# 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

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Lorenzo Mario Amorosa, Francesco Conti, Nicola Quercioli, Flavio Zabini, Tayebeh Lotfi Mahyari, et al.. Reconstruction of SINR Maps from Sparse Measurements using Group Equivariant Non-Expansive Operators. 2026. hal-05456540

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

Lorenzo Mario Amorosa<fnref n="1" />, Member, IEEE, Francesco Conti<fnref n="2" />, Nicola Quercioli<fnref n="1" />,
Flavio Zabini<fnref n="1" />, Member, IEEE, Tayebeh Lotfi Mahyari<fnref n="3" />, Yiqun Ge<fnref n="3" />, and Patrizio Frosini<fnref n="4" />

Corresponding author: Flavio Zabini (email: flavio.zabini2@unibo.it).

This work has been carried out in the framework of the CNIT National Laboratory WiLab and the WiLab-Huawei Joint Innovation Center.

**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.

**INDEX TERMS** Group Equivariant Non-Expansive Operator (GENEO), Machine Learning (ML), Signal Reconstruction, Topological Data Analysis (TDA).

## I. INTRODUCTION

IN CURRENT and future communication systems such as sixth generation (6G) wireless networks, key functionalities like resource management and service provisioning rely heavily on the knowledge of spatially-varying quantities [1], including signal-to-interference-noise ratio (SINR) maps, which indicate the signal quality experienced by users in a wireless network. However, acquiring large volumes of such data is often prohibitively expensive in terms of resource consumption, including power, spectrum, and time [2]. Therefore, it becomes crucial to reconstruct these quantities from a significantly smaller set of measurements.

At the same time, machine learning (ML) is increasingly being leveraged in next-generation wireless networks for tasks such as radio resource scheduling, network traffic forecasting, or beamforming [3]. Neural networks, though powerful, typically involve a large number of parameters that must be learned from data. Introducing a new class of operators capable of performing similar functions with a substantially smaller number of trainable parameters would represent a key advancement. This motivates the idea of reconstructing spatial signals - such as SINR maps of a wireless system in an urban environment - using a novel class

<footnote n="1">¹Department of Electrical, Electronic and Information Engineering (DEI), "Guglielmo Marconi", University of Bologna & WiLab - National Wireless Communication Laboratory (CNIT), Bologna, Italy.</footnote>

<footnote n="2">²Institut national de recherche en sciences et technologies du numérique (Inria), Université Côte d'Azur, France.</footnote>

<footnote n="3">³Huawei Technologies Canada, Ottawa, ON, Canada.</footnote>

<footnote n="4">⁴Department of Computer Science, University of Pisa, Pisa, Italy.</footnote>
of operators known as Group Equivariant Non-Expansive Operators (GENEOs) [4]–[8].

A. BACKGROUND ON SIGNAL RECONSTRUCTION

The problem of reconstructing signals from sampled data has long been studied, initially under the assumption of uniform sampling of band-limited signals, culminating in the classical Whittaker-Kotelnikov-Shannon (WKS) theorem [9]–[11]. When the sampling rate exceeds twice the maximum signal frequency (Nyquist rate), perfect reconstruction is possible via linear time-invariant interpolation. This fundamental result has been extended to multidimensional domains through lattice sampling, as demonstrated by Petersen and Middleton [12]. A more comprehensive theoretical framework, including reconstruction mean square error (MSE) analysis for lattice sampling, is provided in [13].

However, when the goal is to reconstruct a spatially varying physical quantity from a given set of spatial samples, the assumption of regular sampling becomes unrealistic. Practical scenarios, such as wireless sensor networks (WSN) (where sensors measure quantities like temperature, air pollutant concentration, or magnetic fields), or radio mobile systems (where the aim may be to reconstruct SINR maps based on received power measurements from mobile devices whose spatial locations are typically inherently random) involve highly irregular sampling distributions that cannot be represented by a lattice. The fundamental theoretical results for signal reconstruction from irregular spatial samples include the well-known contributions of Landau [14], [15], who established necessary conditions for perfect reconstruction in terms of average sampling density, and the lesser-known but significant theorem by Levinson [16], which, as early as 1940, provided sufficient conditions, though in a purely mathematical setting not yet linked to sampling theory. For realistic WSN or radio-mobile scenarios, where sample positions are perturbed and signals are not strictly band-limited, the reconstruction MSE has been analyzed in [17], under the assumption that the signal lies in a shift invariant (SI) space and that linear interpolation is applied.

