Theory and Methods for Large-Scale Multi-Modal Matrix Data

现代数据采集技术产生了携带丰富信息的新型数据，但也给分析带来了新的挑战。在许多现代数据集中，测量的基本单位可以是记录一组或多组个体之间的相互作用的矩阵甚至更高阶数组。例如，基因共表达网络测量特定器官组织中每对基因之间的平均关联强度。有了在不同发育阶段收集的基因共表达网络，就有可能理解一组基因如何以一致的方式改变它们的行为。作为另一个例子，下一代测序技术能够产生不同尺度的基因表达数据：组织样本数据包括批量组织样本中的基因表达，而单细胞RNA测序数据包含单个细胞的相同基因的表达。在这些例子的启发下，本研究工作旨在为复杂矩阵值数据集开发新的概率工具和统计推理方法，使科学家能够以连贯和高效的方式发现这类数据集中的显著结构。该项目还为研究生提供了研究培训机会。该项目由两部分组成。在第一部分中，PI研究了具有跨层共享潜在结构的多层网络，并开发了有效地组合跨不同层的信息以恢复潜在结构的方法，这在仅有单层可用的情况下是不可能的。预期的结果将提供以矩阵形式描述随机噪声行为的新的概率定理，以及它们的线性组合和高阶函数。在第二部分中，PI研究了与组织和单细胞RNA-SEQ数据相关的一系列推理问题，从计算高效的降维和变量选择入手，然后是组织RNA-SEQ数据中的细胞类型去卷积等下游推理问题。通过开发一种具有可证明的全局收敛性质的无投影、基于梯度的算法，预期的结果将为稀疏主成分分析文献提供重要的补充。细胞类型的反卷积问题将是一个有趣的应用，结合了变量选择、非负矩阵分解和优化的技术。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Modern data acquisition technology produces new types of data that carry rich information but also poses new challenges for analysis. In many modern datasets, the basic unit of measurement can be a matrix or even higher order array recording the interactions among one or multiple groups of individuals. For example, a gene co-expression network measures the average strength of correlation between each pair of genes in a particular organ tissue. With gene co-expression networks collected at different developmental stages, it is possible to understand how groups of genes change their behavior in a coherent way. As another example, next generation sequencing techniques are able to produce gene expression data at different scales: Tissue sample data consists of gene expressions in bulk tissue samples, whereas single cell RNA sequencing data contains expressions of the same genes for individual cells. Motivated by the these examples, this research work aims at developing novel probability tools and statistical inference methods for complex matrix valued datasets, which will enable scientists to uncover salient structures in such datasets in a coherent and efficient way. The project also provides research training opportunities for graduate students. This project consists of two parts. In the first part, the PI studies multiple layer networks with a shared latent structure across layers and develops methods to efficiently combine the information across different layers to recover the latent structure, which would be impossible if only a single layer were available. The expected results will provide new probability theorems describing the behavior of random noises in matrix forms, as well as their linear combinations and higher order functions. In the second part, the PI studies a series of inference problems related to tissue and single cell RNA-seq data, starting from dimensionality reduction and variable selection in a computationally efficient manner, followed by downstream inference problems such as cell type deconvolution in tissue RNA-seq data. The expected results will provide an important addition to the sparse principal components analysis literature, by developing a projection-free, gradient-based algorithm with provable global convergence properties. The cell type deconvolution problem will be an interesting application combining techniques from variable selection, nonnegative matrix factorization, and optimization.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.