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OpenRANet: Neuralized Spectrum Access by Joint Subcarrier and Power Allocation with Optimization-based Deep Learning arxiv.org/abs/2409.12964 .IT .AI

OpenRANet: Neuralized Spectrum Access by Joint Subcarrier and Power Allocation with Optimization-based Deep Learning

The next-generation radio access network (RAN), known as Open RAN, is poised to feature an AI-native interface for wireless cellular networks, including emerging satellite-terrestrial systems, making deep learning integral to its operation. In this paper, we address the nonconvex optimization challenge of joint subcarrier and power allocation in Open RAN, with the objective of minimizing the total power consumption while ensuring users meet their transmission data rate requirements. We propose OpenRANet, an optimization-based deep learning model that integrates machine-learning techniques with iterative optimization algorithms. We start by transforming the original nonconvex problem into convex subproblems through decoupling, variable transformation, and relaxation techniques. These subproblems are then efficiently solved using iterative methods within the standard interference function framework, enabling the derivation of primal-dual solutions. These solutions integrate seamlessly as a convex optimization layer within OpenRANet, enhancing constraint adherence, solution accuracy, and computational efficiency by combining machine learning with convex analysis, as shown in numerical experiments. OpenRANet also serves as a foundation for designing resource-constrained AI-native wireless optimization strategies for broader scenarios like multi-cell systems, satellite-terrestrial networks, and future Open RAN deployments with complex power consumption requirements.

arxiv.org

The Asymptotic Behaviour of Information Leakage Metrics arxiv.org/abs/2409.13003 .IT

The Asymptotic Behaviour of Information Leakage Metrics

Information theoretic leakage metrics quantify the amount of information about a private random variable $X$ that is leaked through a correlated revealed variable $Y$. They can be used to evaluate the privacy of a system in which an adversary, from whom we want to keep $X$ private, is given access to $Y$. Global information theoretic leakage metrics quantify the overall amount of information leaked upon observing $Y$, whilst their pointwise counterparts define leakage as a function of the particular realisation $y$ that the adversary sees, and thus can be viewed as random variables. We consider an adversary who observes a large number of independent identically distributed realisations of $Y$. We formalise the essential asymptotic behaviour of an information theoretic leakage metric, considering in turn what this means for pointwise and global metrics. With the resulting requirements in mind, we take an axiomatic approach to defining a set of pointwise leakage metrics, as well as a set of global leakage metrics that are constructed from them. The global set encompasses many known measures including mutual information, Sibson mutual information, Arimoto mutual information, maximal leakage, min entropy leakage, $f$-divergence metrics, and g-leakage. We prove that both sets follow the desired asymptotic behaviour. Finally, we derive composition theorems which quantify the rate of privacy degradation as an adversary is given access to a large number of independent observations of $Y$. It is found that, for both pointwise and global metrics, privacy degrades exponentially with increasing observations for the adversary, at a rate governed by the minimum Chernoff information between distinct conditional channel distributions. This extends the work of Wu et al. (2024), who have previously found this to be true for certain known metrics, including some that fall into our more general set.

arxiv.org

Convergence of Markov Chains for Constant Step-size Stochastic Gradient Descent with Separable Functions arxiv.org/abs/2409.12243

Convergence of Markov Chains for Constant Step-size Stochastic Gradient Descent with Separable Functions

Stochastic gradient descent (SGD) is a popular algorithm for minimizing objective functions that arise in machine learning. For constant step-sized SGD, the iterates form a Markov chain on a general state space. Focusing on a class of separable (non-convex) objective functions, we establish a "Doeblin-type decomposition," in that the state space decomposes into a uniformly transient set and a disjoint union of absorbing sets. Each of the absorbing sets contains a unique invariant measure, with the set of all invariant measures being the convex hull. Moreover the set of invariant measures are shown to be global attractors to the Markov chain with a geometric convergence rate. The theory is highlighted with examples that show: (1) the failure of the diffusion approximation to characterize the long-time dynamics of SGD; (2) the global minimum of an objective function may lie outside the support of the invariant measures (i.e., even if initialized at the global minimum, SGD iterates will leave); and (3) bifurcations may enable the SGD iterates to transition between two local minima. Key ingredients in the theory involve viewing the SGD dynamics as a monotone iterated function system and establishing a "splitting condition" of Dubins and Freedman 1966 and Bhattacharya and Lee 1988.