Nonetheless, both the theoretical density conditions required for exact reconstruction (a multidimensional generalization of the Nyquist rate) and the practical results on reconstruction MSE depend on the specific realization of the random variables modeling sample positions. When these positions are modeled as a point process, and other elements - such as measurement errors or localization inaccuracies - introduce randomness, perfect reconstruction must be replaced by “sufficiently good” reconstruction, evaluated via metrics like MSE. In this context, a statistical description based on stochastic geometry becomes essential. The simplest example is the Poisson point process (PPP), where points are independently distributed. For PPP-based sampling, analytical expressions for the reconstruction MSE have been derived in [18]. Further studies [19] have explored the impact of practical factors such as quantization, energy

consumption, and communication capacity. More recently, other classes of point processes have been investigated to account for phenomena such as repulsion between samples [20] and clustering [21]. A recent work that combines classical sampling theory (i.e., signal reconstruction via linear convolution) with the theory of spatial point processes - comprehensively addressing the effects of different point process classes (determinantal, Cox, etc.) - can be found in [22].

From a practical standpoint, the results of these stochastic sampling studies can be summarized as follows: ignoring the second-order effects induced by the statistics of the point process (e.g., attraction or repulsion among points), the reconstruction MSE can, under certain regularity conditions, approach zero as a function of the inverse of $\lambda/(2B)^d$, where $\lambda$ is the intensity (i.e., the average number of points per unit space), $B$ is the per-dimension bandwidth of the signal, and $d$ is the spatial dimensionality. This result proves useful when an abundance of samples is available (high $\lambda$) or when the signal varies slowly in space (small $2B$). In contrast, when resources are limited, it becomes essential to achieve acceptable reconstruction quality using as few samples as possible. In such cases, it is necessary to make prior assumptions about the signal - for example, that it belongs to a union of subspaces, that it is a Gaussian random process, or that it has a sparse representation in some basis. Moreover, the reconstruction technique may need to go beyond simple linear interpolation.

Since the early 2000s, the classical linear reconstruction problem has been extended to recover continuous-time functions from sequences of irregular and corrupted discrete samples [23]. In [24], using a Hilbert space formulation, the Shannon sampling procedure is reinterpreted as an orthogonal projection onto the subspace of band-limited functions and extended to shift-invariant function spaces. Geometric approaches, such as oblique dual frame vectors, have been shown to achieve MSE-minimizing reconstruction - and become exact when Levinson's conditions are satisfied - in various practical settings including noise, quantization, and nonlinearities [25]–[29]. More advanced interpolation methods that outperform consistent linear interpolators have been proposed [30], along with iterative algorithms capable of perfect reconstruction in nonlinear cases [31]. However, extending these results to the multidimensional domain remains a significant open challenge.

When the signal is modeled as a Gaussian process, several reconstruction strategies become feasible, often requiring fewer samples than those dictated by classical sampling theory [32]–[37]. Compressed sensing, originally developed for finite-dimensional vectors [38], [39], has also been extended to infinite-dimensional and continuous-time signals with sparse representations. This allows for significant reductions in sampling density while maintaining a target reconstruction MSE [40]–[43]. However, these methods rely on the fundamental assumption that the signal resides in
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an SI subspace or its generalizations (e.g., a union of subspaces), effectively extending the key idea behind the WKS theorem. It is important to note that assuming a basis function representation implicitly defines the topology of the function space (e.g., the Paley-Wiener space for band-limited functions).

## B. MAIN CONTRIBUTION

In this work, we propose GENEOS as a completely novel approach for signal reconstruction. GENEOS were originally developed within topological data analysis (TDA) and central to the theory of group equivariant operators (GEOs) [5]. GEOs are of particular interest because they provide a mathematical formalization of the concept of an observer as a functional agent and, in particular, enable a topological-geometric representation of neural networks. Within the GEO framework, GENEOS play a crucial role, as they allow for the definition of a distance between GEOs without requiring the operators to share the same domains and codomains [8]. Moreover, they exhibit a useful approximation property via finite sets of operators.

Our main contributions are summarized as follows:

1) We propose a GENEO-based approach for reconstructing radio maps in urban wireless networks, including SINR, from extremely sparse samples.

2) We integrate TDA-derived metrics alongside standard statistical metrics to better capture geometric and spatial characteristics of radio signals.

