Distribution-Free Tests of Independence in High Dimensions

We consider the testing of mutual independence among all entries in a $d$-dimensional random vector based on $n$ independent observations. We study two families of distribution-free test statistics, which include Kendall's tau and Spearman's rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and propose tests that control the type I error in the high-dimensional setting where $d>n$. We further show that the two tests are rate-optimal in terms of power against sparse alternatives, and outperform competitors in simulations, especially when $d$ is large.Comment: to appear in Biometrik

Witten Genus and String Complete Intersections

In this note, we prove that the Witten genus of nonsingular string complete intersections in product of complex projective spaces vanishes. Our result generalizes a known result of Landweber and Stong (cf. [HBJ]).Comment: Some materials and references are adde

Studying Maximum Information Leakage Using Karush-Kuhn-Tucker Conditions

When studying the information leakage in programs or protocols, a natural question arises: "what is the worst case scenario?". This problem of identifying the maximal leakage can be seen as a channel capacity problem in the information theoretical sense. In this paper, by combining two powerful theories: Information Theory and Karush-Kuhn-Tucker conditions, we demonstrate a very general solution to the channel capacity problem. Examples are given to show how our solution can be applied to practical contexts of programs and anonymity protocols, and how this solution generalizes previous approaches to this problem

Smooth local solutions to weingarten equations and σk\sigma_k-equations

In this paper, we study the existence of smooth local solutions to Weingarten equations and $\sigma_k$-equations. We will prove that, for $2 \leq k \leq n$, the Weingarten equations and the $\sigma_k$-equations always have smooth local solutions regardless of the sign of the functions in the right-hand side of the equations. We will demonstrate that the associate linearized equations are uniformly elliptic if we choose the initial approximate solutions appropriately