We sincerely thank all the discussants Kjell Doksum and Joan Fujimura (DF); Jianqing Fan (Fan); Peihua Qiu, Kai Yang, and Lu You (QYY); and Yanming Li, Hyokyoung Grace Hong, and Yi Li (LHL) for the thought-provoking and insightful discussions on our paper. We would also like to thank the Editor Fabrizio Ruggeri for processing and organizing the discussion. Ahmed would like to specially thank hi...

Observations from real-world problems are often highdimensional vectors, i.e. made up of many variables. Learning methods, including artificial neural networks, often have difficulties to handle a relatively small number of high-dimensional data. In this paper, we show how concepts gained from our intuition on 2and 3dimensional data can be misleading when used in high-dimensional settings. When...

Clustering in high-dimensional spaces is a difficult problem which is recurrent in many domains, for example in image analysis. The difficulty is due to the fact that highdimensional data usually live in different low-dimensional subspaces hidden in the original space. This paper presents a family of Gaussian mixture models designed for highdimensional data which combine the ideas of subspace c...

With the rapid growth of computational biology and e-commerce applications, high-dimensional data becomes very common. Thus, mining highdimensional data is an urgent problem of great practical importance. However, there are some unique challenges for mining data of high dimensions, including (1) the curse of dimensionality and more crucial (2) the meaningfulness of the similarity measure in the...

Recently, high dimensional classification problems have been ubiquitous due to significant advances in technology. High dimensionality poses significant statistical challenges and renders many traditional classification algorithms impractical to use. In this chapter, we present a comprehensive overview of different classifiers that have been highly successful in handling high dimensional data c...

Low rank representation (LRR) has recently attracted great interest due to its pleasing efficacy in exploring low-dimensional subspace structures embedded in data. One of its successful applications is subspace clustering, by which data are clustered according to the subspaces they belong to. In this paper, at a higher level, we intend to cluster subspaces into classes of subspaces. This is nat...

Speech recognition systems typically contain many Gaussian distributions, and hence a large number of parameters. This makes them both slow to decode speech, and large to store. Techniques have been proposed to decrease the number of parameters. One approach is to share parameters between multiple Gaussians, thus reducing the total number of parameters and allowing for shared likelihood calcula...

Subspace clustering (SC) is a promising technology involving clusters that are identified based on their association with subspaces in high-dimensional spaces. SC can be classified into hard subspace clustering (HSC) and soft subspace clustering (SSC). While HSC algorithms have been studied extensively and are well accepted by the scientific community, SSC algorithms are relatively new. However...