단체복합체에서 조화 공간의 이론과 응용

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 국웅.A harmonic cycle λ, also called a discrete harmonic form, is a solution of the Laplace's equation with the combinatorial Laplace operator obtained from the boundary operators of a simplicial chain complex. By combinatorial Hodge theory, harmonic spaces are isomorphic to homology groups with real coefficients. In particular, if a cell complex has a reduced homology with Betti number β_i = 1 of a specific dimension i, it has a unique harmonic cycle up to scalar multiplication, which we call the standard harmonic cycle. We will present a formula for the standard harmonic cycle λ of a cell complex based on a high-dimensional generalization of cycletrees. Moreover, by using duality, we will define the standard harmonic cocycle λ* and show intriguing combinatorial properties of λ and λ* in relation to (dual) spanning trees, (dual) cycletrees, winding numbers w( · ) and cutting numbers c( · ) in high dimensions. Finally, we will also suggest two application methods; an analysis to detect oscillations by using winding number, and cutting number, and a network embedding method, called harmonic mirroring.조화 사이클 λ는 이산 조화 형식으로도 부르며 라플라시안 방정식의 해이다. 이 라플라시안 방정식은 단체 연쇄복합체의 경계 작용소로부터 만든 조합론적 라플라시안 작용소가 0일 때 생기는 수식이다. 조합론적 호지 이론에 의하여 조화 공간은 실수를 계수로 갖는 호몰로지군과 동형이다. 특히 연쇠복합체의 특정 차원 i에서 축소 호몰리지군의 베티수 β_i가 1이라면, 스칼라곱을 제외하고 불변하는 고유한 조화 사이클을 얻을 수 있고, 이를 표준 조화 사이클이라 부른다. 우리는 표준 조화 사이클 λ을 표현하는 공식을 고차원으로 일반화된 사이클 트리를 바탕으로 나타내었다. 더욱이, 쌍대성을 이용하여 표준 조화 쌍대사이클 λ*를 정의하였고, λ와 λ*의 흥미롭고 조합론적인 성질들을 고차원 상태에서 보였다. 이는 (쌍대) 생성나무와 (쌍대) 사이클 트리, 회전수 w( · ), 자름수 c( · )와의 관계를 갖는다. 마지막으로 우리는 응용을 위한 두 가지 방법론을 제시한다. 회전수와 자름수를 이용한 진동 측정법과 조화 미러링으로 불리는 네트워크 매장 방법이다.1 Introduction 1 2 Preliminaries 3 2.1 Review of nite chain complex and (co)homology 3 2.2 High dimensional spanning trees 4 2.3 Harmonic space and combinatorial Hodge theory 5 3 Cycletree and its minimal cycle 7 3.1 Cycletree 7 3.2 Minimal cycle 9 4 Winding number 14 5 Standard harmonic cycle 17 6 Duality and dual spanning tree 20 7 Dual cycletree and cutting number 25 7.1 Dual cycletree and its minimal cocycle 25 7.2 Cutting number 27 8 Standard harmonic cocycle and relationship 31 9 Application 35 9.1 Oscillation Detection 35 9.2 Harmonic Mirroring 38 Abstract (in Korean) 41 Acknowledgement (in Korean) 42Docto

Combinatorial and Hodge Laplacians: Similarity and Difference

As key subjects in spectral geometry and combinatorial graph theory respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data and in their realization of diffusion and minimization of harmonic measures. It is believed that they also both associate with vector calculus, through the gradient, curl, and divergence, as argued in the popular usage of "Hodge Laplacians on graphs" in the literature. Nevertheless, these Laplacians are intrinsically different in their domains of definitions and applicability to specific data formats, hindering any in-depth comparison of the two approaches. To facilitate the comparison and bridge the gap between the combinatorial Laplacian and Hodge Laplacian for the discretization of continuous manifolds with boundary, we further introduce Boundary-Induced Graph (BIG) Laplacians using tools from Discrete Exterior Calculus (DEC). BIG Laplacians are defined on discrete domains with appropriate boundary conditions to characterize the topology and shape of data. The similarities and differences of the combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are then examined. Through an Eulerian representation of 3D domains as level-set functions on regular grids, we show experimentally the conditions for the convergence of BIG Laplacian eigenvalues to those of the Hodge Laplacian for elementary shapes.Comment: 26 page

