Long-time principal geodesic analysis in director-based dynamics of hybrid mechanical systems

In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior

리만다양체 상의 비모수적 차원축소방법론

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 통계학과, 2022. 8. 오희석.Over the decades, parametric dimension reduction methods have been actively developed for non-Euclidean data analysis. Examples include Fletcher et al., 2004; Huckemann et al., 2010; Jung et al., 2011; Jung et al., 2012; Zhang et al., 2013. Sometimes the methods are not enough to capture the structure of data. This dissertation presents newly developed nonparametric dimension reductions for data observed on manifold, resulting in more flexible fits. More precisely, the main focus is on the generalizations of principal curves into Riemannian manifold. The principal curve is considered as a nonlinear generalization of principal component analysis (PCA). The dissertation consists of four main parts as follows. First, the approach given in Chapter 3 lie in the same lines of Hastie (1984) and Hastie and Stuetzle (1989) that introduced the definition of original principal curve on Euclidean space. The main contributions of this study can be summarized as follows: (a) We propose both extrinsic and intrinsic approaches to form principal curves on spheres. (b) We establish the stationarity of the proposed principal curves on spheres. (c) In extensive numerical studies, we show the usefulness of the proposed method through real seismological data and real Human motion capture data as well as simulated data on 2-sphere, 4-sphere. Secondly, As one of further work in the previous approach, a robust nonparametric dimension reduction is proposed. To this ends, absolute loss and Huber loss are used rather than L2 loss. The contributions of Chapter 4 can be summarized as follows: (a) We study robust principal curves on spheres that are resistant to outliers. Specifically, we propose absolute-type and Huber-type principal curves, which go through the median of data, to robustify the principal curves for a set of data which may contain outliers. (b) For a theoretical aspect, the stationarity of the robust principal curves is investigated. (c) We provide practical algorithms for implementing the proposed robust principal curves, which are computationally feasible and more convenient to implement. Thirdly, An R package 'spherepc' comprehensively providing dimension reduction methods on a sphere is introduced with details for possible reproducible research. To the best of our knowledge, no available R packages offer the methods of dimension reduction and principal curves on a sphere. The existing R packages providing principal curves, such as 'princurve' and 'LPCM', are available only on Euclidean space. In addition, most nonparametric dimension reduction methods on manifold involve somewhat complex intrinsic optimizations. The proposed R package 'spherepc' provides the state-of-the-art principal curve technique on the sphere and comprehensively collects and implements the existing techniques. Lastly, for an effective initial estimate of complex structured