Big Data bring new opportunities to modern society and challenges to data scientists. On the one hand, Big Data hold great promises for discovering subtle population patterns and heterogeneities that are not possible with small-scale data. On the other hand, the massive sample size and high dimensionality of Big Data introduce unique computational and statistical challenges, including scalability and storage bottleneck, noise accumulation, spurious correlation, incidental endogeneity and measurement errors. These challenges are distinguished and require new computational and statistical paradigm. This paper gives overviews on the salient features of Big Data and how these features impact on paradigm change on statistical and computational methods as well as computing architectures. We also provide various new perspectives on the Big Data analysis and computation. In particular, we emphasize on

This article introduces the large portfolio selection using gross-exposure constraints. It shows that with gross-exposure constraints, the empirically selected optimal portfolios based on estimated covariance matrices have similar performance to the theoretical optimal ones and there is no error accumulation effect from estimation of vast covariance matrices. This gives theoretical justification to the empirical results by Jagannathan and Ma. It also shows that the no-short-sale portfolio can be improved by allowing some short positions. The applications to portfolio selection, tracking, and improvements are also addressed. The utility of our new approach is illustrated by simulation and empirical studies on the 100 FamaFrench industrial portfolios and the 600 stocks randomly selected from Russell 3000.

We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$ -forms of a projective bundle. In particular we generalize Bott’s formula for the projective space to a projective bundle over a scheme of characteristic zero.

Suppose we lhave n observationis ftom Y= X+ c, where e is nieasuremiienit error with kiowin distributioni, and the denesity f of X is uniknowii withi thte nloln-paraniietric conistrainit

We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube C_n of rank n. For any indecomposable rigid object M in C_n, we define an analogous X_M of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of M. We show that X_M,X_{M′} satisfy the mutation formula when M,M′ form an exchange pair, and that X_? : M↦X_M gives a bijection from the set of indecomposable rigid objects in C_n to the set of cluster variables of cluster algebra of type C_{n−1}, which induces a bijection between the set of basic maximal rigid objects in C_n and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube Cn encode the combinatorics of the cluster algebra of type B_{n−1} since the combinatorics of cluster algebras of type B_{n−1} or of type C_{n−1} are the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type C.