Nonconvex and Nonsmooth Sparse Optimization via Adaptively Iterative Reweighted Methods

Wang, Hao; Zhang, Fan; Shi, Yuanming; Hu, Yaohua

doi:10.1007/s10898-021-01093-0

Cited by 5 publications

(2 citation statements)

References 38 publications

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“…[32]. Extensive computational studies in [28,37,56,60] revealed that the ℓ q regularization admits a significantly stronger sparsity promoting property than the ℓ 1 regularization in the sense that it guarantees to achieve a more sparse solution from a smaller amount of samples. Biological studies in [27,28,46,47] exhibited that the ℓ 0 and ℓ 1 2 regularizations achieve a more reliable solution in biological sense than the ℓ 1 regularization when applied to infer GRN.…”

Section: Regularization Methodsmentioning

confidence: 99%

“…However, the lower-order regularization problem (1.3) is nonconvex and nonsmooth, and thus it is very difficult in general to design practical algorithms to approach its global solution. Alternatively, tremendous efforts have been devoted to the development of optimization algorithms for approaching a local minimum or a stationary point of problem (1.3), such as smoothing method [14], PGA [28], iterative reweighted minimization method [56] and difference of convex functions algorithm (DCA) [23]. However, limited by the nonconvexity of the lower-order regularization, the convergence theory for the lower-order regularization problem is still far from satisfactory: only convergence to a stationary point was established in the literature [2,28,29]; while there is still no theoretical evidence to guarantee the convergence to a global minimum or the ground true sparse solution.…”

Section: Regularization Methodsmentioning

confidence: 99%

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Joint sparse optimization: lower-order regularization method and application in cell fate conversion

Hu,

Kwok Wai Yu

et al. 2024

Inverse Problems

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Multiple measurement signals are commonly collected in practical applications, and joint sparse optimization adopts the synchronous effect within multiple measurement signals to improve model analysis and sparse recovery capability. In this paper, we investigate the joint sparse optimization problem via $\ell_{p,q}$ regularization ($0\le q \le 1 \le p$) in three aspects: theory, algorithm and application. In the theoretical aspect, we introduce a weak notion of joint restricted Frobenius norm condition associated with the $\ell_{p,q}$ regularization, and apply it to establish an oracle property and a recovery bound for the $\ell_{p,q}$ regularization of joint sparse optimization problem. In the algorithmic aspect, we apply the well-known proximal gradient algorithm to solve the $\ell_{p,q}$ regularization problems, provide analytical formulas for proximal subproblems of certain specific $\ell_{p,q}$ regularizations, and establish the global convergence and linear convergence rate of the proximal gradient algorithm under some mild conditions. More importantly, we propose two types of proximal gradient algorithms with the truncation technique and the continuation technique, respectively, and establish their convergence to the ground true joint sparse solution within a tolerance relevant to the noise level and the recovery bound under the assumption of restricted isometry property. In the aspect of application, we develop a novel method, based on joint sparse optimization with lower-order regularization and proximal gradient algorithm, to infer the master transcription factors for cell fate conversion, which is a powerful tool in developmental biology and regenerative medicine. Numerical results indicate that the novel method facilitates fast identification of master transcription factors, give raise to the possibility of higher successful conversion rate and in the hope of reducing biological experimental cost.

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Section: Regularization Methodsmentioning

confidence: 99%

Section: Regularization Methodsmentioning

confidence: 99%

Joint sparse optimization: lower-order regularization method and application in cell fate conversion

Hu,

Kwok Wai Yu

et al. 2024

Inverse Problems

Self Cite

View full text Add to dashboard Cite

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Doubly iteratively reweighted algorithm for constrained compressed sensing models

Sun

Pong

2023

Comput Optim Appl

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Compressed sensing: a discrete optimization approach

Bertsimas,

Johnson

2024

Mach Learn

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We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. CS is a central problem in Statistics, Operations Research and Machine Learning which arises in applications such as signal processing, data compression, image reconstruction, and multi-label learning. We introduce an $$\ell _2$$ ℓ 2 regularized formulation of CS which we reformulate as a mixed integer second order cone program. We derive a second order cone relaxation of this problem and show that under mild conditions on the regularization parameter, the resulting relaxation is equivalent to the well studied basis pursuit denoising problem. We present a semidefinite relaxation that strengthens the second order cone relaxation and develop a custom branch-and-bound algorithm that leverages our second order cone relaxation to solve small-scale instances of CS to certifiable optimality. When compared against solutions produced by three state of the art benchmark methods on synthetic data, our numerical results show that our approach produces solutions that are on average $$6.22\%$$ 6.22 % more sparse. When compared only against the experiment-wise best performing benchmark method on synthetic data, our approach produces solutions that are on average $$3.10\%$$ 3.10 % more sparse. On real world ECG data, for a given $$\ell _2$$ ℓ 2 reconstruction error our approach produces solutions that are on average $$9.95\%$$ 9.95 % more sparse than benchmark methods ($$3.88\%$$ 3.88 % more sparse if only compared against the best performing benchmark), while for a given sparsity level our approach produces solutions that have on average $$10.77\%$$ 10.77 % lower reconstruction error than benchmark methods ($$1.42\%$$ 1.42 % lower error if only compared against the best performing benchmark). When used as a component of a multi-label classification algorithm, our approach achieves greater classification accuracy than benchmark compressed sensing methods. This improved accuracy comes at the cost of an increase in computation time by several orders of magnitude. Thus, for applications where runtime is not of critical importance, leveraging integer optimization can yield sparser and lower error solutions to CS than existing benchmarks.

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Nonconvex and Nonsmooth Sparse Optimization via Adaptively Iterative Reweighted Methods

Cited by 5 publications

References 38 publications

Joint sparse optimization: lower-order regularization method and application in cell fate conversion

Joint sparse optimization: lower-order regularization method and application in cell fate conversion

Doubly iteratively reweighted algorithm for constrained compressed sensing models

Compressed sensing: a discrete optimization approach

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