Low-Rank Spectral Optimization via Gauge Duality

Friedlander, Michael P.; Macêdo, Ives

doi:10.1137/15m1034283

Cited by 19 publications

(23 citation statements)

References 30 publications

(51 reference statements)

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“…Note that this can also be proven as a consequence of von Neumann's trace inequality [32,48]. This property is used by Friedlander et al [17] for the construction of dual methods for low-rank semidefinite optimization.…”

Section: Example 23 (One Norm)mentioning

confidence: 99%

Polar Alignment and Atomic Decomposition

Fan,

Jeong,

Sun

et al. 2019

Preprint

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Structured optimization uses a prescribed set of atoms to assemble a solution that fits a model to data. Polarity, which extends the familiar notion of orthogonality from linear sets to general convex sets, plays a special role in a simple and geometric form of convex duality. This duality correspondence yields a general notion of alignment that leads to an intuitive and complete description of how atoms participate in the final decomposition of the solution. The resulting geometric perspective leads to variations of existing algorithms effective for large-scale problems. We illustrate these ideas with many examples, including applications in matrix completion and morphological component analysis for the separation of mixtures of signals.

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Section: Example 23 (One Norm)mentioning

confidence: 99%

Polar Alignment and Atomic Decomposition

Fan,

Jeong,

Sun

et al. 2019

Preprint

Self Cite

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“…It is well-known that the SDP-based methods are hardly scalable. In order to overcome this drawback, Friedlander and Macdo [27] considered a dual formulation based on the fact that the dimension of the dual problem grows much more slowly than the one of the primal problem. Another convex approach was addressed by Bahmani-Romberg [2] and independently by Goldstein-Studer [30] to relax quadratic equations of PR and meanwhile maximize the inner product between an "anchor" vector and the unknown.…”

Section: √ N1n2mentioning

confidence: 99%

Total Variation--Based Phase Retrieval for Poisson Noise Removal

Chang¹,

Lou²,

Duan³

et al. 2018

SIAM J. Imaging Sci.

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Phase retrieval plays an important role in vast industrial and scientific applications. We consider a noisy phase retrieval problem in which the magnitudes of the Fourier transform (or a general linear transform) of an underling object are corrupted by Poisson noise, since any optical sensors detect photons, and the number of detected photons follows the Poisson distribution. We propose a variational model for phase retrieval based on a total variation regularization as an image prior and maximum a posteriori estimation of Poisson noise model, which is referred to as "TV-PoiPR". We also propose an efficient numerical algorithm based on alternating direction method of multipliers and establish its convergence. Extensive experiments for coded diffraction, holographic, and ptychographic patterns are conducted using both real and complex-valued images to demonstrate the effectiveness of our proposed methods.

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“…Recently, Friedlander & Macedo [29] have proposed a novel first-order method that is based on gauge duality, rather than Lagrangian duality. This approach converts an SDP into an eigenvalue optimization problem.…”

Section: 1mentioning

confidence: 99%

An Optimal-Storage Approach to Semidefinite Programming using Approximate Complementarity

Ding¹,

Yurtsever²,

Cevher³

et al. 2019

Preprint

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This paper develops a new storage-optimal algorithm that provably solves generic semidefinite programs (SDPs) in standard form. This method is particularly effective for weakly constrained SDPs. The key idea is to formulate an approximate complementarity principle: Given an approximate solution to the dual SDP, the primal SDP has an approximate solution whose range is contained in the eigenspace with small eigenvalues of the dual slack matrix. For weakly constrained SDPs, this eigenspace has very low dimension, so this observation significantly reduces the search space for the primal solution. This result suggests an algorithmic strategy that can be implemented with minimal storage: (1) Solve the dual SDP approximately; (2) compress the primal SDP to the eigenspace with small eigenvalues of the dual slack matrix; (3) solve the compressed primal SDP. The paper also provides numerical experiments showing that this approach is successful for a range of interesting large-scale SDPs.

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Low-Rank Spectral Optimization via Gauge Duality

Cited by 19 publications

References 30 publications

Polar Alignment and Atomic Decomposition

Polar Alignment and Atomic Decomposition

Total Variation--Based Phase Retrieval for Poisson Noise Removal

An Optimal-Storage Approach to Semidefinite Programming using Approximate Complementarity

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