L-CAMP: Extremely Local High-Performance Wavelet Representations in High Spatial Dimension

L-CAMP：高空间维度中的极其局部高性能小波表示

• 批准号: 0602837
• 负责人: Amos Ron
• 金额: $ 32.21万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0602837&HistoricalAwards=false
• 关键词: CAMP; Extremely; Local; Performance; Wavelet

+The mathematical development of wavelet theory and the accompaniedcomputational algorithms reached a mature, satisfactory, level in onedimension. The situation is far less satisfactory in higher dimensions.As a matter of fact, the current approaches for the construction of high-Dwavelet representations scale poorly with the dimension. On the one hand,intrinsic constructions become hopelessly complicated already at relativelylow dimensions. On the other hand, the simple approach of liftingunivariate systems to higher dimensions becomes eventually immenselynon-local. As a result, the construction of effective, efficient, waveletrepresentations in high dimensions remains a major challenge and an elusivetarget. The premise of this proposal is that the only way to meet thischallenge is to fundamentally change the principles of waveletconstructions. The ambitious goal of this project is to developrepresentations that scale correctly with the spatial dimension: constantsin the complexity estimates of the algorithms that are independent of thedimension; linear functionals that have limited, controlled, overlapping intheir supports; and performance grade that does not degrade with the growthof the dimension. The project is expected to contribute in tangible ways toNSF's broad criteria. This is first and foremost due to the intrinsicimportance of the research area, and the fact that the problem attacked inthis project is a major hurdle in the relevant research area. In addition,the initiative offers a highly valuable training and education opportunityto mathematical science students, education in areas that are at theinterface between mathematics on the one hand and science and technology onthe other hand.Our era is marked by breathtaking improvement in sensor acquisitioncapabilities and an explosive increase in communication over wired andwireless channels. As a result of these and similar trends, the correcthandling of massive datasets is, these days, at the core of almost everytechnology that deals with scientific data. The most fundamental issue inthis regard is the fact that the size of the dataset is merely an artifactof the acquisition technology; it is not related to pertinent informationthat the data encode, nor to the actual applications that are sought for.Data representation is the scientific discipline that deals with challengeslike the above. It resolves the above problem by transforming the data intoa new format which allows an efficient and effective extraction ofinformation, storage, transmission, and the like. There is probably no wayto overstate the importance of research on data representation: Developmentof novel data representations is ranked among the top scientific prioritiesof our nation. Indeed, the innovators in this field are richly recognizedfor their contributions: even after restricting attention only to researchon the mathematical and statistical aspects of data representation, onefinds, in the last eight years alone, two Medal of Science awards, as wellas five or more elections to the National Academy of Science. The waveletrepresentation is among the most important contributions of the datarepresentation community to science. It was introduced in order to providean answer to the main shortcoming of the Fourier representation, i.e., thefact that the latter never provides a sparse representation to transientevents. The research community succeeded in constructing very good waveletrepresentations in 1D. Those systems are computed and inverted by fastalgorithms and strike as good balance as can be between performance andlocalness. The same it not true in high spatial dimensions. Armed withthat observation, the intent of this ambitious project is to develop novelclasses of wavelet representations that are efficient and effective in highdimensions.

+小波理论及其相关计算算法的数学发展在一个维度上达到了成熟、令人满意的水平。在更高维度上情况远不那么令人满意。事实上，当前构建高维小波表示的方法随维度的扩展性很差。一方面，内在结构在相对较低的维度上已经变得极其复杂。另一方面，将单变量系统提升到更高维度的简单方法最终会变得非常非局部。因此，构建有效、高效的高维小波表示仍然是一个重大挑战和难以捉摸的目标。 该提案的前提是，应对这一挑战的唯一方法是从根本上改变小波构造的原理。该项目的雄心勃勃的目标是开发与空间维度正确缩放的表示：独立于维度的算法复杂性估计中的常数；其支撑点具有有限、受控、重叠的线性泛函；且性能等级不随维度的增长而降低。该项目预计将以切实的方式为 NSF 的广泛标准做出贡献。这首先是由于该研究领域的内在重要性，并且该项目所解决的问题是相关研究领域的主要障碍。此外，该计划还为数学科学学生提供了非常有价值的培训和教育机会，一方面提供数学与科学技术交叉领域的教育。我们这个时代的标志是传感器采集能力的惊人提高以及有线和无线通道通信的爆炸性增长。由于这些和类似的趋势，如今，正确处理海量数据集几乎成为所有处理科学数据的技术的核心。 这方面最根本的问题是数据集的大小仅仅是采集技术的产物；它与数据编码的相关信息无关，也与所寻求的实际应用无关。数据表示是处理上述挑战的科学学科。 它通过将数据转换成新的格式来解决上述问题，这种新的格式允许高效且有效地提取信息、存储、传输等。 数据表示研究的重要性怎么强调都不为过：开发新颖的数据表示被列为我国科学的首要任务之一。事实上，这一领域的创新者的贡献得到了广泛认可：即使在将注意力仅限于数据表示的数学和统计方面的研究之后，人们发现，仅在过去八年中，就有两次科学奖章奖，以及五次或更多的国家科学院院士选举。 小波表示是数据表示界对科学最重要的贡献之一。引入它是为了解决傅立叶表示的主要缺点，即后者从不为瞬态事件提供稀疏表示。 研究界成功地构建了非常好的一维小波表示。 这些系统通过快速算法进行计算和反演，并在性能和局部性之间取得尽可能好的平衡。在高空间维度中情况并非如此。 有了这一观察结果，这个雄心勃勃的项目的目的是开发在高维上高效且有效的新型小波表示形式。