CIF: Small: Learning Low-Dimensional Representations with Heteroscedastic Data Sources

随着数据收集工作的持续增长，数据的异质性也在不断增长。机器学习方法通常假设数据来自单一来源或具有对每个数据点相同的噪声特征的统一仪器。这个项目将解决学习具有异方差数据的低维数据表示的基本问题，其中来自不同来源的样本具有不同方差的加性噪声。众所周知，经典的线性降维方法，如主成分分析(PCA)，对离群点很敏感，因此高方差噪声会降低PCA学习的表示。然而，如果数据确实有一些信号，那么简单地拒绝异常值的稳健方法是次优的，即使它被噪声淹没了。因此，这个项目的前提是使用学习最佳方式的方法，以纳入每个不同数据源的贡献，无论质量如何高或低，以改进总体学习的表示。许多应用将从这项工作中受益，包括医学成像、环境监测、天文数据分析、计算机视觉和生物信息学。研究人员在这一领域的先前工作表明，当学习受到异质和异方差来源的驱动时--例如，在医学成像中，使用来自多台扫描仪的数据，或者具有不同的辐射水平--通过积极考虑异质性并对其进行建模，将学习到更好的模型。如何在这种异质性面前优化学习，到目前为止还没有人研究过，这项研究的目的是填补这一空白。技术贡献将在三个方向上。首先，研究团队将研究有关数据异质性如何影响主成分分析的开放问题，包括建立学习异方差模型所需的样本复杂性，并评估异方差主元分析问题的优化前景。其次，该团队将扩展异方差主成分分析方法和理论，以考虑子空间并集模型、词典学习模型和转换学习模型。第三，研究人员将考虑数据中的异方差对非线性低维嵌入方法的影响。这项工作将集中在基于远程的方法上，并发展对在具有不同数据源的机器学习中使用距离的基础性理解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

As data-collection efforts continue to grow, so does heterogeneity in data. Machine-learning methods typically assume that data come from a single source or uniform instrumentation with noise characteristics that are the same for every data point. This project will address questions fundamental to learning low-dimensional data representations with heteroscedastic data, wherein samples from different sources have additive noise of different variances. It is well-known that classical linear dimensionality-reduction methods such as principal component analysis (PCA) are sensitive to outliers, so high-variance noise will degrade representations learned by PCA. However, robust methods that simply reject outliers are suboptimal if, indeed, the data do have some signal, even if it is buried in noise. The premise of this project therefore is to use approaches that learn the best way to incorporate the contribution of every different data source, no matter how high- or low-quality, to improve the overall learned representation. Many applications will benefit from the work, including medical imaging, environmental monitoring, astronomical data analysis, computer vision, and bioinformatics. The investigators' prior work in this area indicates that when learning is driven by heterogeneous and heteroscedastic sources – for example, in medical imaging, using data from multiple scanners, or with varying radiation levels – a better model will be learned by actively considering and modeling the heterogeneity. How to optimize learning in the face of such heterogeneity has been so far relatively unstudied, and this research aims to fill that gap. The technical contributions will be in three directions. First, the team of researchers will study open questions regarding how heterogeneity in data affects PCA, including establishing the required sample complexity for learning heteroscedastic models and assessing the optimization landscape of heteroscedastic PCA problems. Second, the team will extend heteroscedastic PCA methods and theory to consider union-of-subspaces models, dictionary learning models, and transform learning models. Third, the investigators will consider how nonlinear low-dimensional embedding methods are affected by heteroscedasticity in the data. The work will focus on distance-based methods and develop a foundational understanding of using distances in machine learning with heterogeneous data sources.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.