Fast singular value thresholding without singular value decomposition

Cai, Jian-Feng; Osher, Stanley

doi:10.4310/maa.2013.v20.n4.a2

Cited by 53 publications

(54 citation statements)

References 71 publications

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“…Applying SVT on large matrices (e.g., 1.5K × 10K dimension) is computationally expensive. To overcome this bottleneck, we apply the algorithm in [9]. Instead of performing SVT, this algorithm solves the dual problem of the original low-rank optimization, which only involves polar decomposition and projection.…”

Section: Update L I and E I ∀I ∈ {1 · · · K − 1}mentioning

confidence: 99%

Learning by Associating Ambiguously Labeled Images

Zeng

Xiao

Jia

et al. 2013

2013 IEEE Conference on Computer Vision and Pattern Recognition

117

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Section: Update L I and E I ∀I ∈ {1 · · · K − 1}mentioning

confidence: 99%

Learning by Associating Ambiguously Labeled Images

Zeng

Xiao

Jia

et al. 2013

2013 IEEE Conference on Computer Vision and Pattern Recognition

117

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“…Efficient methods of solving such problems have been recently studied in the literature [29], [30]. In practice, the separation rank r 0 may not be large 2 .…”

Section: Permuted Rank-penalized Least-squaresmentioning

confidence: 99%

Low separation rank covariance estimation using Kronecker product expansions

Tsiligkaridis

Hero

2013

2013 IEEE International Symposium on Information Theory

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Abstract-This paper presents a new method for estimating high dimensional covariance matrices. Our method, permuted rank-penalized least-squares (PRLS), is based on Kronecker product series expansions of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator. In addition, this framework allows one to tradeoff estimation error for approximation error, thus providing a scalable covariance estimation framework in terms of separation rank, an analog to low rank approximation of covariance matrices [1]. The MSE convergence rates generalize the high dimensional rates recently obtained for the ML Flip-flop algorithm [2], [3].

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“…Note that a numerical solution of (D 1,τ ), resp. (P 1,τ ) without singular value decomposition was proposed in [9].…”

Section: Matrix Completion and Tensor Inpainting Problemsmentioning

confidence: 99%

Homogeneous Penalizers and Constraints in Convex Image Restoration

2012

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Recently convex optimization models were successfully applied for solving various problems in image analysis and restoration. In this paper, we are interested in relations between convex constrained optimization problems of the form argmin{Φ(x) subject to Ψ(x) ≤ τ } and their penalized counterparts argmin{Φ(x) + λΨ(x)}. We recall general results on the topic by the help of an epigraphical projection. Then we deal with the special setting Ψ := L · with L ∈ R m,n and Φ := ϕ(H·), where H ∈ R n,n and ϕ : R n → R ∪ {+∞} meet certain requirements which are often fulfilled in image processing models. In this case we prove by incorporating the dual problems that there exists a bijective function such that the solutions of the constrained problem coincide with those of the penalized problem if and only if τ and λ are in the graph of this function. We illustrate the relation between τ and λ for various problems arising in image processing. In particular, we point out the relation to the Pareto frontier for joint sparsity problems. We demonstrate the performance of the constrained model in restoration tasks of images corrupted by Poisson noise with the I-divergence as data fitting term ϕ and in inpainting models with the constrained nuclear norm. Such models can be useful if we have a priori knowledge on the image rather than on the noise level.

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Fast singular value thresholding without singular value decomposition

Cited by 53 publications

References 71 publications

Learning by Associating Ambiguously Labeled Images

Learning by Associating Ambiguously Labeled Images

Low separation rank covariance estimation using Kronecker product expansions

Homogeneous Penalizers and Constraints in Convex Image Restoration

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