Collaborative Research: Data-driven Path Metrics for Machine Learning

大数据时代给计算和数学带来了前所未有的挑战。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。主要研究者提出将经典降维方法与数据驱动距离相结合，使距离和嵌入过程都依赖于数据。这种新颖的方法在平衡数据的基于密度和几何特征方面具有更大的灵活性，实现了基于密度的几何简化，并在少量维度中深刻地表示数据。与黑箱方法（如深度学习）相比，开发的方法可以进行严格的分析，从而为若干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，研究人员将向密歇根大学的Rogel癌症中心咨询有关肿瘤免疫学和t细胞生物学的科学问题。此外，新的数据分析工具将以开源软件包的形式公开提供。研究人员的方法是由一系列数据依赖的路径指标的分析驱动的。这些指标既对密度敏感，又保持几何不变，其平衡由单个参数p的选择来控制。通过利用数据路径空间，pi将获得基于密度的指标和嵌入，同时避免了密度估计器的显式计算，这在大量维度中可能是不可靠的。pi将提出一个简单而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，并研究这些指标的连续极限和相关的图拉普拉斯算子。通过不断改变参数p， pi将提出创建从多个角度表示数据的数据视频。pi将研究多维尺度和图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指标。pi将在数据驱动指标的更大框架中对路径指标进行上下文化，并特别关注生物数据的分析。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。

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.