CAREER: Nonlinear Factor Analysis for Sensing and Learning

因子分析（FA）工具，如非负矩阵分解（NMF）和独立成分分析（ICA），是许多传感和学习应用的基石，如文档分析、高光谱成像、大脑信号处理和表征学习。FA工具旨在以无监督的方式从数据中发现有意义的潜在信息（例如，文档集合中的突出主题）。然而，经典的FA模型没有考虑数据采集/生成过程中经常发生的未知非线性扭曲，因此在关键场景下往往不能产生合理的结果。该项目将开发一套非线性因素分析（NFA）工具，通过有效和可证明地处理未知的非线性来改变现有的非线性因素分析范式。该项目的结果将显著促进对各种NFA模型的基本特性和计算方面的理解，包括模型可识别性、样本复杂性、噪声鲁棒性和算法收敛性——这些在很大程度上是未知的研究领域。该产品将提高科学和工程中经常出现未知非线性扭曲的广泛传感和学习任务的性能，例如遥感、脑机接口、视觉/图像/文本数据分析、生物信息学、地球科学、生物学和生态学。开发具有视觉吸引力的FA和基于nfa的课程模块和软件的综合教育计划将缓解K-12和大学的“数学焦虑”。大学前拓展计划和本科生研究计划将有效地培养早期对数学的兴趣，并提高代表性不足的学生对STEM学科的参与。这些教育活动将为信号和机器智能领域带来多元化和数学上具有竞争力的未来劳动力。该项目将开发一个统一的分析和计算框架，用于学习各种具有挑战性和现实的NFA模型。具体来说，Thrust I将开发一个统一的基于函数方程的框架，用于各种NFA设置下可证明的无监督非线性模型识别。Thrust II将在理解现实条件下（例如，有限样本和噪声情况）的NFA方面取得重要进展，并将提供具有性能保证的有效NFA优化算法。推力III将仔细评估及时和重要的传感和学习任务所提出的方法，包括高光谱成像、生物传感器信号处理和无监督机器学习。这些重点将在解决NFA关键挑战的理论和算法方面产生基本结果。新的基于函数方程的分析框架为现有工具无法解决的各种NFA模型识别问题提供了理论基础。新的NFA性能表征工具在现实环境下（例如，有限数据）将是现有工作的重大飞跃，所有这些工作都使用过于理想的假设（例如，无限数据）。通过统计分析，神经网络学习和非线性规划的集成计算框架将为NFA问题提供可证明和灵活的算法，这些问题目前都缺乏保证的解决方案。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。

Factor analysis (FA) tools, e.g., nonnegative matrix factorization (NMF) and independent component analysis (ICA), are the cornerstones of many sensing and learning applications, e.g., document analytics, hyperspectral imaging, brain signal processing, and representation learning. FA tools are designed to discover meaningful latent information from data (e.g., prominent topics in a collection of documents) in an unsupervised manner. However, classic FA models do not consider unknown nonlinear distortions that often happen in data acquisition/generation, and thus frequently fail to produce sensible results in critical scenarios. This project will develop a suite of nonlinear factor analysis (NFA) tools that will transform existing FA paradigms by effectively and provably handling unknown nonlinearities. Results from this project will significantly advance the understanding of fundamental properties and computational aspects of various NFA models, including model identifiability, sample complexity, noise robustness and algorithm convergence---which are largely uncharted research territories. The products will boost the performance of a broad spectrum of sensing and learning tasks in science and engineering where unknown nonlinear distortions often arise, e.g., remote sensing, brain-computer interface, vision/image/text data analytics, bioinformatics, geoscience, biology, and ecology. The integrated education plan of developing visually appealing FA and NFA-based course modules and software will alleviate “math anxiety” in K-12 and college. The precollege outreach programs and undergraduate research plans will effectively foster early interest in mathematics and enhance underrepresented students’ participation in STEM disciplines. These education activities will lead to a diversified and mathematically competitive future workforce for signal and machine intelligence.This project will develop a unified analytical and computational framework for learning various challenging and realistic NFA models. Specifically, Thrust I will develop a unified functional equation-based framework for provable unsupervised nonlinear model identification under various NFA settings. Thrust II will make important advances towards understanding NFA under realistic conditions (e.g., finite sample and noisy cases), and will offer effective NFA optimization algorithms with performance guarantees. Thrust III will carefully evaluate the proposed approaches over timely and important sensing and learning tasks including hyperspectral imaging, biosensor signal processing, and unsupervised machine learning. These thrusts will produce fundamental results in both theory and algorithms addressing critical challenges in NFA. The new functional equation-based analytical framework offers a theoretical underpinning for various NFA model identification problems that are beyond the reach of existing tools. The new NFA performance characterization tools under realistic settings (e.g., finite data) will be a substantial leap forward from existing works that all use overly ideal assumptions (e.g., unlimited data). The computational framework through an integration of statistical analysis, neural network learning, and nonlinear programming will offer provable and flexible algorithms for NFA problems, which all currently lack guaranteed solutions.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.