Spectral radii of asymptotic mappings and the convergence speed of the standard fixed point algorithm

Important problems in wireless networks can often be solved by computing fixed points of standard or contractive interference mappings, and the conventional fixed point algorithm is widely used for this purpose. Knowing that the mapping used in the algorithm is not only standard but also contractive (or only contractive) is valuable information because we obtain a guarantee of geometric convergence rate, and the rate is related to a property of the mapping called modulus of contraction. To date, contractive mappings and their moduli of contraction have been identified with case-by-case approaches that can be difficult to generalize. To address this limitation of existing approaches, we show in this study that the spectral radii of asymptotic mappings can be used to identify an important subclass of contractive mappings and also to estimate their moduli of contraction. In addition, if the fixed point algorithm is applied to compute fixed points of positive concave mappings, we show that the spectral radii of asymptotic mappings provide us with simple lower bounds for the estimation error of the iterates. An immediate application of this result proves that a known algorithm for load estimation in wireless networks becomes slower with increasing traffic.Comment: Paper accepted for presentation at ICASSP 201

The role of asymptotic functions in network optimization and feasibility studies

Solutions to network optimization problems have greatly benefited from developments in nonlinear analysis, and, in particular, from developments in convex optimization. A key concept that has made convex and nonconvex analysis an important tool in science and engineering is the notion of asymptotic function, which is often hidden in many influential studies on nonlinear analysis and related fields. Therefore, we can also expect that asymptotic functions are deeply connected to many results in the wireless domain, even though they are rarely mentioned in the wireless literature. In this study, we show connections of this type. By doing so, we explain many properties of centralized and distributed solutions to wireless resource allocation problems within a unified framework, and we also generalize and unify existing approaches to feasibility analysis of network designs. In particular, we show sufficient and necessary conditions for mappings widely used in wireless communication problems (more precisely, the class of standard interference mappings) to have a fixed point. Furthermore, we derive fundamental bounds on the utility and the energy efficiency that can be achieved by solving a large family of max-min utility optimization problems in wireless networks.Comment: GlobalSIP 2017 (to appear

Fixed points of nonnegative neural networks

We consider the existence of fixed points of nonnegative neural networks, i.e., neural networks that take as an input and produce as an output nonnegative vectors. We first show that nonnegative neural networks with nonnegative weights and biases can be recognized as monotonic and (weakly) scalable functions within the framework of nonlinear Perron-Frobenius theory. This fact enables us to provide conditions for the existence of fixed points of nonnegative neural networks, and these conditions are weaker than those obtained recently using arguments in convex analysis. Furthermore, we prove that the shape of the fixed point set of nonnegative neural networks with nonnegative weights and biases is an interval, which under mild conditions degenerates to a point. These results are then used to obtain the existence of fixed points of more general types of nonnegative neural networks. The results of this paper contribute to the understanding of the behavior of autoencoders, and they provide insight into neural networks designed using the loop-unrolling technique, which can be seen as a fixed point searching algorithm. The chief theoretical results of this paper are verified in numerical simulations.Comment: 34 page

Improving Resource Efficiency with Partial Resource Muting for Future Wireless Networks

We propose novel resource allocation algorithms that have the objective of finding a good tradeoff between resource reuse and interference avoidance in wireless networks. To this end, we first study properties of functions that relate the resource budget available to network elements to the optimal utility and to the optimal resource efficiency obtained by solving max-min utility optimization problems. From the asymptotic behavior of these functions, we obtain a transition point that indicates whether a network is operating in an efficient noise-limited regime or in an inefficient interference-limited regime for a given resource budget. For networks operating in the inefficient regime, we propose a novel partial resource muting scheme to improve the efficiency of the resource utilization. The framework is very general. It can be applied not only to the downlink of 4G networks, but also to 5G networks equipped with flexible duplex mechanisms. Numerical results show significant performance gains of the proposed scheme compared to the solution to the max-min utility optimization problem with full frequency reuse.Comment: 8 pages, 9 figures, to appear in WiMob 201

On-line lot-sizing with perceptrons

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Characterization of the weak Pareto boundary of resource allocation problems in wireless networks -- Implications to cell-less systems

We establish necessary and sufficient conditions for a network configuration to provide utilities that are both fair and efficient in a well-defined sense. To cover as many applications as possible with a unified framework, we consider utilities defined in an axiomatic way, and the constraints imposed on the feasible network configurations are expressed with a single inequality involving a monotone norm. In this setting, we prove that a necessary and sufficient condition to obtain network configurations that are efficient in the weak Pareto sense is to select configurations attaining equality in the monotone norm constraint. Furthermore, for a given configuration satisfying this equality, we characterize a criterion for which the configuration can be considered fair for the active links. We illustrate potential implications of the theoretical findings by presenting, for the first time, a simple parametrization based on power vectors of achievable rate regions in modern cell-less systems subject to practical impairments.Comment: Accepted at IEEE ICC 202

Dynamic Uplink/Downlink Resource Management in Flexible Duplex-Enabled Wireless Networks

Flexible duplex is proposed to adapt to the channel and traffic asymmetry for future wireless networks. In this paper, we propose two novel algorithms within the flexible duplex framework for joint uplink and downlink resource allocation in multi-cell scenario, named SAFP and RMDI, based on the awareness of interference coupling among wireless links. Numerical results show significant performance gain over the baseline system with fixed uplink/downlink resource configuration, and over the dynamic TDD scheme that independently adapts the configuration to time-varying traffic volume in each cell. The proposed algorithms achieve two-fold increase when compared with the baseline scheme, measured by the worst-case quality of service satisfaction level, under a low level of traffic asymmetry. The gain is more significant when the traffic is highly asymmetric, as it achieves three-fold increase.Comment: 7 pages, 7 figures, ICC 2017 Worksho

Power and Beam Optimization for Uplink Millimeter-Wave Hotspot Communication Systems

We propose an effective interference management and beamforming mechanism for uplink communication systems that yields fair allocation of rates. In particular, we consider a hotspot area of a millimeter-wave (mmWave) access network consisting of multiple user equipment (UE) in the uplink and multiple access points (APs) with directional antennas and adjustable beam widths and directions (beam configurations). This network suffers tremendously from multi-beam multi-user interference, and, to improve the uplink transmission performance, we propose a centralized scheme that optimizes the power, the beam width, the beam direction of the APs, and the UE - AP assignments. This problem involves both continuous and discrete variables, and it has the following structure. If we fix all discrete variables, except for those related to the UE-AP assignment, the resulting optimization problem can be solved optimally. This property enables us to propose a heuristic based on simulated annealing (SA) to address the intractable joint optimization problem with all discrete variables. In more detail, for a fixed configuration of beams, we formulate a weighted rate allocation problem where each user gets the same portion of its maximum achievable rate that it would have under non-interfered conditions. We solve this problem with an iterative fixed point algorithm that optimizes the power of UEs and the UE - AP assignment in the uplink. This fixed point algorithm is combined with SA to improve the beam configurations. Theoretical and numerical results show that the proposed method improves both the UE rates in the lower percentiles and the overall fairness in the network