Diffusion Multiscale Analysis

在这个建议中，研究者和他的合作者解决了几个问题，这些问题来自于集合的多尺度几何的数学分析，以及函数空间的多尺度分解，这些问题是由流形、图和其他相当一般的度量空间上的扩散半群的作用引起的。虽然这些多尺度几何在微分几何、偏微分方程以及图论的许多分支（应用于计算机科学中的问题）中是部分隐式的（和经典的），但直到最近，研究者和他的合作者才引入了一种非常普遍、有效、连贯和统一的结构。通过引入特殊的小波函数，还构建了反映这些多尺度扩散几何的多尺度函数空间分解。这是小波分析在数学上和计算上的一个意义深远的、长期寻求的推广。研究者和他的合作者已经证明了有效计算这些多尺度分解的算法是存在的，它推广了快速小波变换和快速多极方法，产生了保证高精度的快速多尺度算法。研究者将研究双正交扩散多尺度分解的构造、粗糙集上的多尺度函数逼近、数据集的多尺度扩散分析及其与几何测度理论、多尺度马尔可夫链、偏微分方程数值分析、学习理论、高光谱成像和文档语料库分析的关系。研究者期望这种新颖的多尺度结构对所有这些学科产生影响，就像小波分析对低维信号处理和数值分析的影响一样。本提案强调了多尺度分析几个方面的跨学科性质，以及这些思想、工具、结构在纯数学和应用数学以及计算机科学、物理学、工程学、天文学和统计学等其他学科中的广泛适用性。这些新颖的多尺度技术的引入揭示了图和集新的和有趣的多尺度几何结构，以及有效的计算工具来发现它们。应用范围非常广泛，包括分析和组织大型复杂网络（例如计算机网络，生物调节网络等），用于信息提取的文档语料库，高光谱图像（用于医学应用，目标识别等）以及一般的大型数据集。它还应用于开发用于学习和人工智能的新算法，用于复杂任务的自动化。研究者的目标是加强他现有的合作，并建立新的合作，与其他机构，无论是在美国和国外，跨几个学科，特别是计算机科学，天文学，生物学和医学。他将继续与开发下一代仪器的公司合作，用于高光谱成像。他将继续积极参加多学科和跨学科的会议、研讨会和研究活动，并向多学科的听众有效地沟通和传播思想和技术，使他的工作，包括论文和相应算法的计算机代码，更容易通过电子方式获取。

In this proposal, the investigator and his collaborators address several questions arising from the mathematical analysis of multiscale geometries of sets, and multiscale decomposition of function spaces, that arise from the action of a diffusion semigroup on a manifold, a graph and other rather general metric spaces. While these multiscale geometries are partly implicit (and classical) in differential geometry, in partial differential equations, as well as in many branches in graph theory (with applications to problems in computer science), only recently a very general, yet efficient, coherent and unifying construction has been introduced by the investigator and his collaborators. Multiscale function space decompositions that mirror these multiscale diffusion geometries are also constructed, through the introduction of special wavelet functions. This is a far-reaching, and long sought, generalization of wavelet analysis, both mathematically and computationally. The investigator and his collaborators have shown that algorithms for efficiently computing these multiscale decompositions exist, which generalize the fast wavelet transform and Fast Multipole Methods, yielding fast multiscale algorithms guaranteeing high-precision. The investigator will study the construction of biorthogonal diffusion multiscale decompositions, multiscale function approximation on rough sets, multiscale diffusion analysis of data sets and its relationships with geometric measure theory, multiscale Markov chains, numerical analysis of PDEs, learning theory, hyperspectral imaging and document corpora analysis.The investigator expects this novel multiscale construction to have impact in all these disciplines, in a way similar to the impact wavelet analysis had on low-dimensional signal processing and numerical analysis.The present proposal stresses the inter-disciplinary nature of several aspects of multiscale analysis, and the vast applicability of the ideas, tools, constructions, to pure and applied mathematics, and to other disciplines such as computer science, physics, engineering, astronomy and statistics, among others. The introduction of these novel multiscale techniques reveals new and interesting multiscale geometric structures of graphs and sets, together with effective computational tools to discover them. The range of applications is very wide, and includes the analysis and organization of large and complex networks (e.g. computer networks, biological regulatory networks etc...), document corpora for information extraction, hyperspectral imagery (for applications to medicine, target recognition etc...), and large datasets in general. It has also applications to the development of new algorithms for learning and artificial intelligence, for the automation of complex tasks. The investigator aims at strenghtening his existing collaborations, and establishing new ones, with other institutions, both in the United States and abroad, across several disciplines, in particular computer science, astronomy, biology, and medicine. He will continue his existing collaborations with companies developing next-generation instrumentation, for applications to hyperspectral imaging. He will continue to actively participate in multi- and inter-disciplinary conferences, workshops and research activities, and effectively communicating and disseminating ideas and techniques to multi-disciplinary audiences, making his work, including papers and computer code for the corresponding algorithms, easily accessible electronically.