This paper studies secrecy-capacity of an $n$-dimensional Gaussian wiretap channel under the peak-power constraint. This work determines the largest peak-power constraint $\bar{\mathsf{R}}_n$ such that an input distribution uniformly distributed on a single sphere is optimal; this regime is termed the small-amplitude regime. The asymptotic of $\bar{\mathsf{R}}_n$ as $n$ goes to infinity is completely characterized as a function of noise variance at both receivers. Moreover, the secrecy-capacity is also characterized in a form amenable for computation. Furthermore, several numerical examples are provided, such as the example of the secrecy-capacity achieving distribution outside of the small amplitude regime.
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We introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Quantized GRAND (QGRAND) that can efficiently decode any moderate redundancy block-code of any length in an algorithm that is suitable for highly parallelized implementation in hardware. QGRAND can avail of any level of quantized soft information, is established to be almost capacity achieving, and is shown to provide near maximum likelihood decoding performance when provided with five or more bits of soft information per received bit.
This paper considers the problem of inference in cluster randomized experiments when cluster sizes are non-ignorable. Here, by a cluster randomized experiment, we mean one in which treatment is assigned at the level of the cluster; by non-ignorable cluster sizes we mean that "large" clusters and "small" clusters may be heterogeneous, and, in particular, the effects of the treatment may vary across clusters of differing sizes. In order to permit this sort of flexibility, we consider a sampling framework in which cluster sizes themselves are random. In this way, our analysis departs from earlier analyses of cluster randomized experiments in which cluster sizes are treated as non-random. We distinguish between two different parameters of interest: the equally-weighted cluster-level average treatment effect, and the size-weighted cluster-level average treatment effect. For each parameter, we provide methods for inference in an asymptotic framework where the number of clusters tends to infinity and treatment is assigned using simple random sampling. We additionally permit the experimenter to sample only a subset of the units within each cluster rather than the entire cluster and demonstrate the implications of such sampling for some commonly used estimators. A small simulation study shows the practical relevance of our theoretical results.
We consider M-estimation problems, where the target value is determined using a minimizer of an expected functional of a Levy process. With discrete observations from the Levy process, we can produce a "quasi-path" by shuffling increments of the Levy process, we call it a quasi-process. Under a suitable sampling scheme, a quasi-process can converge weakly to the true process according to the properties of the stationary and independent increments. Using this resampling technique, we can estimate objective functionals similar to those estimated using the Monte Carlo simulations, and it is available as a contrast function. The M-estimator based on these quasi-processes can be consistent and asymptotically normal.
In this work, we focus on the high-dimensional trace regression model with a low-rank coefficient matrix. We establish a nearly optimal in-sample prediction risk bound for the rank-constrained least-squares estimator under no assumptions on the design matrix. Lying at the heart of the proof is a covering number bound for the family of projection operators corresponding to the subspaces spanned by the design. By leveraging this complexity result, we perform a power analysis for a permutation test on the existence of a low-rank signal under the high-dimensional trace regression model. We show that the permutation test based on the rank-constrained least-squares estimator achieves non-trivial power with no assumptions on the minimum (restricted) eigenvalue of the covariance matrix of the design. Finally, we use alternating minimization to approximately solve the rank-constrained least-squares problem to evaluate its empirical in-sample prediction risk and power of the resulting permutation test in our numerical study.
Let $X^{(n)}$ be an observation sampled from a distribution $P_{\theta}^{(n)}$ with an unknown parameter $\theta,$ $\theta$ being a vector in a Banach space $E$ (most often, a high-dimensional space of dimension $d$). We study the problem of estimation of $f(\theta)$ for a functional $f:E\mapsto {\mathbb R}$ of some smoothness $s>0$ based on an observation $X^{(n)}\sim P_{\theta}^{(n)}.$ Assuming that there exists an estimator $\hat \theta_n=\hat \theta_n(X^{(n)})$ of parameter $\theta$ such that $\sqrt{n}(\hat \theta_n-\theta)$ is sufficiently close in distribution to a mean zero Gaussian random vector in $E,$ we construct a functional $g:E\mapsto {\mathbb R}$ such that $g(\hat \theta_n)$ is an asymptotically normal estimator of $f(\theta)$ with $\sqrt{n}$ rate provided that $s>\frac{1}{1-\alpha}$ and $d\leq n^{\alpha}$ for some $\alpha\in (0,1).$ We also derive general upper bounds on Orlicz norm error rates for estimator $g(\hat \theta)$ depending on smoothness $s,$ dimension $d,$ sample size $n$ and the accuracy of normal approximation of $\sqrt{n}(\hat \theta_n-\theta).$ In particular, this approach yields asymptotically efficient estimators in some high-dimensional exponential models.
