Diffusion wavelets
Diffusion wavelets
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DOI：
10.1016/j.acha.2006.04.004
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发表时间：
2006-07-01
影响因子：
2.5
通讯作者：
Maggioni, Mauro
中科院分区：
文献类型：
--
作者：
Coifman, Ronald R.;Maggioni, Mauro
我们本文的目标是表明，许多信号处理，改编的傅立叶和小波分析的工具可以自然地提升为数字数据云，图形和歧管的设置。我们将扩散用作平滑和缩放工具，以实现粗晶片和多尺度分析。给定具有较高等级的较大功率的散射操作员t，我们提出了一个通用的多分辨率构造，用于有效计算，代表和压缩V。这允许直接的多尺度计算，以高精度，高精度，运算符的功能，特别是相关的绿色功能，以压缩形式及其快速应用。这些计算快速的运算符类别包括某些类似于扩散的运算符，在任何维度，图形，图形和非均匀介质中都包括任何维度。我们使用与快速多极方法以及与Calderon-Zygmund和伪差异运算符的小波分析相关的想法，以数值强制强制歧管，图形或数据集的天然分层粗网的出现。例如，对于一系列文本记录，构造导致在不同级别的概括下导致目录结构。像经典的Littlewood-Paley和小波理论一样，操作员的二元能力可用于诱导多解析分析：我们使用高效且稳定的算法，正顺式缩放函数的基础以及与此多分辨率分析相关的波浪构建的基础相应的缩放采样运算符，并使用它们来压缩操作员的相应功能。尽管我们的大多数讨论都涉及对称运算符，并且与光谱频段的本地化有关，但不必考虑操作员及其光谱理论的对称性，因为主要假设是我们占据操作员的权力时的数值级别。 （c）2006 Elsevier Inc.保留所有权利。
Our goal in this paper is to show that many of the tools of signal processing, adapted Fourier and wavelet analysis can be naturally lifted to the setting of digital data clouds, graphs, and manifolds. We use diffusion as a smoothing and scaling tool to enable coarse graining and multiscale analysis. Given a diffusion operator T on a manifold or a graph, with large powers of low rank, we present a general multiresolution construction for efficiently computing, representing and compressing V. This allows a direct multiscale computation, to high precision, of functions of the operator, notably the associated Green's function, in compressed form, and their fast application. Classes of operators for which these computations are fast include certain diffusion-like operators, in any dimension, on manifolds, graphs, and in non-homogeneous media. We use ideas related to the Fast Multipole Methods and to the wavelet analysis of Calderon-Zygmund and pseudo-differential operators, to numerically enforce the emergence of a natural hierarchical coarse graining of a manifold, graph or data set. For example for a body of text documents the construction leads to a directory structure at different levels of generalization. The dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory: we construct, with efficient and stable algorithms, bases of orthonormal scaling functions and wavelets associated to this multiresolution analysis, together with the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. While most of our discussion deals with symmetric operators and relates to localization to spectral bands, the symmetry of the operators and their spectral theory need not be considered, as the main assumption is reduction of the numerical ranks as we take powers of the operator. (C) 2006 Elsevier Inc. All rights reserved.