Although the widely linear least mean square error (WLMMSE) receiver has been an appealing option for multiple-input-multiple-output (MIMO) wireless systems, a statistical understanding on its pose-detection signal-to-interference-plus-noise ratio (SINR) in detail is still missing. To this end, we consider a WLMMSE MIMO transmission system with rectilinear or quasi-rectilinear (QR) signals over the uncorrelated Rayleigh fading channel and investigate the statistical properties of its SINR for an arbitrary antenna configuration with $N_t$ transmit antennas and $N_r$ receive ones. We first derive an analytic probability density function (PDF) of the SINR in terms of the confluent hypergeometric function of the second kind, for WLMMSE MIMO systems with an arbitrary $N_r$ and $N_t=2, 3$. For a more general case in practice, i.e., $N_t>3$, we resort to the moment generating function to obtain an approximate but closed form PDF under some mild conditions, which, as expected, is more Gaussian-like as $2N_r-N_t$ increases. The so-derived PDFs are able to provide key insights into the WLMMSE MIMO receiver in terms of the outage probability, the symbol error rate, and the diversity gain, all presented in closed form. In particular, its diversity gain and the gain improvement over the conventional LMMSE one are explicitly quantified as $N_r-(N_t-1)/2$ and $(N_t-1)/2$, respectively. Finally, Monte Carlo simulations support the analysis.
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Presently, the practice of distributed computing is such that problems exist in a mathematical realm different from their solutions: a problem is presented as a set of requirements on possible process or system behaviors, and its solution is presented as algorithmic pseudocode satisfying the requirements. Here, we present a novel mathematical realm, termed \emph{multiagent transition systems with faults}, that aims to accommodate both distributed computing problems and their solutions. A problem is presented as a specification -- a multiagent transition system -- and a solution as an implementation of the specification by another, lower-level multiagent transition systems, which may be proven to be resilient to a given set of faults. This duality of roles of a multiagent transition system can be exploited all the way from a high-level distributed computing problem description down to an agreed-upon base layer, say TCP/IP, resulting in a mathematical protocol stack where each protocol in the stack both implements the protocol above it and serves as a specification for the protocol below it. Correct implementations are compositional and thus provide also an implementation of the protocol stack as a whole. The framework also offers a formal -- yet natural and expressive -- notions of faults, fault-resilient implementations, and their composition.
While scalable cell-free massive MIMO (CF-mMIMO) shows advantages in static conditions, the impact of its changing serving access point (AP) set in a mobile network is not yet addressed. In this paper we first derive the CPU cluster and AP handover rates of scalable CF-mMIMO as exact numerical results and tight closed form approximations. We then use our closed form handover rate result to analyse the mobility-aware throughput. We compare the mobility-aware spectral efficiency (SE) of scalable CF-mMIMO against distributed MIMO with pure network- and UE-centric AP selection, for different AP densities and handover delays. Our results reveal an important trade-off for future dense networks with low control delay: under moderate to high mobility, scalable CF-mMIMO maintains its advantage for the 95th-percentile users but at the cost of degraded median SE.
Factorization of matrices where the rank of the two factors diverges linearly with their sizes has many applications in diverse areas such as unsupervised representation learning, dictionary learning or sparse coding. We consider a setting where the two factors are generated from known component-wise independent prior distributions, and the statistician observes a (possibly noisy) component-wise function of their matrix product. In the limit where the dimensions of the matrices tend to infinity, but their ratios remain fixed, we expect to be able to derive closed form expressions for the optimal mean squared error on the estimation of the two factors. However, this remains a very involved mathematical and algorithmic problem. A related, but simpler, problem is extensive-rank matrix denoising, where one aims to reconstruct a matrix with extensive but usually small rank from noisy measurements. In this paper, we approach both these problems using high-temperature expansions at fixed order parameters. This allows to clarify how previous attempts at solving these problems failed at finding an asymptotically exact solution. We provide a systematic way to derive the corrections to these existing approximations, taking into account the structure of correlations particular to the problem. Finally, we illustrate our approach in detail on the case of extensive-rank matrix denoising. We compare our results with known optimal rotationally-invariant estimators, and show how exact asymptotic calculations of the minimal error can be performed using extensive-rank matrix integrals.
Unmanned aerial vehicles (UAVs) have attracted a lot of research attention because of their high mobility and low cost in serving as temporary aerial base stations (BSs) and providing high data rates for next-generation communication networks. To protect user privacy while avoiding detection by a warden, we investigate a jammer-aided UAV covert communication system, which aims to maximize the user's covert rate with optimized transmit and jamming power. The UAV is equipped with multi-antennas to serve multi-users simultaneously and enhance the Quality of Service. By considering the general composite fading and shadowing channel models, we derive the exact probability density (PDF) and cumulative distribution functions (CDF) of the signal-to-interference-plusnoise ratio (SINR). The obtained PDF and CDF are used to derive the closed-form expressions for detection error probability and covert rate. Furthermore, the covert rate maximization problem is formulated as a Nash bargaining game, and the Nash bargaining solution (NBS) is introduced to investigate the negotiation among users. To solve the NBS, we propose two algorithms, i.e., particle swarm optimization-based and joint twostage power allocation algorithms, to achieve covertness and high data rates under the warden's optimal detection threshold. All formulated problems are proven to be convex, and the complexity is analyzed. The numerical results are presented to verify the theoretical performance analysis and show the effectiveness and success of achieving the covert communication of our algorithms.
