Concave Perron–Frobenius Theory and applications

Krause, Ulrich

doi:10.1016/s0362-546x(01)00281-4

Cited by 86 publications

(54 citation statements)

References 8 publications

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“…Since (because ) and , we have that for all . The rest of the proof follows from applying Theorem 1 in [24]. That the fixed-point iteration converges geometrically fast follows from the remark after Theorem 1 in [24].…”

Section: A Proof Of Theoremmentioning

confidence: 90%

A Unified Analysis of Max-Min Weighted SINR for MIMO Downlink System

Cai

Quek

Tan

2011

IEEE Trans. Signal Process.

123

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Abstract-This paper studies the max-min weighted signal-to-interference-plus-noise ratio (SINR) problem in the multipleinput-multiple-output (MIMO) downlink, where multiple users are weighted according to priority and are subject to a weighted-sum-power constraint. First, we study the multiple-input-single-output (MISO) and single-input-multipleoutput (SIMO) problems using nonlinear Perron-Frobenius theory. As a by-product, we solve the open problem of convergence for a previously proposed MISO algorithm by Wiesel, Eldar, and Shamai in 2006. Furthermore, we unify our analysis with respect to the previous alternate optimization algorithm proposed by Tan, Chiang, and Srikant in 2009, by showing that our MISO result can, in fact, be derived from their algorithm. Next, we combine our MISO and SIMO results into an algorithm for the MIMO problem. We show that our proposed algorithm is optimal when the channels are rank-one, or when the network is operating in the low signal-to-noise ratio (SNR) region. Finally, we prove the parametric continuity of the MIMO problem in the power constraint, and we use this insight to propose a heuristic initialization strategy for improving the performance of our (generally) suboptimal MIMO algorithm. The proposed initialization strategy exhibits improved performance over random initialization.Index Terms-Beamforming, multiple-input-multiple-output (MIMO), uplink-downlink duality.

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Section: A Proof Of Theoremmentioning

confidence: 90%

A Unified Analysis of Max-Min Weighted SINR for MIMO Downlink System

Cai

Quek

Tan

2011

IEEE Trans. Signal Process.

123

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“…also [3]) under very modest assumptions on the matrix. Other types of non-linear Perron-Frobenius Theorem appear in [7] and [8].…”

Section: Introductionmentioning

confidence: 99%

A Nonlinear Transfer Operator Theorem

Pollicott

2016

J Stat Phys

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“…Concavity appears naturally as a common feature in all examples mentioned above: subhomogeneity of order 1 is a proxy for concavity, sigmoidal functions are concave when restricted to the positive semiaxis, and so are Michaelis-Menten functions. Nonlinear cooperative concave dynamical systems lead to simple theorems for existence and uniqueness of fixed points [12], [14] and, when the trajectories they induce do not diverge, also stability analysis of the fixed point is easily shown. What is additionally shown in the paper is that all these properties can be formulated as spectral conditions on the Jacobian of the system.…”

Section: Introductionmentioning

confidence: 99%

Existence, uniqueness and stability properties of positive equilibria for a class of nonlinear cooperative systems

Abara

Ticozzi

Altafini

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-We provide conditions that guarantee existence, uniqueness and stability of strictly positive equilibria for nonlinear cooperative systems associated to vector fields that are concave or subhomogeneous. This class of positive systems describes well interconnected dynamics that are of key interest for communication, biological, economical and neural network applications. These conditions can be formulated directly in terms of the spectral radius of the Jacobian of the system, and do not require to use constant inputs to move the equilibrium point from the origin to the interior of the positive orthant.

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Concave Perron–Frobenius Theory and applications

Cited by 86 publications

References 8 publications

A Unified Analysis of Max-Min Weighted SINR for MIMO Downlink System

A Unified Analysis of Max-Min Weighted SINR for MIMO Downlink System

A Nonlinear Transfer Operator Theorem

Existence, uniqueness and stability properties of positive equilibria for a class of nonlinear cooperative systems

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