data on manifold, local principal geodesics are first provided and the method is applied to various simulated and real seismological data. For variance stabilization and theoretical investigations for the procedure, nextly, the focus is on the generalization of Kégl (1999); Kégl et al., (2000), which provided the new definition of principal curve on Euclidean space, into generic Riemannian manifolds. Theories including consistency and convergence rate of the procedure by means of empirical risk minimization principle, are further established on generic Riemannian manifolds. The consequences on the real data analysis and simulation study show the promising characteristics of the proposed approach.본 학위 논문은 다양체 자료의 변동성을 더욱 효과적으로 찾아내기 위해, 다양체 자료의 새로운 비모수적 차원축소방법론을 제시하였다. 구체적으로, 주곡선(principal curves) 방법을 일반적인 다양체 공간으로 확장하는 것이 주요 연구 주제이다. 주곡선은 주성분분석(PCA)의 비선형적 확장 중 하나이며, 본 학위논문은 크게 네 가지의 주제로 이루어져 있다. 첫 번째로, Hastie (1984), Hastie and Stuetzle (1989}의 방법을 임의의 차원의 구면으로 표준적인 방식으로 확장한다. 이 연구 주제의 공헌은 다음과 같다. (a) 임의의 차원의 구면에서 내재적, 외재적인 방식의 주곡선 방법을 각각 제안한다. (b) 본 방법의 이론적 성질(정상성)을 규명한다. (c) 지질학적 자료 및 인간 움직임 자료 등의 실제 자료와 2차원, 4차원 구면위의 시뮬레이션 자료에 본 방법을 적용하여, 그 유용성을 보인다. 두 번째로, 첫 번째 주제의 후속 연구 중 하나로서, 두꺼운 꼬리 분포를 가지는 자료에 대하여 강건한 비모수적 차원축소 방법을 제안한다. 이를 위해, L2 손실함수 대신에 L1 및 휴버(Huber) 손실함수를 활용한다. 이 연구 주제의 공헌은 다음과 같다. (a) 이상치에 민감하지 않은 강건화주곡선(robust principal curves)을 정의한다. 구체적으로, 자료의 기하적 중심점을 지나는 L1 및 휴버 손실함수에 대응되는 새로운 주곡선을 제안한다. (b) 이론적인 측면에서, 강건화주곡선의 정상성을 규명한다. (c) 강건화주곡선을 구현하기 위해 계산이 빠른 실용적인 알고리즘을 제안한다. 세 번째로, 기존의 차원축소방법 및 본 방법론을 제공하는 R 패키지를 구현하였으며 이를 다양한 예제 및 설명과 함께 소개한다. 본 방법론의 강점은 다양체 위에서의 복잡한 최적화 방정식을 풀지않고, 직관적인 방식으로 구현 가능하다는 점이다. R 패키지로 구현되어 제공된다는 점이 이를 방증하며, 본 학위 논문의 연구를 재현가능하게 만든다. 마지막으로, 보다 복잡한 구조를 가지는 다양체 자료의 구조를 추정하기위해, 국소주측지선분석(local principal geodesics) 방법을 우선 제안한다. 이 방법을 실제 지질학 자료 및 다양한 모의실험 자료에 적용하여 그 활용성을 보였다. 다음으로, 추정치의 분산안정화 및 이론적 정당화를 위하여 Kégl (1999), Kégl et al., (2000) 방법을 일반적인 리만다양체로 확장한다. 더 나아가, 방법론의 일치성, 수렴속도와 같은 점근적 성질을 비롯하여 비점근적 성질인 집중부등식(concentration inequality)을 통계적학습이론을 이용하여 규명한다.1 Introduction 1 2 Preliminaries 8 2.1 Principal curves 8 2.1 Riemannian manifolds and centrality on manifold 10 2.1 Principal curves on Riemannian manifolds 14 3 Spherical principal curves 15 3.1 Enhancement of principal circle for initialization 16 3.2 Proposed principal curves 25 3.3 Numerical experiments 34 3.4 Proofs 45 3.5 Concluding remarks 62 4 Robust spherical principal curves 64 4.1 The proposed robust principal curves 64 4.2 Stationarity of robust spherical principal curves 72 4.3 Numerical experiments 74 4.4 Summary and future work 80 5 spherepc: An R package for dimension reduction on a sphere 84 5.1 Existing methods 85 5.2 Spherical principal curves 91 5.3 Local principal geodesics 94 5.4 Application 99 5.5 Conclusion 101 6 Local principal curves on Riemannian manifolds 112 6.1 Preliminaries 116 6.2 Local principal geodesics 118 6.3 Local principal curves 125 6.4 Real data analysis 133 6.5 Further work 133 7 Conclusion 139 A. Appendix 141 A.1. Appendix for Chapter 3 141 A.2. Appendix for Chapter 4 145 A.3. Appendix for Chapter 6 152 Abstract in Korean 176 Acknowledgement in Korean 179박

