A relation between conditional entropy and conditional expectation to evaluate secrecy systems

带失真与保密性的二阶图像刻画

Second-Order Image Characterization with Distortion and Secrecy

本文提出二阶图像刻画方法，统一处理失真与保密约束，给出多输出联合波动下的速率界，并应用于多源辅助、高斯描述及隐私放大等场景。

Relations Between Conditional Shannon Entropy and Expectation of $\ell_α$-Norm

The paper examines relationships between the conditional Shannon entropy and the expectation of $\ell_α$-norm for joint probability distributions. More precisely, we investigate the tight bounds of the expectation of $\ell_α$-norm with a fixed conditional Shannon entropy, and vice versa. As applications of the results, we derive the tight bounds between the conditional Shannon entropy and several information measures which are determined by the expectation of $\ell_α$-norm, e.g., the conditional Rényi entropy and the conditional $R$-norm information. Moreover, we apply these results to discret…

The Method of Conditional Expectations for Cubic Metric Reduction in OFDM

High variations in the OFDM signal envelope cause nonlinear distortion in the power amplifier of the transmitter, which is a major drawback. Peak-to-Average Power Ratio (PAPR) and Cubic Metric (CM) are commonly used for quantifying this characteristic of the signal. Despite the reportedly higher accuracy compared to PAPR, limited research has been done on reduction algorithms for CM. In this paper, the Method of Conditional Expectations (CE Method) is used to achieve CM reduction by the Sign Selection approach. Using the CE Method, the amenable mathematical structure of CM is exploited to deve…

与Belfiore和Solé猜想有关的一些结果

Some results related to the conjecture by Belfiore and Solé

在本文的第一部分中，我们考虑了接吻数与保密增益之间的关系。证明了在$n=24m+8k$维偶幺模格上，若最短向量长度为$\geq 2m$，则随着长度为$2m$的向量数的减少，保密增益增加.我们也将在一般的幺模格上证明一个类似的结果。我们还将考虑向量较短的情况。此外，假设Belfiore和Solé的猜想，我们将计算随着向量数量的变化，保密增益的逆之间的差异。我们将通过一个例子证明，存在两个相同维数的格具有相同的最短向量长度和相同的接吻数，但不同的保密增益。最后，我们考虑一些情况下的一个问题由Elkies提供一个答案，一类特殊的格假设猜想的Belfiore和Solé。我们还将得到Gaulter关于猜想的一些结果的条件改进。

A General Derivative Identity for the Conditional Expectation with Focus on the Exponential Family

Consider a pair of random vectors $(\mathbf{X},\mathbf{Y}) $ and the conditional expectation operator $\mathbb{E}[\mathbf{X}|\mathbf{Y}=\mathbf{y}]$. This work studies analytic properties of the conditional expectation by characterizing various derivative identities. The paper consists of two parts. In the first part of the paper, a general derivative identity for the conditional expectation is derived. Specifically, for the Markov chain $\mathbf{U} \leftrightarrow \mathbf{X} \leftrightarrow \mathbf{Y}$, a compact expression for the Jacobian matrix of $\mathbb{E}[\mathbf{U}|\mathbf{Y}=\mathbf{…

Polynomial Approximations of Conditional Expectations in Scalar Gaussian Channels

We consider a channel $Y=X+N$ where $X$ is a random variable satisfying $\mathbb{E}[|X|]<\infty$ and $N$ is an independent standard normal random variable. We show that the minimum mean-square error estimator of $X$ from $Y,$ which is given by the conditional expectation $\mathbb{E}[X \mid Y],$ is a polynomial in $Y$ if and only if it is linear or constant; these two cases correspond to $X$ being Gaussian or a constant, respectively. We also prove that the higher-order derivatives of $y \mapsto \mathbb{E}[X \mid Y=y]$ are expressible as multivariate polynomials in the functions $y \mapsto \mat…