Resampling Methods for High-Dimensional and Large-Scale Data

重抽样方法是一类广泛的工具，用于衡量统计结果的可变性，例如，允许研究人员确定实验结果是否显著。在过去的几十年里，这些方法得到了广泛的研究，它们已经成为统计学实践的基础--很大程度上是因为它们可以在依赖相对较少的假设的情况下解决复杂的问题。然而，关于重采样方法在现代数据分析中的表现仍有许多需要了解的地方，在现代数据分析中，观测往往具有大量的特征(高维数据)，或者在数据量如此之大以至于超过计算资源(大规模数据)的情况下。在这两个具有挑战性的环境下，本研究将扩展重采样方法的适用范围，并将以下面讨论的两个研究主题为指导。第一，在高维数据环境下，与估计和预测问题相比，对推理问题(包括测试和可信区间)的理解仍然不发达。鉴于重采样方法是一种通用的推理方法，了解它们如何受到低维结构和正则化效果的影响是很重要的。特别是，建议的研究将研究涉及结构化协方差矩阵的高维模型中的重采样方法的性能。其次，在大规模数据环境下，随机化算法因其能够产生快速近似解而受到越来越多的关注。虽然这类算法的输出是随机的，但它们的波动往往可以以更大的计算量为代价来减少。随机化算法的这一一般特性导致了在精度和计算成本之间进行优化折衷的问题。为了找到解决方案，拟议的研究将调查如何使用重采样方法来衡量一系列流行的随机算法的权衡。

Resampling methods are a broad class of tools that serve to measure the variability of statistical results, for example, allowing a researcher to determine whether or not the outcome of an experiment is significant. Over the course of the last few decades, these methods have been extensively studied, and they have become fundamental to the practice of statistics - in large part because they can solve complex problems while relying on relatively few assumptions. Nevertheless, much remains to be understood about the performance of resampling methods in the context of modern data analysis, where observations tend to have large numbers of features (high-dimensional data), or where the quantity of data is so large that it outstrips computational resources (large-scale data). In both of these challenging settings, the proposed research will extend the applicability of resampling methods, and these efforts will be guided by two research themes discussed below.First, in the setting of high-dimensional data, the understanding of inference problems, including tests and confidence intervals, remains underdeveloped in comparison with estimation and prediction problems. Given that resampling methods are a general-purpose approach to inference, it is important to know how they are influenced by the effects of low-dimensional structure and regularization. In particular, the proposed research will study the performance of resampling methods in high-dimensional models involving structured covariance matrices. Second, in the setting of large-scale data, randomized algorithms have received growing attention for their ability to produce fast approximate solutions. Although the outputs of such algorithms are random, their fluctuations can often be reduced at the expense of greater computation. This general trait of randomized algorithms leads to the problem of optimizing a tradeoff between precision and computational cost. Towards a solution, the proposed research will investigate how resampling methods can be used to measure this tradeoff for a collection of popular randomized algorithms.