CRII: CIF: RUI: Exploiting Geometry in Robust Signal Processing and Feature Extraction

在任何量化的努力中，都必须进行测量。人们很早就知道，根据来源的不同，这些测量结果可能具有不同程度的可靠性。通常，这些不准确的测量形成了“噪声”的概念。每秒从数百万个不同的来源收集数十亿个数据观测，产生了无数的问题；该项目旨在开发新的方法来识别和抵消噪声以及由此导致的数值和统计工具的不稳定性。这种从数据观测中识别和提取系统异常值的方法在网络安全、网络和隐私的应用中是有用的。项目通过几个现有的问题框架来激发这些技术；几十年来，这些方法在实践和理论上都取得了成功，但在存在系统性异常值时仍然存在缺陷。在这个项目中开发的工具几乎可以被任何数据科学或机器学习的实践者使用。这些想法，特别是处理异常值检测和拒绝，也与通信，医学，地质学和社会选择有关。最近的研究表明，这种非常嘈杂的模型可以通过采用数据的几何角度的技术来处理，并施加在现实世界中通常无法实现的结构约束，但保留了数据的足够特征，以便可以应用数据处理和统计分析的经典概念。在本项目中，将开发新的实用算法，用于使用浮体作为数据分散度量的特征提取。这种几何结构已被证明可以捕获足够的数据集结构，从而仍然可以准确地提取重要的结构特征，同时允许大量数据损坏。该工具使经典算法（如独立成分分析（ICA）和主成分分析（PCA））能够处理具有重尾噪声的数据，否则这些数据将导致数值不稳定。此外，该项目将使用广义中心极限定理（GCLT）来确定更一般的假设，这些假设可以让人们利用强大的统计工具和优化技术。这种方法的灵活性在于，当底层模型本身不完全符合鲁棒估计例程的参数时，可以使用GCLT来证明数值算法的收敛性。最后，该项目将涉及进一步应用ICA作为数据分析算法约简的工具。它已经被证明能够实现有效的学习算法，并证明一些几何学习问题（例如学习半空间交叉点）可以通过统计技术来解决。这很好地补充了上述研究方向，其目的是通过主要使用几何工具来打破统计问题中的算法障碍。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In any quantitative endeavor, one must make measurements. One learns early and often that these measurements can have, depending on the source, varying degrees of reliability. Typically, these inaccurate measurements form a notion of "noise". Gathering billions of data observations per second, from millions of different sources, creates a myriad of problems; this project seeks to develop new ways to identify and counteract noise and the resulting instability in numerical and statistical tools. Such identification and extraction of systematic outliers from the data observations can be useful in applications of cyber-security, networking, and privacy. The project motivates these techniques through the lens of several existing problem frameworks; these have been successful in practice and in theory for decades, but still have shortcomings in the presence of systematic outliers. The tools developed during this project could be employed by virtually any practitioner of data science or machine learning. The ideas, specifically dealing with outlier detection and rejection, also have relevance to communication, medicine, geology, and social choice.It has been shown recently that such very noisy models can be handled by techniques that take a geometric perspective of the data, and impose structural constraints that are not typically realized in the real-world, but preserve enough features of the data so that classical notions of data processing and statistical analysis may be applied. In this project, new practical algorithms will be developed for feature extraction that use the floating body as a measure of data dispersion. This geometric structure has been proven to capture enough structure of a data set that one can still accurately extract important structural features, while allowing for significant portions of the data to be corrupt. This tool enables classical algorithms such as Independent Component Analysis (ICA) and Principal Component Analysis (PCA) to operate on data that has heavy-tailed noise, which would otherwise cause numerical instability. Further, the project will use the Generalized Central Limit Theorem (GCLT) to identify more general assumptions that can allow one to take advantage of robust statistical tools and optimization techniques. The flexibility of this is that one can use the GCLT to prove convergence of numerical algorithms when the underlying model itself does not exactly fit the parameters of robust estimation routines. Finally, the project will involve furthering applications of using ICA as a tool in algorithmic reductions for data analysis. It has already been shown to enable efficient learning algorithms, and demonstrating that some geometric learning problems (e.g. learning halfspace intersections) can be tackled by statistical techniques. This nicely complements the above research directions, which aim to break algorithmic barriers in statistical problems by using primarily geometric tools.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.