LEAPS-MPS: Uncovering and Exploiting Multiscale Structures in Big Data Using Diffusion-Based Representation and Optimal Sampling

信息消费者需要有效的数据处理方法，科学家有兴趣了解数据的模式和结构，以解释各种现象发生的原因。新兴大数据分析研究的重点已转向替代数据结构，尤其是（组合）图和网络。尽管图形和网络需要数学上复杂的算法，但它们具有内置的链接，可以突出显示节点之间的重要关系，并使它们在应用程序中更易于解释。该项目引入了新的理论和算法来解决图数据挑战的各个方面，这些挑战对社会各个领域都有影响，包括医疗专业人员用于诊断和治疗疾病的成像工具，以及从描述大脑中基因、RNA 和蛋白质之间相互作用的监管网络到硬件和传感器技术的应用。所探索的方法扩展了谐波分析算法系列，这些算法产生了诸如用于商业应用中使用的图像的基于小波的工具和用于电报通信的采样定理等工具。该项目还涉及培训和指导本科生，特别是来自较少接触科学和技术职业的弱势群体的学生。其目的是激励他们为先进数学科学领域的职业生涯做好准备，并提高美国 STEM 劳动力的全球竞争力。其他活动包括通过公开讲座和协调外展活动促进 STEM 领域的公平和包容性。应用谐波分析工具将在这个为期两年的项目中发挥核心作用。主要研究目标包括：（i）提高图和网络上数据信号多尺度表示的质量，（ii）优化高维、非带限信号的采样策略，以及（iii）设计压缩算法以展示图数据集的潜力。该分析结合了框架理论、扩散几何和定向多尺度建模；将使用数值线性代数和计算调和分析来开发新颖的数值技术。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Consumers of information desire efficient methods for processing data, and scientists are interested in understanding patterns and structures in data to explain why various phenomena occur. The focus of emerging big data analytics research has shifted to alternative data structures, especially (combinatorial) graphs, and networks. Although graphs and networks require mathematically complex algorithms, they have built-in links that highlight important relationships between nodes and make them simpler to interpret in applications. This project introduces new theories and algorithms to address aspects of graph data challenges that have impacts on diverse segments of society including imaging tools used by medical professionals to diagnose and treat diseases, and on applications ranging from the regulatory networks describing the interactions between genes, RNA, and proteins in the brain, to hardware and sensor technology. The methods explored expand the family of harmonic analysis algorithms that produced tools such as the wavelet-based ones for images used in commercial applications and sampling theorems for telegraph communication. The project also involves training and mentoring undergraduate students, particularly students from underrepresented groups with less exposure to careers in science and technology. The intent is to motivate and prepare them for careers in advanced mathematical sciences and increase the global competitiveness of the US-based STEM workforce. Other activities include promoting equity and inclusion in STEM fields through public lectures and coordinated outreach activities. Applied harmonic analysis tools will play a central role in the two-year project. The primary research objectives include: (i) enhancing the quality of multiscale representations of data signals on graphs and networks, (ii) the optimization of sampling strategies for high-dimensional, non-bandlimited signals, and (iii) the design of compression algorithms to demonstrate the potential for datasets on graphs. The analysis combines frame theory, diffusion geometry, and directional multiscale modeling; novel numerical techniques will be developed using numerical linear algebra and computational harmonic analysis.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.