Multihop relaying is a potential technique to mitigate channel impairments in optical wireless communications (OWC). In this paper, multiple fixed-gain amplify-and-forward (AF) relays are employed to enhance the OWC performance under the combined effect of atmospheric turbulence, pointing errors, and fog. We consider a long-range OWC link by modeling the atmospheric turbulence by the Fisher-Snedecor ${\cal{F}}$ distribution, pointing errors by the generalized non-zero boresight model, and random path loss due to fog. We also consider a short-range OWC system by ignoring the impact of atmospheric turbulence. We derive novel upper bounds on the probability density function (PDF) and cumulative distribution function (CDF) of the end-to-end signal-to-noise ratio (SNR) for both short and long-range multihop OWC systems by developing exact statistical results for a single-hop OWC system under the combined effect of ${\cal{F}}$-turbulence channels, non-zero boresight pointing errors, and fog-induced fading. Based on these expressions, we present analytical expressions of outage probability (OP) and average bit-error-rate (ABER) performance for the considered OWC systems involving single-variate Fox's H and Meijer's G functions. Moreover, asymptotic expressions of the outage probability in high SNR region are developed using simpler Gamma functions to provide insights on the effect of channel and system parameters. The derived analytical expressions are validated through Monte-Carlo simulations, and the scaling of the OWC performance with the number of relay nodes is demonstrated with a comparison to the single-hop transmission.
In this paper, a new communication-efficient federated learning (FL) framework is proposed, inspired by vector quantized compressed sensing. The basic strategy of the proposed framework is to compress the local model update at each device by applying dimensionality reduction followed by vector quantization. Subsequently, the global model update is reconstructed at a parameter server (PS) by applying a sparse signal recovery algorithm to the aggregation of the compressed local model updates. By harnessing the benefits of both dimensionality reduction and vector quantization, the proposed framework effectively reduces the communication overhead of local update transmissions. Both the design of the vector quantizer and the key parameters for the compression are optimized so as to minimize the reconstruction error of the global model update under the constraint of wireless link capacity. By considering the reconstruction error, the convergence rate of the proposed framework is also analyzed for a smooth loss function. Simulation results on the MNIST and CIFAR-10 datasets demonstrate that the proposed framework provides more than a 2.5% increase in classification accuracy compared to state-of-art FL frameworks when the communication overhead of the local model update transmission is less than 0.1 bit per local model entry.
Universal coding of integers~(UCI) is a class of variable-length code, such that the ratio of the expected codeword length to $\max\{1,H(P)\}$ is within a constant factor, where $H(P)$ is the Shannon entropy of the decreasing probability distribution $P$. However, if we consider the ratio of the expected codeword length to $H(P)$, the ratio tends to infinity by using UCI, when $H(P)$ tends to zero. To solve this issue, this paper introduces a class of codes, termed generalized universal coding of integers~(GUCI), such that the ratio of the expected codeword length to $H(P)$ is within a constant factor $K$. First, the definition of GUCI is proposed and the coding structure of GUCI is introduced. Next, we propose a class of GUCI $\mathcal{C}$ to achieve the expansion factor $K_{\mathcal{C}}=2$ and show that the optimal GUCI is in the range $1\leq K_{\mathcal{C}}^{*}\leq 2$. Then, by comparing UCI and GUCI, we show that when the entropy is very large or $P(0)$ is not large, there are also cases where the average codeword length of GUCI is shorter. Finally, the asymptotically optimal GUCI is presented.
We present a pipelined multiplier with reduced activities and minimized interconnect based on online digit-serial arithmetic. The working precision has been truncated such that $p<n$ bits are used to compute $n$ bits product, resulting in significant savings in area and power. The digit slices follow variable precision according to input, increasing upto $p$ and then decreases according to the error profile. Pipelining has been done to achieve high throughput and low latency which is desirable for compute intensive inner products. Synthesis results of the proposed designs have been presented and compared with the non-pipelined online multiplier, pipelined online multiplier with full working precision and conventional serial-parallel and array multipliers. For $8, 16, 24$ and $32$ bit precision, the proposed low power pipelined design show upto $38\%$ and $44\%$ reduction in power and area respectively compared to the pipelined online multiplier without working precision truncation.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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