Intelligent reflecting surfaces (IRSs) enable multiple-input multiple-output (MIMO) transmitters to modify the communication channels between the transmitters and receivers. In the presence of eavesdropping terminals, this degree of freedom can be used to effectively suppress the information leakage towards such malicious terminals. This leads to significant potential secrecy gains in IRS-aided MIMO systems. This work exploits these gains via a tractable joint design of downlink beamformers and IRS phase-shifts. In this respect, we consider a generic IRS-aided MIMO wiretap setting and invoke fractional programming and alternating optimization techniques to iteratively find the beamformers and phase-shifts that maximize the achievable weighted secrecy sum-rate. Our design concludes two low-complexity algorithms for joint beamforming and phase-shift tuning. Performance of the proposed algorithms are numerically evaluated and compared to the benchmark. The results reveal that integrating IRSs into MIMO systems not only boosts the secrecy performance of the system, but also improves the robustness against passive eavesdropping.
The distributed convex optimization problem over the multi-agent system is considered in this paper, and it is assumed that each agent possesses its own cost function and communicates with its neighbours over a sequence of time-varying directed graphs. However, due to some reasons there exist communication delays while agents receive information from other agents, and we are going to seek the optimal value of the sum of agents' loss functions in this case. We desire to handle this problem with the push-sum distributed dual averaging (PS-DDA) algorithm. It is proved that this algorithm converges and the error decays at a rate $\mathcal{O}\left(T^{-0.5}\right)$ with proper step size, where $T$ is iteration span. The main result presented in this paper also illustrates the convergence of the proposed algorithm is related to the maximum value of the communication delay on one edge. We finally apply the theoretical results to numerical simulations to show the PS-DDA algorithm's performance.
Presently, the practice of distributed computing is such that problems exist in a mathematical realm different from their solutions: a problem is presented as a set of requirements on possible process or system behaviors, and the solution is presented as algorithmic pseudocode satisfying the requirements. Here, we present a novel mathematical realm, termed \emph{multiagent transition systems}, that aims to accommodate both distributed computing problems and their solutions. A problem is presented as a specification -- a multiagent transition system -- and a solution as an implementation of the specification by another, lower-level multiagent transition systems. This duality of roles of a multiagent transition system can be exploited all the way from a high-level distributed computing problem description down to an agreed-upon base layer, say TCP/IP, resulting in a mathematical protocol stack where each protocol is implemented by the one below it. Correct implementations are compositional and thus provide also an implementation of the protocol stack as a whole. The framework also offers a formal, yet natural, notion of faults and their resilience. We present two illustrations of the power of the approach: A multiagent transition systems specifying a centralized single-chain protocol and a distributed longest-chain protocol, show an implementation of this protocol by the longest-chain protocol, and conclude -- via the compositionality of correct implementations -- that the distributed longest-chain protocol is universal for centralized multiagent transition systems. Second, we describe a DAG-based blockchain consensus protocol stack that addresses each of the key tasks of a blockchain protocol -- dissemination, equivocation-exclusion, and ordering -- by a different layer of the stack. Additional applications of this mathematical framework are underway.
LU and Cholesky matrix factorization algorithms are core subroutines used to solve systems of linear equations (SLEs) encountered while solving an optimization problem. Standard factorization algorithms are highly efficient but remain susceptible to the accumulation roundoff errors, which can lead solvers to return feasibility and optimality certificates that are actually invalid. This paper introduces a novel approach for solving sequences of closely related SLEs encountered in nonlinear programming efficiently and without roundoff errors. Specifically, it introduces rank-one update algorithms for the roundoff-error-free (REF) factorization framework, a toolset built on integer-preserving arithmetic that has led to the development and implementation of fail-proof SLE solution subroutines for linear programming. The formal guarantees of the proposed algorithms are formally established through the derivation of theoretical insights. Their computational advantages are supported with computational experiments, which demonstrate upwards of 75x-improvements over exact factorization run-times on fully dense matrices with over one million entries. A significant advantage of the proposed methodology is that the length of any coefficient calculated via the associated algorithms is bounded polynomially in the size of the inputs without having to resort to greatest common divisor operations, which are required by and thereby hinder an efficient implementation of exact rational arithmetic approaches.
Bayesian bandit algorithms with approximate inference have been widely used in practice with superior performance. Yet, few studies regarding the fundamental understanding of their performances are available. In this paper, we propose a Bayesian bandit algorithm, which we call Generalized Bayesian Upper Confidence Bound (GBUCB), for bandit problems in the presence of approximate inference. Our theoretical analysis demonstrates that in Bernoulli multi-armed bandit, GBUCB can achieve $O(\sqrt{T}(\log T)^c)$ frequentist regret if the inference error measured by symmetrized Kullback-Leibler divergence is controllable. This analysis relies on a novel sensitivity analysis for quantile shifts with respect to inference errors. To our best knowledge, our work provides the first theoretical regret bound that is better than $o(T)$ in the setting of approximate inference. Our experimental evaluations on multiple approximate inference settings corroborate our theory, showing that our GBUCB is consistently superior to BUCB and Thompson sampling.
Methods that align distributions by minimizing an adversarial distance between them have recently achieved impressive results. However, these approaches are difficult to optimize with gradient descent and they often do not converge well without careful hyperparameter tuning and proper initialization. We investigate whether turning the adversarial min-max problem into an optimization problem by replacing the maximization part with its dual improves the quality of the resulting alignment and explore its connections to Maximum Mean Discrepancy. Our empirical results suggest that using the dual formulation for the restricted family of linear discriminators results in a more stable convergence to a desirable solution when compared with the performance of a primal min-max GAN-like objective and an MMD objective under the same restrictions. We test our hypothesis on the problem of aligning two synthetic point clouds on a plane and on a real-image domain adaptation problem on digits. In both cases, the dual formulation yields an iterative procedure that gives more stable and monotonic improvement over time.
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