This paper introduces a two-stage generative AI (GenAI) framework tailored for temporal spectrum cartography in low-altitude economy networks (LAENets). LAENets, characterized by diverse aerial devices such as UAVs, rely heavily on wireless communication technologies while facing challenges, including spectrum congestion and dynamic environmental interference. Traditional spectrum cartography methods have limitations in handling the temporal and spatial complexities inherent to these networks. Addressing these challenges, the proposed framework first employs a Reconstructive Masked Autoencoder (RecMAE) capable of accurately reconstructing spectrum maps from sparse and temporally varying sensor data using a novel dual-mask mechanism. This approach significantly enhances the precision of reconstructed radio frequency (RF) power maps. In the second stage, the Multi-agent Diffusion Policy (MADP) method integrates diffusion-based reinforcement learning to optimize the trajectories of dynamic UAV sensors. By leveraging temporal-attention encoding, this method effectively manages spatial exploration and exploitation to minimize cumulative reconstruction errors. Extensive numerical experiments validate that this integrated GenAI framework outperforms traditional interpolation methods and deep learning baselines by achieving 57.35% and 88.68% reconstruction error reduction, respectively. The proposed trajectory planner substantially improves spectrum map accuracy, reconstruction stability, and sensor deployment efficiency in dynamically evolving low-altitude environments.
Spectrum cartography (SC) focuses on estimating the radio power propagation map of multiple emitters across space and frequency using limited sensor measurements. Recent advances in SC have shown that leveraging learned deep generative models (DGMs) as structural constraints yields state-of-the-art performance. By harnessing the expressive power of neural networks, these structural "priors" capture intricate patterns in radio maps. However, training DGMs requires substantial data, which is not always available, and distribution shifts between training and testing data can further degrade performance. To address these challenges, this work proposes using untrained neural networks (UNNs) for SC. UNNs, commonly applied in vision tasks to represent complex data without training, encode structural information of data in neural architectures. In our approach, a custom-designed UNN represents radio maps under a spatio-spectral domain factorization model, leveraging physical characteristics to reduce sample complexity of SC. Experiments show that the method achieves performance comparable to learned DGM-based SC, without requiring training data.
Radio map estimation (RME), also known as spectrum cartography, aims to reconstruct the strength of radio interference across different domains (e.g., space and frequency) from sparsely sampled measurements. To tackle this typical inverse problem, state-of-the-art RME methods rely on handcrafted or data-driven structural information of radio maps. However, the former often struggles to model complex radio frequency (RF) environments and the latter requires excessive training -- making it hard to quickly adapt to in situ sensing tasks. This work presents a spatio-spectral RME approach based on plug-and-play (PnP) denoising, a technique from computational imaging. The idea is to leverage the observation that the denoising operations of signals like natural images and radio maps are similar -- despite the nontrivial differences of the signals themselves. Hence, sophisticated denoisers designed for or learned from natural images can be directly employed to assist RME, avoiding using radio map data for training. Unlike conventional PnP methods that operate directly in the data domain, the proposed method exploits the underlying physical structure of radio maps and proposes an ADMM algorithm that denoises in a latent domain. This design significantly improves computational efficiency and enhances noise robustness. Theoretical aspects, e.g., recoverability of the complete radio map and convergence of the ADMM algorithm are analyzed. Synthetic and real data experiments are conducted to demonstrate the effectiveness of our approach.
In Spectrum cartography (SC), the generation of exposure maps for radio frequency electromagnetic fields (RF-EMF) spans dimensions of frequency, space, and time, which relies on a sparse collection of sensor data, posing a challenging ill-posed inverse problem. Cartography methods based on models integrate designed priors, such as sparsity and low-rank structures, to refine the solution of this inverse problem. In our previous work, EMF exposure map reconstruction was achieved by Generative Adversarial Networks (GANs) where physical laws or structural constraints were employed as a prior, but they require a large amount of labeled data or simulated full maps for training to produce efficient results. In this paper, we present a method to reconstruct EMF exposure maps using only the generator network in GANs which does not require explicit training, thus overcoming the limitations of GANs, such as using reference full exposure maps. This approach uses a prior from sensor data as Local Image Prior (LIP) captured by deep convolutional generative networks independent of learning the network parameters from images in an urban environment. Experimental results show that, even when only sparse sensor data are available, our method can produce accurate estimates.
Spectrum cartography (SC), also known as radio map estimation (RME), aims at crafting multi-domain (e.g., frequency and space) radio power propagation maps from limited sensor measurements. While early methods often lacked theoretical support, recent works have demonstrated that radio maps can be provably recovered using low-dimensional models -- such as the block-term tensor decomposition (BTD) model and certain deep generative models (DGMs) -- of the high-dimensional multi-domain radio signals. However, these existing provable SC approaches assume that sensors send real-valued (full-resolution) measurements to the fusion center, which is unrealistic. This work puts forth a quantized SC framework that generalizes the BTD and DGM-based SC to scenarios where heavily quantized sensor measurements are used. A maximum likelihood estimation (MLE)-based SC framework under a Gaussian quantizer is proposed. Recoverability of the radio map using the MLE criterion are characterized under realistic conditions, e.g., imperfect radio map modeling and noisy measurements. Simulations and real-data experiments are used to showcase the effectiveness of the proposed approach.
Radio maps characterize quantities of interest in radio communication environments, such as the received signal strength and channel attenuation, at every point of a geographical region. Radio map estimation typically entails interpolative inference based on spatially distributed measurements. In this tutorial article, after presenting some representative applications of radio maps, the most prominent radio map estimation methods are discussed. Starting from simple regression, the exposition gradually delves into more sophisticated algorithms, eventually touching upon state-of-the-art techniques. To gain insight into this versatile toolkit, illustrative toy examples will also be presented.
