A Broyden Class of Quasi-Newton Methods for Riemannian Optimization

Huang, Wen; Gallivan, Kyle A.; Absil, Pierre-Antoine

doi:10.1137/140955483

Cited by 152 publications

(157 citation statements)

References 24 publications

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“…It should be noted that for any fixed partition of t r , this algorithm provides the exact optimal γ that is restricted to the graph on this partition. Recently, an optimization method of performing optimization methods on Riemannian manifolds was developed [42], with one method being the Broyden-FletcherGoldfarb-Shanno (BFGS) algorithm. The BFGS algorithm is a faster alternative to dynamic programming.…”

Section: Discussionmentioning

confidence: 99%

Variability of Ventricular Repolarization Dispersion Quantified by Time-Warping the Morphology of the T-Waves

Ramírez

Orini

Tucker

et al. 2017

IEEE Trans. Biomed. Eng.

109

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Abstract-Objective: We propose two electrocardiogram (ECG)-derived markers of T-wave morphological variability in the temporal, dw, and amplitude, da, domains. Two additional markers, d NL w and d NL a , restricted to measure the non-linear information present within dw and da are also proposed. Methods: We evaluated the accuracy of the proposed markers in capturing T-wave time and amplitude variations in 3 situations: (1) In a simulated set up with presence of additive Laplacian noise, (2) when modifying the spatio-temporal distribution of electrical repolarization with an electro-physiological cardiac model and (3) in ECG records from healthy subjects undergoing a tilt table test. Results: The metrics dw, da, d NL w and d NL a followed T-wave time and amplitude induced variations under different levels of noise, were strongly associated with changes in the spatiotemporal dispersion of repolarization, and showed to provide additional information to differences in the heart rate, QT and Tpe intervals, and in the T-wave width and amplitude. Conclusion: The proposed ECG-derived markers robustly quantify T-wave morphological variability, being strongly associated with changes in the dispersion of repolarization. Significance: The proposed ECG-derived markers can help to quantify the variability in the dispersion of ventricular repolarization, showing a great potential to be used as arrhythmic risk predictors in clinical situations.

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Section: Discussionmentioning

confidence: 99%

Variability of Ventricular Repolarization Dispersion Quantified by Time-Warping the Morphology of the T-Waves

Ramírez

Orini

Tucker

et al. 2017

IEEE Trans. Biomed. Eng.

109

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“…The optimization problem (14) is equivalent to (11), meaning that both (11) and (14) have the same Karush-Kuhn-Tucker (KKT) points. To find a KKT point for the optimization problem (11), we solve (14) by using a proper optimization algorithm on the Stiefel manifold, that is, the superlinearly convergent BFGS algorithm of [5] for solving (15) As discussed in [5] and [6], PSDEIV-Rr converges globally to a local solution of (11) with an asymptotic superlinear convergence rate.…”

Section: Mathematical Solutionmentioning

confidence: 99%

A New Error in Variables Model for Solving Positive Semi-Definite Linear Systems with Applications to Well-known Control Problems

Bagherpour¹,

Mahdavi–Amiri²

2017

Proceedings of the 4th International Conference of Control, Dynamic Systems, and Robotics (CDSR'17)

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-We have recently presented a competitive method to solve overdetermined linear systems of equations with multiple right hand side vectors, where the unknown matrix is symmetric and positive definite. The coefficient and the right hand side matrices are respectively named as data and target matrices. There are also physical contexts, such as modeling a deformable structure and computing the correlation matrix in finance or insurance/reinsurance industries, where a symmetric positive semi-definite solution of an overdetermined linear system of equations needs to be computed. Several methods have been proposed for solving such problems with the assumption that the data matrix is unrealistically error free. In real measurements, however, both the data and target matrices may contain error. Thus, for practical purposes, an error in variables model appears to be more appropriate. Here, we define a new error function to consider error in both data and target matrices and propose an iterative algorithm to minimize the defined error. We illustrate that our proposed approach turns to be efficient in attaining accurate solutions. For this purpose, we generate random test problems and show that our proposed algorithm computes the solution faster than the existing methods. Moreover, the standard deviation in error matrix is considerably lower when we use our proposed algorithm. This feature makes our method more efficient and desirable in real applications.

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“…It is well-known that the steepest descent method may have slow convergence, see e.g., [11]. In this paper, we apply a faster algorithm, a limited-memory version of Riemannian BFGS (Broyden-Fletcher-Goldfarb-Shanno) method (LRBFGS), which is introduced in [3] and shown to outperform many other state-of-the-art Riemannian algorithms for many large-scale problems, e.g., [11], [4], [5].…”

Section: A Riemannian Approachmentioning

confidence: 99%