Generalization of Mrs. Gerber's Lemma

Mrs. Gerber's Lemma (MGL) hinges on the convexity of $H(p*H^{-1}(u))$, where $H(u)$ is the binary entropy function. In this work, we prove that $H(p*f(u))$ is convex in $u$ for every $p\in [0,1]$ provided $H(f(u))$ is convex in $u$, where $f(u) : (a, b) \to [0, \frac12]$. Moreover, our result subsumes MGL and simplifies the original proof. We show that the generalized MGL can be applied in binary broadcast channel to simplify some discussion.Comment: Accepted by Communications in Information and System

Higher Order Derivatives in Costa's Entropy Power Inequality

Let $X$ be an arbitrary continuous random variable and $Z$ be an independent Gaussian random variable with zero mean and unit variance. For $t~>~0$, Costa proved that $e^{2h(X+\sqrt{t}Z)}$ is concave in $t$, where the proof hinged on the first and second order derivatives of $h(X+\sqrt{t}Z)$. Specifically, these two derivatives are signed, i.e., $\frac{\partial}{\partial t}h(X+\sqrt{t}Z) \geq 0$ and $\frac{\partial^2}{\partial t^2}h(X+\sqrt{t}Z) \leq 0$. In this paper, we show that the third order derivative of $h(X+\sqrt{t}Z)$ is nonnegative, which implies that the Fisher information $J(X+\sqrt{t}Z)$ is convex in $t$. We further show that the fourth order derivative of $h(X+\sqrt{t}Z)$ is nonpositive. Following the first four derivatives, we make two conjectures on $h(X+\sqrt{t}Z)$: the first is that $\frac{\partial^n}{\partial t^n} h(X+\sqrt{t}Z)$ is nonnegative in $t$ if $n$ is odd, and nonpositive otherwise; the second is that $\log J(X+\sqrt{t}Z)$ is convex in $t$. The first conjecture can be rephrased in the context of completely monotone functions: $J(X+\sqrt{t}Z)$ is completely monotone in $t$. The history of the first conjecture may date back to a problem in mathematical physics studied by McKean in 1966. Apart from these results, we provide a geometrical interpretation to the covariance-preserving transformation and study the concavity of $h(\sqrt{t}X+\sqrt{1-t}Z)$, revealing its connection with Costa's EPI.Comment: Second version submitted. https://sites.google.com/site/chengfancuhk

Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly perturbed version $\mathbf{M}$ is observed. The noise matrix $\mathbf{M}-\mathbf{M}^{\star}$ is composed of zero-mean independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise, for example, when we have two independent samples for each entry of $\mathbf{M}^{\star}$ and arrange them into an {\em asymmetric} data matrix $\mathbf{M}$. The aim is to estimate the leading eigenvalue and eigenvector of $\mathbf{M}^{\star}$. We demonstrate that the leading eigenvalue of the data matrix $\mathbf{M}$ can be $O(\sqrt{n})$ times more accurate --- up to some log factor --- than its (unadjusted) leading singular value in eigenvalue estimation. Further, the perturbation of any linear form of the leading eigenvector of $\mathbf{M}$ --- say, entrywise eigenvector perturbation --- is provably well-controlled. This eigen-decomposition approach is fully adaptive to heteroscedasticity of noise without the need of careful bias correction or any prior knowledge about the noise variance. We also provide partial theory for the more general rank-$r$ case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be beneficial.Comment: accepted to Annals of Statistics, 2020. 37 page

d+id' Chiral Superconductivity in Bilayer Silicene

We investigate the structure and physical properties of the undoped bilayer silicene through first-principles calculations and find the system is intrinsically metallic with sizable pocket Fermi surfaces. When realistic electron-electron interaction turns on, the system is identified as a chiral d+id' topological superconductor mediated by the strong spin fluctuation on the border of the antiferromagnetic spin density wave order. Moreover, the tunable Fermi pocket area via strain makes it possible to adjust the spin density wave critical interaction strength near the real one and enables a high superconducting critical temperature

Compressing Inertial Motion Data in Wireless Sensing Systems – An Initial Experiment

The use of wireless inertial motion sensors, such as accelerometers, for supporting medical care and sport’s training, has been under investigation in recent years. As the number of sensors (or their sampling rates) increases, compressing data at source(s) (i.e. at the sensors), i.e. reducing the quantity of data that needs to be transmitted between the on-body sensors and the remote repository, would be essential especially in a bandwidth-limited wireless environment. This paper presents a set of compression experiment results on a set of inertial motion data collected during running exercises. As a starting point, we selected a set of common compression algorithms to experiment with. Our results show that, conventional lossy compression algorithms would achieve a desirable compression ratio with an acceptable time delay. The results also show that the quality of the decompressed data is within acceptable range

MatchZoo: A Learning, Practicing, and Developing System for Neural Text Matching

Text matching is the core problem in many natural language processing (NLP) tasks, such as information retrieval, question answering, and conversation. Recently, deep leaning technology has been widely adopted for text matching, making neural text matching a new and active research domain. With a large number of neural matching models emerging rapidly, it becomes more and more difficult for researchers, especially those newcomers, to learn and understand these new models. Moreover, it is usually difficult to try these models due to the tedious data pre-processing, complicated parameter configuration, and massive optimization tricks, not to mention the unavailability of public codes sometimes. Finally, for researchers who want to develop new models, it is also not an easy task to implement a neural text matching model from scratch, and to compare with a bunch of existing models. In this paper, therefore, we present a novel system, namely MatchZoo, to facilitate the learning, practicing and designing of neural text matching models. The system consists of a powerful matching library and a user-friendly and interactive studio, which can help researchers: 1) to learn state-of-the-art neural text matching models systematically, 2) to train, test and apply these models with simple configurable steps; and 3) to develop their own models with rich APIs and assistance