Curvature corrected tangent space-based approximation of manifold-valued data

When generalizing schemes for real-valued data approximation or decomposition to data living in Riemannian manifolds, tangent space-based schemes are very attractive for the simple reason that these spaces are linear. An open challenge is to do this in such a way that the generalized scheme is applicable to general Riemannian manifolds, is global-geometry aware and is computationally feasible. Existing schemes have been unable to account for all three of these key factors at the same time. In this work, we take a systematic approach to developing a framework that is able to account for all three factors. First, we will restrict ourselves to the -- still general -- class of symmetric Riemannian manifolds and show how curvature affects general manifold-valued tensor approximation schemes. Next, we show how the latter observations can be used in a general strategy for developing approximation schemes that are also global-geometry aware. Finally, having general applicability and global-geometry awareness taken into account we restrict ourselves once more in a case study on low-rank approximation. Here we show how computational feasibility can be achieved and propose the curvature-corrected truncated higher-order singular value decomposition (CC-tHOSVD), whose performance is subsequently tested in numerical experiments with both synthetic and real data living in symmetric Riemannian manifolds with both positive and negative curvature

Doctor of Philosophy in Computing

dissertationAn important area of medical imaging research is studying anatomical diffeomorphic shape changes and detecting their relationship to disease processes. For example, neurodegenerative disorders change the shape of the brain, thus identifying differences between the healthy control subjects and patients affected by these diseases can help with understanding the disease processes. Previous research proposed a variety of mathematical approaches for statistical analysis of geometrical brain structure in three-dimensional (3D) medical imaging, including atlas building, brain variability quantification, regression, etc. The critical component in these statistical models is that the geometrical structure is represented by transformations rather than the actual image data. Despite the fact that such statistical models effectively provide a way for analyzing shape variation, none of them have a truly probabilistic interpretation. This dissertation contributes a novel Bayesian framework of statistical shape analysis for generic manifold data and its application to shape variability and brain magnetic resonance imaging (MRI). After we carefully define the distributions on manifolds, we then build Bayesian models for analyzing the intrinsic variability of manifold data, involving the mean point, principal modes, and parameter estimation. Because there is no closed-form solution for Bayesian inference of these models on manifolds, we develop a Markov Chain Monte Carlo method to sample the hidden variables from the distribution. The main advantages of these Bayesian approaches are that they provide parameter estimation and automatic dimensionality reduction for analyzing generic manifold-valued data, such as diffeomorphisms. Modeling the mean point of a group of images in a Bayesian manner allows for learning the regularity parameter from data directly rather than having to set it manually, which eliminates the effort of cross validation for parameter selection. In population studies, our Bayesian model of principal modes analysis (1) automatically extracts a low-dimensional, second-order statistics of manifold data variability and (2) gives a better geometric data fit than nonprobabilistic models. To make this Bayesian framework computationally more efficient for high-dimensional diffeomorphisms, this dissertation presents an algorithm, FLASH (finite-dimensional Lie algebras for shooting), that hugely speeds up the diffeomorphic image registration. Instead of formulating diffeomorphisms in a continuous variational problem, Flash defines a completely new discrete reparameterization of diffeomorphisms in a low-dimensional bandlimited velocity space, which results in the Bayesian inference via sampling on the space of diffeomorphisms being more feasible in time. Our entire Bayesian framework in this dissertation is used for statistical analysis of shape data and brain MRIs. It has the potential to improve hypothesis testing, classification, and mixture models

Spherical Principal Curves

This paper presents a new approach for dimension reduction of data observed in a sphere. Several dimension reduction techniques have recently developed for the analysis of non-Euclidean data. As a pioneer work, Hauberg (2016) attempted to implement principal curves on Riemannian manifolds. However, this approach uses approximations to deal with data on Riemannian manifolds, which causes distorted results. In this study, we propose a new approach to construct principal curves on a sphere by a projection of the data onto a continuous curve. Our approach lies in the same line of Hastie and Stuetzle (1989) that proposed principal curves for Euclidean space data. We further investigate the stationarity of the proposed principal curves that satisfy the self-consistency on a sphere. Results from real data analysis with earthquake data and simulation examples demonstrate the promising empirical properties of the proposed approach

Principal subbundles for dimension reduction

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank $k$ tangent subbundle on $\mathbb{R}^d$, $k<d$, which we call a principal subbundle. This determines a sub-Riemannian metric on $\mathbb{R}^d$. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold $M$, construction of a representation of the point-cloud in $\mathbb{R}^k$, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold

Convex algebraic geometry of curvature operators

We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic set is a spectrahedron or a spectrahedral shadow; in particular, for $n\geq5$, these give new counter-examples to the Helton--Nie Conjecture. Moreover, efficient algorithms are provided if $n=4$ to test membership in such a set. For $n\geq5$, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra

Mathematical Imaging and Surface Processing

Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images. This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains