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Reconstruction of SINR Maps from Sparse Measurements using Group Equivariant Non-Expansive Operators

Lorenzo Mario Amorosa, Francesco Conti, Nicola Quercioli, Flavio Zabini, Tayebeh Lotfi Mahyari, Yiqun Ge, Patrizio Frosini

TL;DR

The paper tackles reconstructing high-resolution SINR maps from extremely sparse measurements in next-generation wireless networks by introducing GENEOs, a class of group-equivariant non-expansive operators that embed domain symmetries directly into the reconstruction process. By combining traditional metrics like MSE with topological measures such as the 1-Wasserstein distance, the approach preserves the geometric structure of SINR maps while remaining robust to data scarcity. The authors construct a large pattern library and demonstrate, on realistic ray-tracing urban datasets for Munich and Paris, that GENEO-based reconstruction achieves competitive pixel-wise accuracy and dramatically better topological fidelity than strong baselines like 1-KNN and U-Net. This work highlights the practical value of symmetry-aware, topology-guided operators for generating structurally faithful radio maps essential for efficient network management under sparse sensing constraints.

Abstract

As sixth generation (6G) wireless networks evolve, accurate signal-to-interference-noise ratio (SINR) maps are becoming increasingly critical for effective resource management and optimization. However, acquiring such maps at high resolution is often cost-prohibitive, creating a severe data scarcity challenge. This necessitates machine learning (ML) approaches capable of robustly reconstructing the full map from extremely sparse measurements. To address this, we introduce a novel reconstruction framework based on Group Equivariant Non-Expansive Operators (GENEOs). Unlike data-hungry ML models, GENEOs are low-complexity operators that embed domain-specific geometric priors, such as translation invariance, directly into their structure. This provides a strong inductive bias, enabling effective reconstruction from very few samples. Our key insight is that for network management, preserving the topological structure of the SINR map, such as the geometry of coverage holes and interference patterns, is often more critical than minimizing pixel-wise error. We validate our approach on realistic ray-tracing-based urban scenarios, evaluating performance with both traditional statistical metrics (mean squared error (MSE)) and, crucially, a topological metric (1-Wasserstein distance). Results show that while maintaining competitive MSE, our method dramatically outperforms established ML baselines in topological fidelity. This demonstrates the practical advantage of GENEOs for creating structurally accurate SINR maps that are more reliable for downstream network optimization tasks.

Reconstruction of SINR Maps from Sparse Measurements using Group Equivariant Non-Expansive Operators

TL;DR

The paper tackles reconstructing high-resolution SINR maps from extremely sparse measurements in next-generation wireless networks by introducing GENEOs, a class of group-equivariant non-expansive operators that embed domain symmetries directly into the reconstruction process. By combining traditional metrics like MSE with topological measures such as the 1-Wasserstein distance, the approach preserves the geometric structure of SINR maps while remaining robust to data scarcity. The authors construct a large pattern library and demonstrate, on realistic ray-tracing urban datasets for Munich and Paris, that GENEO-based reconstruction achieves competitive pixel-wise accuracy and dramatically better topological fidelity than strong baselines like 1-KNN and U-Net. This work highlights the practical value of symmetry-aware, topology-guided operators for generating structurally faithful radio maps essential for efficient network management under sparse sensing constraints.

Abstract

As sixth generation (6G) wireless networks evolve, accurate signal-to-interference-noise ratio (SINR) maps are becoming increasingly critical for effective resource management and optimization. However, acquiring such maps at high resolution is often cost-prohibitive, creating a severe data scarcity challenge. This necessitates machine learning (ML) approaches capable of robustly reconstructing the full map from extremely sparse measurements. To address this, we introduce a novel reconstruction framework based on Group Equivariant Non-Expansive Operators (GENEOs). Unlike data-hungry ML models, GENEOs are low-complexity operators that embed domain-specific geometric priors, such as translation invariance, directly into their structure. This provides a strong inductive bias, enabling effective reconstruction from very few samples. Our key insight is that for network management, preserving the topological structure of the SINR map, such as the geometry of coverage holes and interference patterns, is often more critical than minimizing pixel-wise error. We validate our approach on realistic ray-tracing-based urban scenarios, evaluating performance with both traditional statistical metrics (mean squared error (MSE)) and, crucially, a topological metric (1-Wasserstein distance). Results show that while maintaining competitive MSE, our method dramatically outperforms established ML baselines in topological fidelity. This demonstrates the practical advantage of GENEOs for creating structurally accurate SINR maps that are more reliable for downstream network optimization tasks.

Paper Structure

This paper contains 25 sections, 3 theorems, 21 equations, 18 figures.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND ON SIGNAL RECONSTRUCTION
  3. MAIN CONTRIBUTION
  4. ORGANIZATION
  5. MATHEMATICAL BACKGROUND AND STATE OF THE ART
  6. BACKGROUND ON TOPOLOGICAL DATA ANALYSIS
  7. DEEP LEARNING AND TOPOLOGICAL DATA ANALYSIS APPLIED TO SIGNAL RECONSTRUCTION
  8. THEORETICAL FOUNDATIONS OF GENEOS
  9. GENEOS FOR SIGNAL RECONSTRUCTION
  10. NUMERICAL RESULTS
  11. SIGNAL RECONSTRUCTION
  12. SCENARIOS
  13. Munich
  14. Paris
  15. GENEO IMPLEMENTATION
  16. ...and 10 more sections

Key Result

Proposition 1

$0 \le \hat{c}_{S,P}(x,y) \le \mathcal{A}_{S,P}(x,y) \le 1$ for every $(x,y) \in \mathbb{R}^2$.

Figures (18)

  • Figure 1: Example of a 2D signal and its corresponding persistence diagram, with (a) a synthetic 2D signal with pixel intensities varying from 1 to 10 and (b) its persistence diagram. In the persistence diagram, red points indicate the birth and death of connected components, while blue points indicate the birth and death of one-dimensional holes.
  • Figure 2: The persistence diagrams $\mathcal{D}_1$ and $\mathcal{D}_2$ are matched through their optimal $1$-Wasserstein matching.
  • Figure 3: Visualization of (a) $\hat{\varphi}$ values and (b) $\hat{\psi}$ values. In (a), each point $p$ such that $\hat{\psi}(p) = 1$ is colored based on $\hat{\varphi}(p) \in [0, 1]$; remaining areas are shown in white. In (b), the color of each point $p$ represents the intensity of $\hat{\psi}(p)$ which takes only binary values in this example. The color purple indicates the zero value, while the yellow indicates the ones.
  • Figure 4: Illustrative toy example, where we present (a) a ground truth $\varphi$, (b) its sampled version $\hat{\varphi}$, and (c)-(f) four patterns $h_1\chi, \dots, h_4\chi$.
  • Figure 5: Heatmaps of the coefficients $\,\hat{c}_{S,P_i}$ for $i=1,2,3,4$, where $S = (\hat{\varphi},\hat{\psi}) \quad\text{and}\quad P_i = (h_i,\chi) \,,$ with $\hat{\varphi},h_i,\chi$ as defined in Fig. \ref{['fig:toy']}. Sub-figures (a)–(d) correspond to $i=1,2,3,4$, respectively.
  • ...and 13 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 1
  • ...and 2 more