반응 억제의 개인차와 관련한 대규모 휴지기 뇌네트워크의 특성

학위논문 (박사) -- 서울대학교 대학원 : 인문대학 협동과정 인지과학전공, 2020. 8. 이동수.Response inhibition is one of the essential cognitive functions and suppresses inappropriate responses for goal-directed behavior. When a brain is cognitively engaged, it enters a cognitive state that task-positive regions are activated, and the default mode network is deactivated (DMN). In contrast, DMN is activated, and task-positive regions are deactivated at rest. The transition between the states is important for the cognitive function, and recent studies have found that the salience network (SN) plays a crucial role in detecting and processing a salient signal and suppressing DMN at rest. It can be assumed that there exists optimized connectivity to perform response inhibition successfully and that it will also appear in resting-state requiring no cognitive effort. It was hypothesized that lower functional connectivity within SN and higher functional connectivity within DMN and greater anti-correlation between then is related to better response inhibition. The response inhibition of individuals was measured by the stop-signal task and the Stroop task. The correlation between intra-/inter-component functional connectivity derived from independent component analysis with dual regression and task performances were examined to test the hypothesis. The intra-/inter-component structural connectivity analysis using diffusion tensor imaging was conducted to provide a deeper understanding of functional connectivity. Topological characteristics of inter-component functional connectivity were also examined using the minimum spanning tree (MST) of each individual to provide a heuristic insight from the topological view. The results indicate that the functional connectivity within SN, but not DMN components, and the functional and structural connectivity between SN and DMN components are critical to elucidate individual differences in response inhibition. Higher structural connectivity but low functional connectivity of SN at rest was an important feature for superior response inhibition. The stronger structural connectivity and stronger anti-correlation between SN and DMN components were also indicative of better response inhibition. MST of a subject with the best performance showed direct connections between SN and anterior DMN/pDMN, whereas the MST of the one with the worst performance does not. These intra-/inter components connectivities reflect the organization of the brain that enables competent response inhibition and account for individual differences. This study might suggest that the individuals characteristics of large-scale network components at rest provide evidence to illustrate response inhibition of an individual without any experimental scan.반응 억제는 가장 주요한 인지 기능 중 하나이며 이상행동을 동반하는 다양한 정신 질환과도 깊은 관련이 있다. 따라서 이와 관련된 신경적 특성을 탐구하는 것은 매우 중요하다. 우리의 뇌는 어떠한 인지 기능을 수행할 때, 작업 관련 영역들을 활성화하고 자기 참조적 처리를 하는 디폴트 모드 네트워크 영역들은 비활성화한다. 휴지기에는 반대로 작업 관련 영역들은 비활성화하고 디폴트 모드 네트워크 영역은 활성화한다. 이처럼 인지 기능을 수행하기 위해서는 상태를 효율적으로 전환하는 것이 중요하다. 현출성 네트워크 (salience network)는 어떠한 과제를 할 때 중요한 자극을 탐지하여 처리하며 또한 디폴트 모드 네트워크의 활성을 억제하기 때문에 상태 간 전환에 핵심적인 역할을 하는 대규모 뇌네트워크이다. 