Convex Functions and Optimization Methods on Riemannian Manifolds

Udrişte, Constantin

doi:10.1007/978-94-015-8390-9

Cited by 489 publications

(407 citation statements)

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“…A common method to optimize on Euclidean spaces, namely gradient steepest descent, may be readily extended to smooth manifolds [21]. To this purpose, let us consider the differential equation on the manifold Y : …”

Section: A Survey Of Some Geometrical Conceptsmentioning

confidence: 99%

Learning the Fréchet Mean over the Manifold of Symmetric Positive-Definite Matrices

Fiori

2009

Cogn Comput

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The present manuscript tackles the problem of learning the average of a set of symmetric positive-definite (SPD) matrices. Averages are computed via the notion of Fréchet mean, and the associated metric dispersion is interpreted as the variance of the patterns around the Fréchet mean. Also, the problem of continuous interpolation of two SPD patterns is tackled within the manuscript. The property of volume conservation of the Fréchet mean and of the considered interpolatory scheme for SPD matrices is discussed as well. The paper describes several applications where the technique could be readily exploited, including in machine learning, intelligent control, pattern classification, speech emotion classification and diffusion tensor data analysis in medicine.

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Section: A Survey Of Some Geometrical Conceptsmentioning

confidence: 99%

Learning the Fréchet Mean over the Manifold of Symmetric Positive-Definite Matrices

Fiori

2009

Cogn Comput

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“…There are some advantages for a generalization of optimization methods from Euclidean spaces to Riemannian manifolds, because nonconvex and nonsmooth constrained optimization problems can be seen as convex and smooth unconstrained optimization problems from the Riemannian geometry point of view; see, for example, [15], [11], [12]. Nemeth [10] and Wang et al [16] studied monotone and accretive vector fields on Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…We recall some fundamental definitions, basic properties and notations needed for a comprehensive reading of this paper. These can be found in any textbook on Riemannian geometry, for example, [13], [15].…”

Section: Introductionmentioning

confidence: 99%

Equilibrium and mixed equilibrium problems under weak monotonicity on Hadamard manifolds

2018

Commun. Optim. Theory

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“…The distance between a query point Y and a convex hull model C SP D can be defined as the geodesic distance from a point to a convex set in the manifold [145]: Now, we are ready to define the nearest convex hull classifier on SPD manifolds. Similar to…”

Section: Formulations For Manifold Convex Hullmentioning

confidence: 99%

Fast and accurate image and video analysis on Riemannian manifolds

Zhao¹

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Recent times have seen increasing attention on representing images and videos on Riemannian manifolds. Such representations offer new means of preserving intrinsic data structures, which Euclidean-based representations often fail to capture. Preserving the intrinsic structure can bring superior benefits in terms of richer representations and robustness to variations. However, to intrinsically operate on the manifold incurs expensive computation complexity because of using non-linear operators. Furthermore, it is difficult to extend the existing computer vision techniques which were originally developed in Euclidean space into the manifold. To address these issues, recent research often uses extrinsic approaches such as tangent space projection and kernelised approaches. Unfortunately, these methods are not free of drawbacks in terms of low accuracy and high computational load. To that end, this research studies approaches for analysing manifold features which achieve superior performance from the manifold structures whilst computational complexity can be massively reduced. This thesis illustrates the general steps of manifold approaches for image and video analysis, here called manifold scheme, followed by discussions on the possible ways to reduce the computational complexities. Guided by the manifold scheme, this research further proposes two frameworks to analyse manifold features: random projection on Riemannian manifolds and convex hull on Symmetric Positive Definite manifolds. To solve the problems posed by each framework, we present three solutions. Through experiments on several computer vision tasks, we verified that our proposed methods can massively reduce the computational load, while still achieving competitive performance. In addition, our manifold scheme shows another possible way to reduce the computational load of manifold approaches through the generating manifold model using effective lower-level features with low dimensionality. Thus, this thesis proposes a landmark manifold model generated by effective landmark points for facial emotion recognition, which shows competitive performance with much lower computational complexity compared to state-of-the-art manifold methods. Design of experiments (10%)

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Convex Functions and Optimization Methods on Riemannian Manifolds

Cited by 489 publications

References 0 publications

Learning the Fréchet Mean over the Manifold of Symmetric Positive-Definite Matrices

Learning the Fréchet Mean over the Manifold of Symmetric Positive-Definite Matrices

Equilibrium and mixed equilibrium problems under weak monotonicity on Hadamard manifolds

Fast and accurate image and video analysis on Riemannian manifolds

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