Principal component analysis (PCA) is a widely employed statistical tool used primarily for dimensionality reduction. However, it is known to be adversely affected by the presence of outlying observations in the sample, which is quite common. Robust PCA methods using M-estimators have theoretical benefits, but their robustness drop substantially for high dimensional data. On the other end of the spectrum, robust PCA algorithms solving principal component pursuit or similar optimization problems have high breakdown, but lack theoretical richness and demand high computational power compared to the M-estimators. We introduce a novel robust PCA estimator based on the minimum density power divergence estimator. This combines the theoretical strength of the M-estimators and the minimum divergence estimators with a high breakdown guarantee regardless of data dimension. We present a computationally efficient algorithm for this estimate. Our theoretical findings are supported by extensive simulations and comparisons with existing robust PCA methods. We also showcase the proposed algorithm's applicability on two benchmark data sets and a credit card transactions data set for fraud detection.

Annotation of cells in single-cell clustering requires a homogeneous grouping of cell populations. Since single-cell data are susceptible to technical noise, the quality of genes selected prior to clustering is of crucial importance in the preliminary steps of downstream analysis. Therefore, interest in robust gene selection has gained considerable attention in recent years. We introduce sc-REnF [robust entropy based feature (gene) selection method], aiming to leverage the advantages of $R{\prime}{e}nyi$ and $Tsallis$ entropies in gene selection for single cell clustering. Experiments demonstrate that with tuned parameter ($q$), $R{\prime}{e}nyi$ and $Tsallis$ entropies select genes that improved the clustering results significantly, over the other competing methods. sc-REnF can capture relevancy and redundancy among the features of noisy data extremely well due to its robust objective function. Moreover, the selected features/genes can able to determine the unknown cells with a high accuracy. Finally, sc-REnF yields good clustering performance in small sample, large feature scRNA-seq data. Availability: The sc-REnF is available at https://github.com/Snehalikalall/sc-REnF.

Entropy and relative or cross entropy measures are two very fundamental concepts in information theory and are also widely used for statistical inference across disciplines. The related optimization problems, in particular the maximization of the entropy and the minimization of the cross entropy or relative entropy (divergence), are essential for general logical inference in our physical world. In this paper, we discuss a two parameter generalization of the popular Rényi entropy and associated optimization problems. We derive the desired entropic characteristics of the new generalized entropy measure including its positivity, expandability, extensivity and generalized (sub-)additivity. More importantly, when considered over the class of sub-probabilities, our new family turns out to be scale-invariant; this property does not hold for most existing generalized entropy measures. We also propose the corresponding cross entropy and relative entropy measures and discuss their geometric properties including generalized Pythagorean results over β-convex sets. The maximization of the new entropy and the minimization of the corresponding cross or relative entropy measures are carried out explicitly under the non-extensive (`third-choice') constraints given by the Tsallis' normalized q-expectations which also correspond to the β-linear family of probability distributions. Important properties of the associated forward and reverse projection rules are discussed along with their existence and uniqueness. In this context, we have come up with, for the first time, a class of entropy measures - a subfamily of our two-parameter generalization - that leads to the classical (extensive) exponential family of MaxEnt distributions under the non-extensive constraints; this discovery has been illustrated through the useful concept of escort distributions and can potentially be important for future research in information theory. Other members of the new entropy family, however, lead to the power-law type generalized q-exponential MaxEnt distributions which is in conformity with Tsallis' nonextensive theory. Therefore, our new family indeed provides a wide range of entropy and associated measures combining both the extensive and nonextensive MaxEnt theories under one umbrella.

We propose a sparse regression method based on the non-concave penalized density power divergence loss function which is robust against infinitesimal contamination in very high dimensionality. Present methods of sparse and robust regression are based on ℓ1-penalization, and their theoretical properties are not well-investigated. In contrast, we use a general class of folded concave penalties that ensure sparse recovery and consistent estimation of regression coefficients. We propose an alternating algorithm based on the Concave-Convex procedure to obtain our estimate, and demonstrate its robustness properties using influence function analysis. Under some conditions on the fixed design matrix and penalty function, we prove that this estimator possesses large-sample oracle properties in an ultrahigh-dimensional regime. The performance and effectiveness of our proposed method for parameter estimation and prediction compared to state-of-the-art are demonstrated through simulation studies.

Minimum divergence methods are popular tools in a variety of statistical applications. We consider tubular model adequacy tests, and demonstrate that the new divergences that are generated in the process are very useful in robust statistical inference. In particular, we show that the family of S-divergences can be alternatively developed using the tubular model adequacy tests; a further application of the paradigm generates a larger superfamily of divergences. We describe the properties of this larger class and its potential applications in robust inference. Along the way, the failure of the first order influence function analysis in capturing the robustness of these procedures is also established.