Imputing single cell RNA sequencing data: Mathematical, statistical and computational challenges

新的单细胞RNA测序（scRNA-seq）技术可以同时测量所有基因的表达水平。 3万个基因分布在数千到数百万个细胞中。scRNA-seq数据的分析已经导致了 生物学的基本进展，包括发现新的细胞类型，检测 相似的细胞，以及细胞发育轨迹的重建。单细胞测量包括 微量RNA的扩增并导致具有许多零的极其稀疏的数据矩阵，而一些 这些零是由于数据缺失（辍学），其他零代表真正的生物学活性。然而，许多scRNA-seq 插补方法对所有观察到的零条目进行相同处理，导致插补矩阵经常高估 转录活性其他试图区分生物零和辍学的方法缺乏严格的 理论保障。该提案的目标是开发模型，支持数学理论， 计算工具，明确考虑到真正的生物零点的存在。矩阵插补 这一约束既涉及计算挑战，也涉及随机矩阵理论中的理论问题， 高维统计这些方法包括秩估计和低秩稀疏矩阵的部分恢复 观察到的数据，并在存在辍学和零的双聚类，我们计划开发新的方法的基础上， 非光滑连续优化，并获得伴随的统计保证，我们还计划开发 集成学习方法巧妙地将多个插补算法的输出进行联合收割机组合。最后我们希望 通过对极大极小化率和信息较低的研究， 界限。为了应对这些挑战，我们将在我们富有希望的初步成果和 光谱方法，高维统计，矩阵分析，数值优化和基因组学的研究者。

Novel single cell RNA sequencing (scRNA-seq) technologies can simultaneously measure the expression levels of all 30,000 genes over thousands to millions of individual cells. The analysis of scRNA-seq data has already led to fundamental advances in biology, including discovery of new cell types, detection of subtle differences between similar cells, and reconstruction of cellular developmental trajectories. Single- cell measurements involve amplification of tiny amounts of RNA and result in extremely sparse data matrices with many zeros, While some of these zeros are due to missing data (dropouts), others represent true biological inactivity. Yet, many scRNA-seq imputation methods treat all observed zero entries identically, leading to imputed matrices that often overestimate transcriptional activity. Other methods that do attempt to distinguish biological zeros from dropouts lack rigorous theoretical guarantees. The goals of this proposal are to develop models, supporting mathematical theory, and computational tools that explicitly take the existence of true biological zeros into account. Matrix imputation under this constraint involves both computational challenges as well as theoretical questions in random matrix theory and high dimensional statistics. These include rank estimation and low rank sparse matrix recovery from partially observed data, and biclustering in the presence of dropouts and zeros, We plan to develop novel approaches based on non-smooth continuous optimization, and derive accompanying statistical guarantees, We also plan to develop ensemble learning approaches that cleverly combine the outputs of multiple imputation algorithms. Finally, we hope to gain important insights regarding recovery from such data via a study of minimax rates and information lower bounds. To address these challenges, we will build on our promising preliminary results and the joint expertise of the investigators in spectral methods, high dimensional statistics, matrix analysis, numerical optimization, and genomics.