Projection-like Retractions on Matrix Manifolds

Absil, P.-A.; Malick, Jérôme

doi:10.1137/100802529

Cited by 235 publications

(241 citation statements)

References 17 publications

(32 reference statements)

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“…We remark that, in the context of [5,6], PM induces a second-order retraction onM near A given by ∆ ∈ TM(A) → PM(A + ∆) ∈M, which locally fits the exponential mapping up to second order.…”

Section: Algorithm 36 (Projection Onto Fixed-rank Matrix Manifold VImentioning

confidence: 96%

“…With P N at hand, the global minimizer of the sparse matrix subproblem is computed as in step 1 of Algorithm 3.3 below. On the other hand, such a global minimizer does not necessarily guarantee a sufficient decrease in the objective as required by condition (6). When the global minimizer fails to fulfill condition (6), we resort to a local minimizer as specified by step 3 in Algorithm 3.3.…”

Section: Sparse Matrix (B-)subproblemmentioning

confidence: 99%

“…It is known [21,6] that for any point A on the smooth manifoldM, the projection PM is a smooth diffeomorphism in a neighborhood of A, and moreover the differentiation rule…”

Section: Algorithm 36 (Projection Onto Fixed-rank Matrix Manifold VImentioning

confidence: 99%

“…Since ∆ ∈ TM(A) → PM(A + ∆) ∈M is a second-order retraction onM near A (see Example 18 in [6]), we have the following Taylor expansion:…”

Section: Lim Supmentioning

confidence: 99%

“…Let {A k } ⊂M be generated by Algorithm 3.7 along with some sequence {B k } ⊂ N satisfying condition (6). Then the following statements hold true:…”

mentioning

confidence: 99%

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Robust Principal Component Pursuit via Inexact Alternating Minimization on Matrix Manifolds

Hintermüller

2014

J Math Imaging Vis

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Robust principal component pursuit (RPCP) refers to a decomposition of a data matrix into a low-rank component and a sparse component. In this work, instead of invoking a convex-relaxation model based on the nuclear norm and the 1 -norm as is typically done in this context, RPCP is solved by considering a least-squares problem subject to rank and cardinality constraints. An inexact alternating minimization scheme, with guaranteed global convergence, is employed to solve the resulting constrained minimization problem. In particular, the low-rank matrix subproblem is resolved inexactly by a tailored Riemannian optimization technique, which favorably avoids singular value decompositions in full dimension. For the overall method, a corresponding q-linear convergence theory is established. The numerical experiments show that the newly proposed method compares competitively with a popular convex-relaxation based approach.

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Section: Algorithm 36 (Projection Onto Fixed-rank Matrix Manifold VImentioning

confidence: 96%

Section: Sparse Matrix (B-)subproblemmentioning

confidence: 99%

“…It is known [21,6] that for any point A on the smooth manifoldM, the projection PM is a smooth diffeomorphism in a neighborhood of A, and moreover the differentiation rule…”

Section: Algorithm 36 (Projection Onto Fixed-rank Matrix Manifold VImentioning

confidence: 99%

“…Since ∆ ∈ TM(A) → PM(A + ∆) ∈M is a second-order retraction onM near A (see Example 18 in [6]), we have the following Taylor expansion:…”

Section: Lim Supmentioning

confidence: 99%

“…Let {A k } ⊂M be generated by Algorithm 3.7 along with some sequence {B k } ⊂ N satisfying condition (6). Then the following statements hold true:…”

mentioning

confidence: 99%

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Robust Principal Component Pursuit via Inexact Alternating Minimization on Matrix Manifolds

Hintermüller

2014

J Math Imaging Vis

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Nonconvex Optimization via MM Algorithms: Convergence Theory

Lange

Won

Landeros

et al. 2021

Wiley StatsRef: Statistics Reference Online

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The majorization-minimization (MM) principle is an extremely general framework for deriving optimization algorithms. It includes the expectation-maximization (EM) algorithm, proximal gradient algorithm, concave-convex procedure, quadratic lower bound algorithm, and proximal distance algorithm as special cases. Besides numerous applications in statistics, optimization, and imaging, the MM principle finds wide applications in large scale machine learning problems such as matrix completion, discriminant analysis, and nonnegative matrix factorizations. When applied to nonconvex optimization problems, MM algorithms enjoy the advantages of convexifying the objective function, separating variables, numerical stability, and ease of implementation. However, compared to the large body of literature on other optimization algorithms, the convergence analysis of MM algorithms is scattered and problem specific. This survey presents a unified treatment of the convergence of MM algorithms. With modern applications in mind, the results encompass non-smooth objective functions and nonasymptotic analysis.

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Tensor completion using geodesics on Segre manifolds

Lars

Veken

Vannieuwenhoven

2022

Numerical Linear Algebra App

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We propose a Riemannian conjugate gradient algorithm for approximating incomplete tensors by canonical polyadic decompositions of low rank. Our main contribution consists of an explicit expression for an almost everywhere complete set of geodesics of the Segre manifold of rank-1 tensors. These are leveraged in our Riemannian optimization algorithm over a geometrically convenient parametrization of rank-r tensors to move in the direction of a tangent vector over the r-fold product of Segre manifolds. We apply our method to movie rating predictions in a recommender system for the MovieLens dataset, and for identifying pure fluorophores via fluorescent spectroscopy with missing data. In this last application, we can recover the tensor decomposition from only 6.5% of the data. In our numerical experiments, the proposed Riemannian conjugate gradient algorithm was competitive with a state-of-the-art quasi-Newton method with truncated conjugate gradient inner solves from Tensorlab in terms of accuracy and could reduce the computation time by half.

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Projection-like Retractions on Matrix Manifolds

Cited by 235 publications

References 17 publications

Robust Principal Component Pursuit via Inexact Alternating Minimization on Matrix Manifolds

Robust Principal Component Pursuit via Inexact Alternating Minimization on Matrix Manifolds

Nonconvex Optimization via MM Algorithms: Convergence Theory

Tensor completion using geodesics on Segre manifolds

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