CRII: CIF: Generalizations for Matrix and Tensor Estimation

矩阵和张量估计是数据科学和机器学习中处理缺失数据的核心构件。它们已被广泛应用于许多领域，包括社会计算、计算机视觉和计算生物学。该项目旨在推动这些技术的发展，以处理在真实世界数据集中看到的模型变化；特别是，数据收集很少是统一的，并且经常存在交互数据和协变量或特征信息的混合。例如，生物数据集可能包含单个基因的已知属性，以及有关基因如何相互作用的信息。交互作用数据是从真实实验中收集的，因此可能是高度非均匀分布的。该项目开发的技术可以在实验数据较少的情况下对该数据集进行更有效的预测。该项目的技术目标包括推广均匀抽样模型之外的矩阵和张量估计理论和算法，并设计结合辅助信息和矩阵交互数据的最优高效算法。该方法重点研究了基于相似度的协同过滤算法。对于这些模型的每一种变化，研究人员计划表征信息理论阈值和最小最大最优估计错误率，设计和分析计算和统计上有效的算法，并提供置信度集来量化估计的不确定性。这些结果将极大地增加矩阵和张量估计方法的灵活性，用于顺序决策和高维科学数据分析。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。

Matrix and tensor estimation are core building blocks in data science and machine learning for dealing with missing data. They have been used widely across many domains, including social computing, computer vision, and computational biology. This project seeks to advance these techniques to handle model variations that are seen in real world datasets; in particular, data collection is rarely uniform, and there is often a mix of interaction data and covariate or feature information. As an example, a biological dataset might contain known properties of individual genes, in addition to information about how genes interact. The interaction data is collected from real experiments and thus may be highly non-uniformly distributed. The techniques developed from this project could enable more efficient predictions over this dataset given less experimental data.The technical goals of this project involve generalizing matrix and tensor estimation theory and algorithms beyond uniform sampling models, and designing optimally efficient algorithms that incorporate side information together with matrix interaction data. The approach proposed focuses on similarity based collaborative filtering algorithms. For each of these model variations, the researchers plan to characterize information theoretic thresholds and minimax optimal estimation error rates, design and analyze computationally and statistically efficient algorithms, and provide confidence sets to quantify uncertainty of estimates. These results will greatly increase the flexibility of matrix and tensor estimation methods to be used for sequential decision making and high dimensional scientific data analyses.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.