Collaborative Research: MSPA-MCS: Sparse Multivariate Data Analysis

该提案开发和研究了经典多变量数据分析方法的稀疏变体。它主要关注稀疏主成分分析(PCA)和相关的稀疏典型相关分析(CCA)，但也打算探索对应分析和判别分析等方法的稀疏变体。开发稀疏多变量分析算法的动机是它们有可能产生比经典分析更可解释和更稳健的统计结果，同时在统计效率和表达能力方面牺牲尽可能少的东西。研究人员给出了稀疏主成分分析的一种凸松弛算法，它是一个大规模半定规划。该研究首先研究了这种松弛的理论和实际性能，以及求解相应半定程序的大规模实例所涉及的计算复杂性。下一步，重点是将这些结果推广到上面提到的其他多变量数据分析方法。主成分分析(PCA)是一种经典的统计工具，用于研究具有大量变量(气象记录、基因表达系数、利率曲线、社会网络等)的实验数据。它主要用作降维工具：主成分分析生成一组简化的合成变量，以获取关于数据的最大信息量。这使得可以在三维图形上表示包含数千个变量的数据集，同时仍然捕获原始数据的大部分特征，从而使可视化和解释变得更容易。不幸的是，主成分分析的主要缺点是这些新的合成变量是所有原始变量的加权和，这使得它们的物理解释变得困难。这项研究将研究计算稀疏主成分分析的算法，即计算新的合成变量，这些变量只是几个问题变量的加权和，同时保持原始数据集的大部分特征。稀疏主成分分析是一个困难的组合问题，但研究人员已经提出了一种松弛算法，可以利用凸优化中的最新结果有效地求解。研究人员计划研究这种松弛的理论和实际表现，并将这些结果扩展到其他统计方法。

This proposal develops and studies sparse variants of classic multivariate data analysis methods. It primarily focuses on sparse principal component analysis (PCA) and the related sparse canonical correlation analysis (CCA), but also intends to explore sparse variants of methods such as correspondence analysis and discriminant analysis. The motivation for developing sparse multivariate analysis algorithms is their potential for yielding statistical results that are more interpretable and more robust than classical analyses, while giving up as little as possible in the way of statistical efficiency and expressive power. The investigators have derived a convex relaxation for sparse PCA as a large-scale semidefinite program. The proposed research first studies the theoretical and practical performance of this relaxation as well as the computational complexity involved in solving large-scale instances of the corresponding semidefinite programs. In a next step, it focuses on extending these results to the other multivariate data analysis methods cited above. Principal Component Analysis (or PCA) is a classic statistical tool used to study experimental data with a very large number of variables (meteorological records, gene expression coefficients, the interest rate curve, social networks, etc). It is primarily used as a dimensionality reduction tool: PCA produces a reduced set of synthetic variables that captures a maximum amount of information on the data. This makes it possible to represent data sets with thousands of variables on a three dimensional graph while still capturing most of the features of the original data, thus making visualization and interpretation easier. Unfortunately, the key shortcoming of PCA is that these new synthetic variables are a weighted sum of all the original variables making their physical interpretation difficult. The proposed research will study algorithms for computing sparse PCA, i.e., computing new synthetic variables that are the weighted sum of only a few problem variables while keeping most of the features of the original data set. Sparse PCA is a hard combinatorial problem but the investigators have produced a relaxation that can be solved efficiently using recent results in convex optimization. The investigators plan to study the theoretical and practical performance of this relaxation and extend these results to other statistical methods.