Spectral embedding and spectral clustering are common methods for non-linear dimensionality reduction and clustering of complex high dimensional datasets. In this paper we provide a diffusion based probabilistic analy...
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(纸本)9783540737490
Spectral embedding and spectral clustering are common methods for non-linear dimensionality reduction and clustering of complex high dimensional datasets. In this paper we provide a diffusion based probabilistic analysis of algorithms that use the normalized graph Laplacian. Given the pairwise adjacency matrix of all points in a dataset, we define a random walk on the graph of points and a diffusion distance between any two points. We show that the diffusion distance is equal to the Euclidean distance in the embedded space with all eigenvectors of the normalized graph Laplacian. This identity shows that characteristic relaxation times and processes of the random walk on the graph are the key concept that governs the properties of these spectral clustering and spectral embedding algorithms. Specifically, for spectral clustering to succeed, a necessary condition is that the mean exit times from each cluster need to be significantly larger than the largest (slowest) of all relaxation times inside all of the individual clusters. For complex, multiscale data, this condition may not hold and multiscale methods need to be developed to handle such situations.
Much of the community detection literature studies structural communities, communities defined solely by the connectivity patterns of the network. Often networks contain additional metadata which can inform community ...
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Much of the community detection literature studies structural communities, communities defined solely by the connectivity patterns of the network. Often networks contain additional metadata which can inform community detection such as the grade and gender of students in a high school social network. In this work, we introduce a tuning parameter to the content map equation that allows users of the Infomap community detection algorithm to control the metadata's relative importance for identifying network structure. On synthetic networks, we show that our algorithm can overcome the structural detectability limit when the metadata are well aligned with community structure. On real-world networks, we show how our algorithm can achieve greater mutual information with the metadata at a cost in the traditional map equation. Our tuning parameter, like the focusing knob of a microscope, allows users to “zoom in” and “zoom out” on communities with varying levels of focus on the metadata.
In this paper we present a result which modifies Moreau's theorem [1] on linear consensus problem over time-varying directed networks. Our result uses integral connectivity on varying-length intervals to character...
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In the quest for understanding problems of supermaneuverability for combat aircraft, a study is presented about the role the dihedral-effect can have in fuselage-reorientation maneuvers that involve high angles-of-att...
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In some cases time-optimal aircraft roll-maneuvers exhibit a nonintuitive feature, in which the extremal rollcontrol causes the vehicle to initially roll the "wrongway". We present an explanation of this, an...
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In some cases time-optimal aircraft roll-maneuvers exhibit a nonintuitive feature, in which the external roll control causes the vehicle to initially roll the "wrong way". We present an explanation of this, ...
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Sensitivity based methods for shape optimization of fluid flow governed by the Navier-Stokes equations are considered. The crucial components of an algorithm for this task are the proper parametrization of the boundar...
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Sensitivity based methods for shape optimization of fluid flow governed by the Navier-Stokes equations are considered. The crucial components of an algorithm for this task are the proper parametrization of the boundary, the efficient calculation of the sensitivities with respect to the boundary parameters, the selection of a good optimization algorithm, and an efficient scheme for state calculations. Some of these issues are discussed, together with pitfalls and potential solutions.< >
The problem of interest is how heading reversal maneuvers should be performed by combat aircraft in a time-optimal manner. The focus of this study is the relationship between the singular points and other structural p...
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This paper reviews Frechet sensitivity analysis for partial differential equations with variations in distributed parameters. The Frechet derivative provides a linear map between parametric variations and the lineariz...
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This paper reviews Frechet sensitivity analysis for partial differential equations with variations in distributed parameters. The Frechet derivative provides a linear map between parametric variations and the linearized response of the solution. We propose a methodology based on representations of the Frechet derivative operator to find those variations that lead to the largest changes to the solution (the most significant variations). This includes an algorithm for computing these variations that only requires the action of the Frechet operator on a given direction (the Gateaux derivative) and its adjoint. This algorithm is applicable since it does not require an approximation of the entire Frechet operator, but only typical sensitivity analysis software for partial differential equations. The proposed methodology can be utilized to find worst case distributed disturbances and is thus applicable to uncertainty quantification and the optimal placement of sensors and actuators.
The classic Hegselmann-Krause (HK) model for opinion dynamics consists of a set of agents on the real line, each one instructed to move, at every time step, to the mass center of the agents within a fixed distance R. ...
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