Exposed faces and duality for symmetric and unitarily invariant norms

Sá, Eduardo Marques de

doi:10.1016/0024-3795(94)90499-5

Cited by 26 publications

(22 citation statements)

References 16 publications

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“…(11), we define the Partial Singular Value Thresholding (PSVT) operator P N,τ [·]. Before defining the PSVT, we first introduce the von Neumann's lemma (see details in de Sá et al [14]). [14]).…”

Section: Solving a *mentioning

confidence: 99%

See 1 more Smart Citation

Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications

Tai

Bazin

et al. 2016

IEEE Trans. Pattern Anal. Mach. Intell.

221

164

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Robust Principal Component Analysis (RPCA) via rank minimization is a powerful tool for recovering underlying low-rank structure of clean data corrupted with sparse noise/outliers. In many low-level vision problems, not only it is known that the underlying structure of clean data is low-rank, but the exact rank of clean data is also known. Yet, when applying conventional rank minimization for those problems, the objective function is formulated in a way that does not fully utilize a priori target rank information about the problems. This observation motivates us to investigate whether there is a better alternative solution when using rank minimization. In this paper, instead of minimizing the nuclear norm, we propose to minimize the partial sum of singular values, which implicitly encourages the target rank constraint. Our experimental analyses show that, when the number of samples is deficient, our approach leads to a higher success rate than conventional rank minimization, while the solutions obtained by the two approaches are almost identical when the number of samples is more than sufficient. We apply our approach to various low-level vision problems, e.g., high dynamic range imaging, motion edge detection, photometric stereo, image alignment and recovery, and show that our results outperform those obtained by the conventional nuclear norm rank minimization method.

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Section: Solving a *mentioning

confidence: 99%

“…Before defining the PSVT, we first introduce the von Neumann's lemma (see details in de Sá et al [14]). [14]). For any matrices B, Z ∈ R m×n and vectors of the singular values σ(·), the following equality holds: (13) where U n denotes the set of n×n unitary matrices, and A, B = Tr(A T B), for any matrix A ∈ R m×n .…”

Section: Solving a *mentioning

confidence: 99%

Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications

Tai

Bazin

et al. 2016

IEEE Trans. Pattern Anal. Mach. Intell.

221

164

View full text Add to dashboard Cite

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“…In [7] these results are generalized to arbitrary unitarily invariant norms when F = 0. See [3] for a discussion regarding the proof and the original formulation of von Neumann.…”

Section: I(a) = R K (A)mentioning

confidence: 99%

Convex envelopes for fixed rank approximation

Андерссон

Carlsson

2017

Optim Lett

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A convex envelope for the problem of finding the best approximation to a given matrix with a prescribed rank is constructed. This convex envelope allows the usage of traditional optimization techniques when additional constraints are added to the finite rank approximation problem. Expression for the dependence of the convex envelope on the singular values of the given matrix is derived and global minimization properties are derived. The corresponding proximity operator is also studied.

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“…According to de Sá [5] this optimality is hidden in the original von Neumann's proof of (1), but this is not that simple. Lewis [10] studied (1) under Lie theoretical terms.…”

Section: 3mentioning

confidence: 99%