Topological Properties of Manifolds

Gauld, David

doi:10.2307/2319220

Cited by 16 publications

(11 citation statements)

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“…where ρ is the density of the material. Using a finite-difference numerical solution with explicit time integration, one can solve the system of the hyperbolic differential equations (29) and (30). However, such a solution scheme can be computationally expensive since prohibitively small timesteps and are required in order to satisfy the Courant-Friedrichs-Lewy (CFL) condition [30].…”

Section: Elasto-plastic Materials Modelmentioning

confidence: 99%

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Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations

Giovanis¹,

Shields²

2018

Journal of Computational Physics

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This paper addresses uncertainty quantification (UQ) for problems where scalar (or low-dimensional vector) response quantities are insufficient and, instead, full-field (very high-dimensional) responses are of interest. To do so, an adaptive stochastic simulation-based methodology is introduced that refines the probability space based on Grassmann manifold variations. The proposed method has a multi-element character discretizing the probability space into simplex elements using a Delaunay triangulation. For every simplex, the high-dimensional solutions corresponding to its vertices (sample points) are projected onto the Grassmann manifold. The pairwise distances between these points are calculated using appropriately defined metrics and the elements with large total distance are sub-sampled and refined. As a result, regions of the probability space that produce significant changes in the full-field solution are accurately resolved. An added benefit is that an approximation of the solution within each element can be obtained by interpolation on the Grassmann manifold without the need to develop a mathematical surrogate model. The method is applied to study the probability of shear band formation in a bulk metallic glass using the shear transformation zone theory. *

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Section: Elasto-plastic Materials Modelmentioning

confidence: 99%

“…The doubly infinite Grassmannian is non-Hausdorff and therefore nonmetrizable [29]. Consequently, to define a metric between any pair of subspaces of arbitrary dimensions is to define one on G(∞, ∞).…”

Section: Distance Between Subspaces Of Different Dimensionmentioning

confidence: 99%

Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations

Giovanis¹,

Shields²

2018

Journal of Computational Physics

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“…The weakness of SMOTE is due to the topological structure of a manifold; it will only produce an accurate kNN set if the query instances are close together (Gauld 2008). Similarly, a straight-line distance will only be accurate on a manifold when it is applied to measure the distance between instances in the same local neighbourhood.…”

Section: Limitations Of Smote On Manifoldsmentioning

confidence: 99%

Manifold-based synthetic oversampling with manifold conformance estimation

2017

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Classification domains such as those in medicine, national security and the environment regularly suffer from a lack of training instances for the class of interest. In many cases, classification models induced under these conditions have poor predictive performance on the important minority class. Synthetic oversampling can be applied to mitigate the impact of imbalance by generating additional training instances. In this field, the majority of research has focused on refining the SMOTE algorithm. We note, however, that the generative bias of SMOTE is not appropriate for the large class of learning problems that conform to the manifold property. These are high-dimensional problems, such as image and spectral classification, with implicit feature spaces that are lower-dimensional than their physical data spaces. We show that ignoring this can lead to instances being generated in erroneous regions of the data space. We propose a general framework for manifold-based synthetic oversampling that helps users to select a domain-appropriate manifold learning method, such as PCA or autoencoder, and apply it to model and generate additional training samples. We evaluate data generation on theoretical distributions and image classification tasks that are standard in the manifold learning literature, and empirically show its positive impact on the classification of high-dimensional image and gamma-ray spectra tasks, along with 16 UCI datasets.

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“…Indeed in the remaining part of this proof, we restrict our attention to x in C, t ≥ t 0 and d with L ∞ norm smaller than D c , so there is no difference on [t 0 , +∞) between solutions of (2) and solutions of (8).…”

Section: Introduction and Motivationsmentioning

confidence: 99%

“…Acknowledgements: The authors wish to thank Prof. JeanMichel Coron for pointing out reference [8]. The first author wishes to thank Prof. Rantzer and Prof. Sepulchre for useful discussions on the subject of this note.…”

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confidence: 99%

Stability robustness in the presence of exponentially unstable isolated equilibria

Angeli

Praly

2010

49th IEEE Conference on Decision and Control (CDC)

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Abstract-This note studies nonlinear systems evolving on manifolds with a finite number of asymptotically stable equilibria and a Lyapunov function which strictly decreases outside equilibrium points. If the linearizations at unstable equilibria have at least one eigenvalue with positive real part, then almost global asymptotic stability turns out to be robust with respect to sufficiently small disturbances in the L∞ norm. Applications of this result are shown in the study of almost global Input-toState stability.

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Topological Properties of Manifolds

Cited by 16 publications

References 0 publications

Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations

Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations

Manifold-based synthetic oversampling with manifold conformance estimation

Stability robustness in the presence of exponentially unstable isolated equilibria

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