arxiv.org

Universal localizations, Atiyah conjectures and graphs of groups arxiv.org/abs/2409.12268

Universal localizations, Atiyah conjectures and graphs of groups

Let $G$ be a countable group that is the fundamental group of a graph of groups with finite edge groups and vertex groups that satisfy the strong Atiyah conjecture over $K \subseteq \mathbb{C}$ a field closed under complex conjugation. Assume that the orders of finite subgroups of $G$ are bounded above. We show that $G$ satisfies the strong Atiyah conjecture over $K$. In particular, this implies that the strong Atiyah conjecture is closed under free products. Moreover, we prove that the $\ast$-regular closure of $K[G]$ in $\mathcal{U}(G)$, $\mathcal{R}_{\small K[G]}$, is a universal localization of the graph of rings associated to the graph of groups, where the rings are the corresponding $\ast$-regular closures. As a result, we obtain that the algebraic and center-valued Atiyah conjecture over $K$ are also closed under the graph of groups construction provided that the edge groups are finite. We also infer some consequences on the structure of the $K_0$ and $K_1$-groups of $\mathcal{R}_{\small K[G]}$. The techniques developed allow us to prove that $K[G]$ fulfills the strong, algebraic and center-valued Atiyah conjectures and that $\mathcal{R}_{\small K[G]}$ is the universal localization of $K[G]$ over the set of all matrices that become invertible in $\mathcal{U}(G)$ if $G$ lies in a certain class of groups $\mathcal{T}_{\small \mathcal{VLI}}$, which contains in particular virtually-{locally indicable} groups that are the fundamental group of a graph of virtually free groups.

arxiv.org

Percolation at the uniqueness threshold via subgroup relativization arxiv.org/abs/2409.12283

Percolation at the uniqueness threshold via subgroup relativization

We study percolation on nonamenable groups at the uniqueness threshold $p_u$, the critical value that separates the phase in which there are infinitely many infinite clusters from the phase in which there is a unique infinite cluster. The number of infinite clusters at $p_u$ itself is a subtle question, depending on the choice of group, with only a relatively small number of examples understood. In this paper, we do the following: 1. Prove non-uniqueness at $p_u$ in a new class of examples, namely those groups that contain an amenable, $wq$-normal subgroup of exponential growth. Concrete new examples to which this result applies include lamplighters over nonamenable base groups. 2. Prove a co-heredity property of a certain strong form of non-uniqueness at $p_u$, stating that this property is inherited from a $wq$-normal subgroup to the entire group. Remarkably, this co-heredity property is the same as that proven for the vanishing of the first $\ell^2$ Betti number by Peterson and Thom (Invent. Math. 2011), supporting the conjecture that the two properties are equivalent. Our proof is based on the method of subgroup relativization, and relies in particular on relativized versions of uniqueness monotonicity, the equivalence of non-uniqueness and connectivity decay, the sharpness of the phase transition, and the Burton-Keane theorem. As a further application of the relative Burton-Keane theorem, we resolve a question of Lyons and Schramm (Ann. Probab. 1999) concerning intersections of random walks with percolation clusters.

arxiv.org

Series expansions for SPDEs with symmetric $\alpha$-stable L\'evy noise arxiv.org/abs/2409.12286

Series expansions for SPDEs with symmetric $α$-stable Lévy noise

In this article, we examine a stochastic partial differential equation (SPDE) driven by a symmetric $α$-stable (S$α$S) Lévy noise, that is multiplied by a linear function $σ(u)=u$ of the solution. The solution is interpreted in the mild sense. For this models, in the case of the Gaussian noise, the solution has an explicit Wiener chaos expansion, and is studied using tools from Malliavin calculus. These tools cannot be used for an infinite-variance Lévy noise. In this article, we provide sufficient conditions for the existence of a solution, and we give an explicit series expansion of this solution. To achieve this, we use the multiple stable integrals, which were developed in Samorodnitsky and Taqqu (1990, 1991), and originate from the LePage series representation of the noise. To give a meaning to the stochastic integral which appears in the definition of solution, we embed the space-time Lévy noise into a Lévy basis, and use the stochastic integration theory (Bichteler and Jacod 1983, Bichteler 2002) with respect to this object, as in other studies of SPDEs with heavy-tailed noise: Chong (2017a), Chong (2017b), Chong, Dalang and Humeau (2019). As applications, we consider the heat and wave equations with linear multiplicative noise, also called the parabolic/hyperbolic Anderson models.

arxiv.org
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