Spectrum cartography constructs maps of metrics such as channel gain or received signal power across a geographic area of interest using spatially distributed sensor measurements. Applications of these maps include network planning, interference coordination, power control, localization, and cognitive radios to name a few. Since existing spectrum cartography techniques require accurate estimates of the sensor locations, their performance is drastically impaired by multipath affecting the positioning pilot signals, as occurs in indoor or dense urban scenarios. To overcome such a limitation, this paper introduces a novel paradigm for spectrum cartography, where estimation of spectral maps relies on features of these positioning signals rather than on location estimates. Specific learning algorithms are built upon this approach and offer a markedly improved estimation performance than existing approaches relying on localization, as demonstrated by simulation studies in indoor scenarios.
The spectrum cartography (SC) technique constructs multi-domain (e.g., frequency, space, and time) radio frequency (RF) maps from limited measurements, which can be viewed as an ill-posed tensor completion problem. Model-based cartography techniques often rely on handcrafted priors (e.g., sparsity, smoothness and low-rank structures) for the completion task. Such priors may be inadequate to capture the essence of complex wireless environments -- especially when severe shadowing happens. To circumvent such challenges, offline-trained deep neural models of radio maps were considered for SC, as deep neural networks (DNNs) are able to "learn" intricate underlying structures from data. However, such deep learning (DL)-based SC approaches encounter serious challenges in both off-line model learning (training) and completion (generalization), possibly because the latent state space for generating the radio maps is prohibitively large. In this work, an emitter radio map disaggregation-based approach is proposed, under which only individual emitters' radio maps are modeled by DNNs. This way, the learning and generalization challenges can both be substantially alleviated. Using the learned DNNs, a fast nonnegative matrix factorization-based two-stage SC method and a performance-enhanced iterative optimization algorithm are proposed. Theoretical aspects -- such as recoverability of the radio tensor, sample complexity, and noise robustness -- under the proposed framework are characterized, and such theoretical properties have been elusive in the context of DL-based radio tensor completion. Experiments using synthetic and real-data from indoor and heavily shadowed environments are employed to showcase the effectiveness of the proposed methods.
Spectrum maps, which provide RF spectrum metrics such as power spectral density for every location in a geographic area, find numerous applications in wireless communications such as interference control, spectrum management, resource allocation, and network planning to name a few. Spectrum cartography techniques construct these maps from a collection of measurements collected by spatially distributed sensors. Due to the nature of the propagation of electromagnetic waves, spectrum maps are complicated functions of the spatial coordinates. For this reason, model-free approaches have been preferred. However, all existing schemes rely on some interpolation algorithm unable to learn from data. This work proposes a novel approach to spectrum cartography where propagation phenomena are learned from data. The resulting algorithms can therefore construct a spectrum map from a significantly smaller number of measurements than existing schemes since the spatial structure of shadowing and other phenomena is previously learned from maps in other environments. Besides the aforementioned new paradigm, this is also the first work to perform spectrum cartography with deep neural networks. To exploit the manifold structure of spectrum maps, a deep network architecture is proposed based on completion autoencoders.
Spectrum cartography aims at estimating power propagation patterns over a geographical region across multiple frequency bands (i.e., a radio map)---from limited samples taken sparsely over the region. Classic cartography methods are mostly concerned with recovering the aggregate radio frequency (RF) information while ignoring the constituents of the radio map---but fine-grained emitter-level RF information is of great interest. In addition, many existing cartography methods work explicitly or implicitly assume random spatial sampling schemes that may be difficult to implement, due to legal/privacy/security issues. The theoretical aspects (e.g., identifiability of the radio map) of many existing methods are also unclear. In this work, we propose a joint radio map recovery and disaggregation method that is based on coupled block-term tensor decomposition. Our method guarantees identifiability of the individual radio map of \textit{each emitter} (thereby that of the aggregate radio map as well), under realistic conditions. The identifiability result holds under a large variety of geographical sampling patterns, including a number of pragmatic systematic sampling strategies. We also propose effective optimization algorithms to carry out the formulated radio map disaggregation problems. Extensive simulations are employed to showcase the effectiveness of the proposed approach.
Signal processing is rich in inherently continuous and often nonlinear applications, such as spectral estimation, optical imaging, and super-resolution microscopy, in which sparsity plays a key role in obtaining state-of-the-art results. Coping with the infinite dimensionality and non-convexity of these problems typically involves discretization and convex relaxations, e.g., using atomic norms. Nevertheless, grid mismatch and other coherence issues often lead to discretized versions of sparse signals that are not sparse. Even if they are, recovering sparse solutions using convex relaxations requires assumptions that may be hard to meet in practice. What is more, problems involving nonlinear measurements remain non-convex even after relaxing the sparsity objective. We address these issues by directly tackling the continuous, nonlinear problem cast as a sparse functional optimization program. We prove that when these problems are non-atomic, they have no duality gap and can therefore be solved efficiently using duality and~(stochastic) convex optimization methods. We illustrate the wide range of applications of this approach by formulating and solving problems from nonlinear spectral estimation and robust classification.
This chapter deals with decentralized learning algorithms for in-network processing of graph-valued data. A generic learning problem is formulated and recast into a separable form, which is iteratively minimized using the alternating-direction method of multipliers (ADMM) so as to gain the desired degree of parallelization. Without exchanging elements from the distributed training sets and keeping inter-node communications at affordable levels, the local (per-node) learners consent to the desired quantity inferred globally, meaning the one obtained if the entire training data set were centrally available. Impact of the decentralized learning framework to contemporary wireless communications and networking tasks is illustrated through case studies including target tracking using wireless sensor networks, unveiling Internet traffic anomalies, power system state estimation, as well as spectrum cartography for wireless cognitive radio networks.