따라서 이와 관련된 연결적 특성이 인지 기능과 밀접한 관련이 있으며, 그러한 특성은 휴지기의 연결성에도 반영되어 있을 것이라 가정하였다. 즉, 본 연구에서는 반응 억제의 개인차를 휴지기 대규모 뇌네트워크들의 특성을 통해 설명할 수 있을 것이며, 특히 현출성 네트워크의 낮은 기능적 연결성, 디폴트 모드 네트워크의 높은 기능적 연결성, 그리고 그 둘 간의 높은 기능적 역 상관 (anti-correlation)이 반응 억제에 우수한 사람들의 특징적인 휴지기 연결성일 것이라 가설을 세웠다. 개인의 반응 억제는 정지 신호 과제와 스트룹 과제를 통해 측정하였으며, 휴지기 대규모 뇌네트워크들의 특성들과 어떠한 상관을 갖는지 알아보았다. 즉, 기능적 뇌네트워크 내의 연결성과 두 뇌네트워크 간 연결성이 과제 수행과 어떠한 상관을 보이는지를 알아보았다. 또한 기능적 연결성에 대한 보다 깊은 이해를 위해 확산 텐서 영상과 트랙토그래피 기법을 사용하여 구조적 연결성과 반응 억제와의 상관을 알아보았다. 반응 억제와 관련된 토폴로지 특성 역시 함께 알아보기 위해 참여자들의 미니멈 스패닝 트리(MST: minimum spanning tree)를 계산하였다. 분석 결과, 현출성 네트워크, 그리고 현출성 네트워크와 디폴트 모드 네트워크 간의 연결성을 통해 반응 억제의 개인차를 설명할 수 있었다. 현출성 네트워크의 성분 내 구조적 연결성은 강하지만 휴지기의 기능적 연결성이 약한 참여자들일수록 반응 억제 수행이 우수했다. 현출성 네트워크와 디폴트 모드 네트워크 간의 구조적 연결성과 기능적 역 상관은 모두 높을수록 우수한 반응 억제를 보였다. 또한 두 네트워크 간 구조적 연결성이 높을수록 기능적 역 상관이 높은 것으로 나타났다. 토폴로지 분석에서는 가장 수행이 좋은 참여자의 MST는 현출성 네트워크와 디폴트 모드 네트워크들 간에 직접적인 연결이 관찰되었으나 수행이 가장 나쁜 참여자에서는 그러한 직접적인 연결이 관찰되지 않았다. 분석 결과, 휴지기의 현출성 네트워크 내 연결성, 그리고 현출성 네트워크와 디폴트 모드 네트워크 간의 기능적 역 상관과 구조적 연결성이 반응 억제의 개인 차이를 설명하였으나, 디폴트 모드 네트워크 내의 연결성은 그렇지 못했다. 이 연구는 과제 수행 중이 아닌 휴지기 동안의 뇌네트워크의 특성들을 통해 반응 억제의 개인차를 설명할 수 있음을 보여준다.1. Introduction 1 1.1. Response inhibition and its neural correlates 1 1.1.1. Cognitive tasks to measure response inhibition 1 1.1.2. The neural correlates of response inhibition 2 1.1.3. Response inhibition and resting-state brain 3 1.2. Investigations on large-scale networks underlying response inhibition 4 1.2.1. Resting-state networks and response inhibition 4 1.2.2. Structural connectivity 6 1.2.3. Topological characteristics 7 1.2.4. The aim of the present study 8 2. Methods 9 2.1. Subjects 9 2.2. Behavioral tasks to assess response inhibition 11 2.3. Brain imaging data acquisition and preprocessing 14 2.3.1. Resting-state fMRI 14 2.3.2. Diffusion tensor imaging 15 2.4. Resting-state networks and functional connectivity analysis 16 2.4.1. Group independent component analysis to identify resting-state networks 16 2.4.2. Dual regression to obtain subject-specific data of components 17 2.4.3. Estimation of subject-specific intra-/inter-component functional connectivity 21 2.5. Structural connectivity analysis 21 2.5.1. Structural connectivity and response inhibition 21 2.5.2. Relationship between functional connectivity and structural connectivity 22 2.6. Topological data analysis 25 2.6.1. Minimum spanning tree 25 3. Results 27 3.1. The performances of behavioral tasks 27 3.2. Intra-component connectivity and response inhibition 30 3.3. Inter-component connectivity and response inhibition 35 3.4. Relationship between functional connectivity and structural connectivity 41 3.5. Minimum spanning tree 43 4. Discussion 46 4.1. Resting-state network and cognition 46 4.2. Salience network and response inhibition 47 4.3. Connectivity and structural connectivity between SN and DMN 50 4.3.1. Functional connectivity between SN and DMN 50 4.3.2. Structural connectivity between SN and DMN 51 4.3.3. Topological characteristics between SN and DMN 53 4.4. Limitations of the study 54 5. Conclusion 56 References 57 국문 초록 70Docto

