# Deep Spectrum Cartography: Completing Radio Map Tensors Using Learned Neural Models

arXiv: https://arxiv.org/abs/2105.00177 · HTML: https://mlsanity.com/paper/2105.00177/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 52 references that are in the map.
- First posted on arXiv 1 May 2021.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications](https://mlsanity.com/paper/1803.01257/) (2018)
- [On Identifiability of Nonnegative Matrix Factorization](https://mlsanity.com/paper/1709.00614/) (2017)
- [On Tensor Completion via Nuclear Norm Minimization](https://mlsanity.com/paper/1405.1773/) (2014)
- [Spectrum Cartography via Coupled Block-Term Tensor Decomposition](https://mlsanity.com/paper/1911.12468/) (2019)
- [Learning Power Spectrum Maps from Quantized Power Measurements](https://mlsanity.com/paper/1606.02679/) (2016)
- [Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective](https://mlsanity.com/paper/2006.08183/) (2020)
- [Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization](https://mlsanity.com/paper/1302.4385/) (2013)
- [Block-Randomized Stochastic Proximal Gradient for Low-Rank Tensor Factorization](https://mlsanity.com/paper/1901.05529/) (2019)
- [Self-Dictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search are Related](https://mlsanity.com/paper/1409.4320/) (2014)
- [Tensor Completion from Regular Sub-Nyquist Samples](https://mlsanity.com/paper/1903.00435/) (2019)
- [Data-Driven Spectrum Cartography via Deep Completion Autoencoders](https://mlsanity.com/paper/1911.12810/) (2019)

## Built on by (most cited first, 14 of 14)

- [RadioDiff-3D: A 3D×3D Radio Map Dataset and Generative Diffusion Based Benchmark for 6G Environment-Aware Communication](https://mlsanity.com/paper/2507.12166/) (2025)
- [Generative AI on SpectrumNet: An Open Benchmark of Multiband 3D Radio Maps](https://mlsanity.com/paper/2408.15252/) (2024)
- [Quantized Radio Map Estimation Using Tensor and Deep Generative Models](https://mlsanity.com/paper/2303.01770/) (2023)
- [Constructing Indoor Region-based Radio Map without Location Labels](https://mlsanity.com/paper/2308.16759/) (2023)
- [Radio Map Estimation via Latent Domain Plug-and-Play Denoising](https://mlsanity.com/paper/2501.13472/) (2025)
- [Sparse Gain Radio Map Reconstruction With Geometry Priors and Uncertainty-Guided Measurement Selection](https://mlsanity.com/paper/2604.05788/) (2026)
- [Blind Radio Mapping via Spatially Regularized Bayesian Trajectory Inference](https://mlsanity.com/paper/2512.13701/) (2025)
- [Generative Spectrum Cartography: Unified Reconstruction and Active Sensing via Diffusion Models](https://mlsanity.com/paper/2512.20108/) (2025)
- [When Bayesian Tensor Completion Meets Multioutput Gaussian Processes: Functional Universality and Rank Learning](https://mlsanity.com/paper/2512.21486/) (2025)
- [Geometry-Aware Cross-Height Channel Knowledge Map Prediction for UAV-Assisted Communications With Uncertainty-Guided 3D Sensing](https://mlsanity.com/paper/2607.00887/) (2026)
- [Deep Multi-Emitter Spectrum Occupancy Mapping that is Robust to the Number of Sensors, Noise and Threshold](https://mlsanity.com/paper/2212.10444/) (2022)
- [Downlink MIMO Channel Estimation from Bits: Recoverability and Algorithm](https://mlsanity.com/paper/2411.16043/) (2024)
- [RadioGen3D: 3D Radio Map Generation via Adversarial Learning on Large-Scale Synthetic Data](https://mlsanity.com/paper/2602.18744/) (2026)
- [Active Learning for Low-Altitude Radio Map Construction via Plug-and-Play Flow Matching](https://mlsanity.com/paper/2609.27486/) (2026)

## Abstract (arXiv)

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.