Proximal Point Algorithm On Riemannian Manifolds

Ferreira, O. P.; Oliveira, Paulo Roberto

doi:10.1080/02331930290019413

Cited by 267 publications

(190 citation statements)

References 10 publications

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“…Moreover, supposing +∞ k=0 1/λ k = +∞ and that f has a minimizer, the authors proved convergence of the sequence (f (x k )) to the minimum value and convergence of the sequence (x k ) to a minimizer point. With the result of convergence of the sequence generated by the proximal algorithm (1.2) (see Theorem 4.1), from a mathematical point -of -view, our paper generalizes the work of Ferreira and Oliveira [17], using Finslerian distances instead of Riemannian distances.…”

Section: Introductionmentioning

confidence: 93%

“…Since inf f > −∞, M is a Hadamard manifold and using similar arguments the case Riemannian, see Ferreira and Oliveira [17], we have that for any r > 0,x ∈ M the function…”

Section: Convergencementioning

confidence: 99%

“…This method was considered, in Riemannian context, by Ferreira and Oliveira [17], in the particular case where M is a Hadamard manifold, d(x k , ·) is a Riemannian distance, domf = M and f is convex. They proved that, for each k ∈ N , the function f (·) + 2 (x k , ·) : M → R is 1-coercive and, consequently, that the sequence {x k } is well defined, with x k+1 being uniquely determined.…”

Section: Introductionmentioning

confidence: 99%

“…In recent years, concepts and techniques of mathematical programming in Euclidean spaces have been extended to Riemannian contexts , not only for theoretical purposes but also to obtain effective algorithms, for example ( [29,17,30,31]). Far as we know, in Kristály et al [32] is presented the first work that models optimization problems in Finsler environments, but effective implementation was in Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off

Neto

Oliveira

Soares³

et al. 2013

J Optim Theory Appl

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In this paper we consider minimization problems with constraints. We will show that if the set of constraints is a Finslerian manifold of non positive flag curvature, and the objective function is differentiable and satisfies the property Kurdyka-Lojasiewicz, then the proximal point method is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converges to a minimizer point. We show how tools of Finslerian geometry, more specifically non symmetrical metrics, can be used to solve nonconvex constrained problems in Euclidean spaces. As an application, we give one result on the speed of decision and making and costs to change.

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Section: Introductionmentioning

confidence: 93%

“…Since inf f > −∞, M is a Hadamard manifold and using similar arguments the case Riemannian, see Ferreira and Oliveira [17], we have that for any r > 0,x ∈ M the function…”

Section: Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off

Neto

Oliveira

Soares³

et al. 2013

J Optim Theory Appl

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“…This method was originally introduced by Martinet [24] and further developed and studied by Rockafellar [27,28]. The convergence of the generated sequence {x k } has been attracted the attention of many authors for the convex case [11,15,17,18,23,24,27,28]. However, there are not many works for the case where f is not convex, we cite [3,26].…”

Section: Introductionmentioning

confidence: 99%

A proximal point algorithm with a ϕ-divergence for quasiconvex programming

Cunha¹,

Neto²,

Oliveira³

2010

Optimization

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We use the proximal point method with the ϕ-divergence given by ϕ(t) = t−log t− 1 for the minimization of quasiconvex functions subject to nonnegativity constraints. We establish that the sequence generated by our algorithm is well-defined in the sense that it exists and it is not cyclical. Without any assumption of boundedness level to the objective function, we obtain that the sequence converges to a stationary point. We also prove that when the regularization parameters go to zero, the sequence converges to an optimal solution.

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An extension of fast iterative shrinkage‐thresholding algorithm to Riemannian optimization for sparse principal component analysis

Huang

2021

Numerical Linear Algebra App

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Sparse principal component analysis (PCA), an important variant of PCA, attempts to find sparse loading vectors when conducting dimension reduction. This paper considers the nonsmooth Riemannian optimization problem associated with the ScoTLASS model1 for sparse PCA which can impose orthogonality and sparsity simultaneously. A Riemannian proximal method is proposed in the work of Chen et al.9 for the efficient solution of this optimization problem. In this paper, two acceleration schemes are introduced. First and foremost, we extend the FISTA method from the Euclidean space to the Riemannian manifold to solve sparse PCA, leading to the accelerated Riemannian proximal gradient method. Since the Riemannian optimization problem for sparse PCA is essentially nonconvex, a restarting technique is adopted to stabilize the accelerated method without sacrificing the fast convergence. Second, a diagonal preconditioner is proposed for the Riemannian proximal subproblem which can further accelerate the convergence of the Riemannian proximal methods. Numerical evaluations establish the computational advantages of the proposed methods over the existing proximal gradient methods on a manifold. Additionally, a short result concerning the convergence of the Riemannian subgradients of a sequence is established, which, together with the result in the work of Chen et al.,9 can show the stationary point convergence of the Riemannian proximal methods.

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Proximal Point Algorithm On Riemannian Manifolds

Cited by 267 publications

References 10 publications

Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off

Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off

A proximal point algorithm with a ϕ-divergence for quasiconvex programming

An extension of fast iterative shrinkage‐thresholding algorithm to Riemannian optimization for sparse principal component analysis

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