Efficient 11-Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences

We present efficient algorithms for approximately solving systems of linear equations in $1$-Laplacians of well-shaped simplicial complexes up to high precision. $1$-Laplacians, or higher-dimensional Laplacians, generalize graph Laplacians to higher-dimensional simplicial complexes and play a key role in computational topology and topological data analysis. Previously, nearly-linear time approximate solvers were developed for simplicial complexes with known collapsing sequences and bounded Betti numbers, such as those triangulating a three-ball in $\mathbb{R}^3$ (Cohen, Fasy, Miller, Nayyeri, Peng, and Walkington [SODA'2014], Black, Maxwell, Nayyeri, and Winkelman [SODA'2022], Black and Nayyeri [ICALP'2022]). Furthermore, Nested Dissection provides quadratic time exact solvers for more general systems with nonzero structures representing well-shaped simplicial complexes embedded in $\mathbb{R}^3$. We generalize the specialized solvers for $1$-Laplacians to simplicial complexes with additional geometric structures but without collapsing sequences and bounded Betti numbers, and we improve the runtime of Nested Dissection. We focus on simplicial complexes that meet two conditions: (1) each individual simplex has a bounded aspect ratio, and (2) they can be divided into "disjoint" and balanced regions with well-shaped interiors and boundaries. Our solvers draw inspiration from the Incomplete Nested Dissection for stiffness matrices of well-shaped trusses (Kyng, Peng, Schwieterman, and Zhang [STOC'2018]).Comment: 45 pages, 3 figures, ESA 202

A COLLISION AVOIDANCE SYSTEM FOR AUTONOMOUS UNDERWATER VEHICLES

The work in this thesis is concerned with the development of a novel and practical collision avoidance system for autonomous underwater vehicles (AUVs). Synergistically, advanced stochastic motion planning methods, dynamics quantisation approaches, multivariable tracking controller designs, sonar data processing and workspace representation, are combined to enhance significantly the survivability of modern AUVs. The recent proliferation of autonomous AUV deployments for various missions such as seafloor surveying, scientific data gathering and mine hunting has demanded a substantial increase in vehicle autonomy. One matching requirement of such missions is to allow all the AUV to navigate safely in a dynamic and unstructured environment. Therefore, it is vital that a robust and effective collision avoidance system should be forthcoming in order to preserve the structural integrity of the vehicle whilst simultaneously increasing its autonomy. This thesis not only provides a holistic framework but also an arsenal of computational techniques in the design of a collision avoidance system for AUVs. The design of an obstacle avoidance system is first addressed. The core paradigm is the application of the Rapidly-exploring Random Tree (RRT) algorithm and the newly developed version for use as a motion planning tool. Later, this technique is merged with the Manoeuvre Automaton (MA) representation to address the inherent disadvantages of the RRT. A novel multi-node version which can also address time varying final state is suggested. Clearly, the reference trajectory generated by the aforementioned embedded planner must be tracked. Hence, the feasibility of employing the linear quadratic regulator (LQG) and the nonlinear kinematic based state-dependent Ricatti equation (SDRE) controller as trajectory trackers are explored. The obstacle detection module, which comprises of sonar processing and workspace representation submodules, is developed and tested on actual sonar data acquired in a sea-trial via a prototype forward looking sonar (AT500). The sonar processing techniques applied are fundamentally derived from the image processing perspective. Likewise, a novel occupancy grid using nonlinear function is proposed for the workspace representation of the AUV. Results are presented that demonstrate the ability of an AUV to navigate a complex environment. To the author's knowledge, it is the first time the above newly developed methodologies have been applied to an A UV collision avoidance system, and, therefore, it is considered that the work constitutes a contribution of knowledge in this area of work.J&S MARINE LT

Pattern Recognition

Pattern recognition is a very wide research field. It involves factors as diverse as sensors, feature extraction, pattern classification, decision fusion, applications and others. The signals processed are commonly one, two or three dimensional, the processing is done in real- time or takes hours and days, some systems look for one narrow object class, others search huge databases for entries with at least a small amount of similarity. No single person can claim expertise across the whole field, which develops rapidly, updates its paradigms and comprehends several philosophical approaches. This book reflects this diversity by presenting a selection of recent developments within the area of pattern recognition and related fields. It covers theoretical advances in classification and feature extraction as well as application-oriented works. Authors of these 25 works present and advocate recent achievements of their research related to the field of pattern recognition