A survey of high dimension low sample size asymptotics

Testing homogeneity: the trouble with sparse functional data

Testing the homogeneity between two samples of functional data is an important task. While this is feasible for intensely measured functional data, we explain why it is challenging for sparsely measured functional data and show what can be done for such data. In particular, we show that testing the marginal homogeneity based on point-wise distributions is feasible under some mild constraints and propose a new two-sample statistic that works well with both intensively and sparsely measured functional data. The proposed test statistic is formulated upon energy distance, and the convergence rate of the test statistic to its population version is derived along with the consistency of the associated permutation test. The aptness of our method is demonstrated on both synthetic and real data sets.

James-Stein for the leading eigenvector

Recent research identifies and corrects bias, such as excess dispersion, in the leading sample eigenvector of a factor-based covariance matrix estimated from a high-dimension low sample size (HL) data set. We show that eigenvector bias can have a substantial impact on variance-minimizing optimization in the HL regime, while bias in estimated eigenvalues may have little effect. We describe a data-driven eigenvector shrinkage estimator in the HL regime called "James-Stein for eigenvectors" (JSE) and its close relationship with the James-Stein (JS) estimator for a collection of averages. We show, both theoretically and with numerical experiments, that, for certain variance-minimizing problems of practical importance, efforts to correct eigenvalues have little value in comparison to the JSE correction of the leading eigenvector. When certain extra information is present, JSE is a consistent estimator of the leading eigenvector.

Polynomial whitening for high-dimensional data

The inverse square root of a covariance matrix is often desirable for performing data whitening in the process of applying many common multivariate data analysis methods. Direct calculation of the inverse square root is not available when the covariance matrix is either singular or nearly singular, as often occurs in high dimensions. We develop new methods, which we broadly call polynomial whitening, to construct a low-degree polynomial in the empirical covariance matrix which has similar properties to the true inverse square root of the covariance matrix (should it exist). Our method does not suffer in singular or near-singular settings, and is computationally tractable in high dimensions. We demonstrate that our construction of low-degree polynomials provides a good substitute for high-dimensional inverse square root covariance matrices, in both $$d < N$$ d < N and $$d \ge N$$ d ≥ N cases. We offer examples on data whitening, outlier detection and principal component analysis to demonstrate the performance of the proposed method.

Resolution of the curse of dimensionality in single-cell RNA sequencing data analysis

Single-cell RNA sequencing (scRNA-seq) can determine gene expression in numerous individual cells simultaneously, promoting progress in the biomedical sciences. However, scRNA-seq data are high-dimensional with substantial technical noise, including dropouts. During analysis of scRNA-seq data, such noise engenders a statistical problem known as the curse of dimensionality (COD). Based on high-dimensional statistics, we herein formulate a noise reduction method, RECODE (resolution of the curse of dimensionality), for high-dimensional data with random sampling noise. We show that RECODE consistently resolves COD in relevant scRNA-seq data with unique molecular identifiers. RECODE does not involve dimension reduction and recovers expression values for all genes, including lowly expressed genes, realizing precise delineation of cell fate transitions and identification of rare cells with all gene information. Compared with representative imputation methods, RECODE employs different principles and exhibits superior overall performance in cell-clustering, expression value recovery, and single-cell-level analysis. The RECODE algorithm is parameter-free, data-driven, deterministic, and high-speed, and its applicability can be predicted based on the variance normalization performance. We propose RECODE as a powerful strategy for preprocessing noisy high-dimensional data.

diproperm: An R Package for the DiProPerm Test

High-dimensional low sample size (HDLSS) data sets frequently emerge in many biomedical applications. The direction-projection-permutation (DiProPerm) test is a two-sample hypothesis test for comparing two high-dimensional distributions. The DiProPerm test is exact, i.e., the type I error is guaranteed to be controlled at the nominal level for any sample size, and thus is applicable in the HDLSS setting. This paper discusses the key components of the DiProPerm test, introduces the diproperm R package, and demonstrates the package on a real-world data set.