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

小波理论的数学发展和所提出的计算算法在一维上已经达到了成熟的、令人满意的水平。而在高维情况下，情况就不那么令人满意了，事实上，现有的构造高维小波表示的方法在维数上的伸缩性很差。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高维的简单方法最终会变得非常非局部。因此，在高维度上构建有效的、高效的、波长表示仍然是一个重大的挑战和难以捉摸的目标。 这一建议的前提是，迎接这一挑战的唯一途径是从根本上改变小波构造的原则。这个项目的宏伟目标是开发与空间维度正确缩放的表示：恒定的独立于维度的算法的复杂性估计;在其支持下具有有限，受控，重叠的线性泛函;以及不会随着维度的增长而降低的性能等级。该项目预计将有助于以切实的方式对NSF的广泛标准。这首先是由于该研究领域的内在重要性，并且该项目中攻击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专业的学生提供了一个非常有价值的培训和教育机会，即在数学与科学技术之间的界面上进行教育。我们的时代标志着传感器采集能力的惊人进步以及有线和无线信道通信的爆炸性增长。由于这些和类似的趋势，大规模数据集的正确处理是，这些天来，几乎所有处理科学数据的技术的核心。 在这方面，最根本的问题是数据集的大小仅仅是采集技术的一个产物;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一种新的格式来解决上述问题，该格式允许高效和有效地提取信息、存储、传输等。 数据表示研究的重要性怎么强调都不为过：开发新的数据表示方法是我们国家最优先考虑的科学问题之一。事实上，这一领域的创新者因其贡献而得到广泛认可：即使将注意力仅限于数据表示的数学和统计方面的研究，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选举。 小波变换是数据表示界对科学最重要的贡献之一。它的引入是为了回答傅立叶表示的主要缺点，即，事实上，后者从来没有提供一个稀疏表示瞬态事件。 研究团体成功地在1D中构建了非常好的waveletrepresentation。 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。 武装与观察，这个雄心勃勃的项目的目的是开发新颖的类小波表示，是高效和有效的高维。

+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.