoising matrix factorization for high-dimensional time series forecasting

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