Collaborative Research: Data-driven Path Metrics for Machine Learning

大数据时代带来了前所未有的计算和数学挑战。传统的机器学习算法往往缺乏可扩展的计算复杂性，而现代方法缺乏坚实的数学基础。此外，高数据维度给传统的数据分析方法带来了挑战。主要研究者（PI）建议将联合收割机经典的降维方法与数据驱动的距离相结合，使得距离和嵌入过程都是数据相关的。这种新颖的方法允许更大的灵活性，在平衡数据的密度为基础的几何特征，实现了基于密度的几何简化，并有见地地表示在一个小的维度数的数据。与深度学习等黑箱方法相比，所开发的方法可以进行严格的分析，为几个统计和机器学习任务提供强有力的理论保证。这项研究将为癌症免疫基因组学提供计算工具，研究人员将与密歇根大学Rogel癌症中心就肿瘤免疫学和T细胞生物学相关的科学问题进行咨询。此外，将以开放源码软件包的形式向公众提供新的数据分析工具。研究人员的方法是由一系列数据依赖路径度量的分析驱动的。这些指标既是密度敏感的，又是几何保持的，其平衡取决于单个参数p的选择。通过利用数据路径的空间，PI将获得基于密度的指标和嵌入，同时避免密度估计器的显式计算，这可能是不可靠的在很多维度上。PI将提出一个简单但高度灵活的数据模型，该模型不假设数据是从流形或流形集合中采样的，并研究这些度量的连续极限和相关的图形拉普拉斯算子。通过不断改变参数p，PI将建议创建从多个角度表示数据的数据视频。PI将研究多维缩放和图形拉普拉斯嵌入作为获得基于路径的低维表示的机制，并将探索具有可扩展计算复杂性的快速算法来近似这些度量。PI将在数据驱动指标的更大框架中将路径指标置于情境中，并特别关注生物数据的分析。该奖项反映了NSF的法定使命，并被认为值得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来支持。

The era of big data has introduced unprecedented computational and mathematical challenges. Traditional machine learning algorithms often lack scalable computational complexity, while modern approaches lack solid mathematical foundations. Moreover, high data dimensionality creates challenges for traditional methods of data analysis. The principal investigators (PIs) propose to combine classic dimension reduction methods with data-driven distances, so that both the distance and embedding procedure are data dependent. This novel approach allows for greater flexibility in balancing the density-based and geometric features of the data, achieves a density-based simplification of geometry, and insightfully represents the data in a small number of dimensions. In contrast to black box methods such as deep learning, the developed methodology can be rigorously analyzed to derive strong theoretical guarantees for several statistical and machine learning tasks. This research will contribute computational tools for cancer immunogenomics and the investigators will consult with the Rogel Cancer Center at the University of Michigan for scientific questions related to tumor immunology and T-cell biology. In addition, new data analysis tools will be made publicly available in an open source software package. The investigators' approach is driven by the analysis of a family of data-dependent path metrics. These metrics are both density-sensitive and geometry-preserving, with the balance governed by the choice of a single parameter p. By utilizing the space of paths through data, the PIs will obtain density based metrics and embeddings while avoiding the explicit computation of a density estimator, which may be unreliable in a large number of dimensions. The PIs will propose a simple yet highly flexible data model which does not assume the data is sampled from a manifold or collection of manifolds, and investigate the continuous limit of these metrics and an associated graph Laplacian operator. By continuously varying the parameter p, the PIs will propose to create data videos which represent the data from multiple perspectives. The PIs will investigate both multidimensional scaling and graph Laplacian embeddings as mechanisms for obtaining path-based low dimensional representations, and will explore fast algorithms with scalable computational complexity for approximating these metrics. The PIs will contextualize path metrics in the larger frame work of data-driven metrics and focus specifically on the analysis of biological data.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.