3) We provide a comprehensive empirical evaluation using realistic wireless scenarios generated with the Sionna RT ray tracing tool [44], showing competitive performance with respect to state-of-the-art ML methods.

## C. ORGANIZATION

This paper is organized as follows. In Sec. II, we introduce theoretical concepts about TDA, related work on signal reconstruction, and mathematical foundations of GENEOS. In Sec. III, we develop our proposed GENEOS for signal reconstruction. In Sec. IV, we numerically evaluate our approach in urban scenarios leveraging ray-tracing simulations in Sionna RT and comparing with relevant ML baselines. Finally, Sec. V provides the conclusions of the paper.

# II. MATHEMATICAL BACKGROUND AND STATE OF THE ART

## A. BACKGROUND ON TOPOLOGICAL DATA ANALYSIS

TDA [45] is a branch of applied mathematics that studies the shape of digital data and leverages this information across various fields of artificial intelligence. Unlike traditional statistical methods, TDA applies concepts from topology to analyze data, enabling the investigation of global structures and properties that remain invariant under continuous deformations.

In practice, TDA analyzes connected components, holes, and higher-dimensional cavities. The mathematical foundation of TDA is persistent homology [46], with its precursor being the size function [47]. Persistent homology not only identifies topological features in the data - such as the connected components and cavities mentioned above - but also quantifies their persistence across multiple scales. A common assumption is that features with longer persistence are more relevant for characterizing the underlying shape of the data.

TDA has achieved remarkable success in real-world applications, spanning medicine [48]–[50], materials science [51], [52], finance [53], [54], computer vision [55], and more. Recently, TDA has also been applied to wireless networks to study coverage properties [56], [57].

Since TDA focuses on the shape of data, it is well-suited for analyzing wireless networks, which often exhibit geometric patterns in signal propagation. For this reason, a metric from the TDA framework - the 1-Wasserstein distance (1-W) defined in Definition 1 - is employed alongside the MSE, to evaluate how well the original signal is preserved during reconstruction. While MSE captures differences in signal intensity, the 1-Wasserstein distance assesses the preservation of the signal's shape.

The concept of GENEOS was originally introduced to reduce the invariance of topological descriptors in the context of classical TDA. Such invariance, in fact, holds for any possible homeomorphism applied to the data domain and results in a notion of shape equivalence that is too generic for many applications. The use of GENEOS makes it possible to restrict this invariance to the action of transformation groups chosen arbitrarily [58]. Subsequently, the theory of GENEOS has been applied in the field of ML as a geometric-topological technique for obtaining an interpretable approximation of functional operators and neural networks [5]–[8].

In the remainder of this section, we introduce the notion of persistence diagram to capture the concept of shape, briefly recalling fundamental definitions and concepts of TDA. We refer the interested reader to [45], [46] for further details. Formally, given a topological space or simplicial complex $X$, a filtration is a finite nested sequence of subspaces or subcomplexes $\emptyset = X_0 \subseteq X_1 \subseteq \dots \subseteq X_n = X$. For each $i = 0, \dots, n$ we can compute the homology $H(X_i)$ and keep track of how the respecting features persist during the filtration process. In particular, we say that a topological feature *borns* at time $i$ if it appears in $H(X_i)$ but not in $H(X_{i-1})$. Similarly, a topological feature *dies* at time $j$ if it appears in $H(X_j)$ but not in $H(X_{j+1})$. Informally, we can track the evolution of components and holes in the sublevels of the filtration, keeping record of their creation and disappearance. We collect the pairs (birth time, death time) of components and holes in the so-called persistence diagram (PD). Mathematically, a PD is a multiset, i.e. a set where each point has a multiplicity. In literature, all points
<figure><img src="image_8.png" /><figcaption>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.</figcaption></figure>

of the diagonal are added to the PD with infinite multiplicity for technical reasons.

In this work, the data considered are 2D signals, which can be naturally interpreted as real-valued functions $f$ defined on a topological space or a simplicial complex. The filtration is constructed considering sets of the form $X_t = f^{-1}((-\infty, t])$, where $t$ ranges in $\mathbb{R}$ and $f^{-1}((-\infty, t])$ is the inverse image of the interval $(-\infty, t]$. We refer to $\dot{X}_t$ as a sublevel set. As illustrated in Fig. 1, TDA provides a powerful framework for quantifying structural features in 2D signals through persistent homology. In Fig. 1(a) we visualize an example of synthetic 2D signal with pixel intensities varying from 1 to 10 and in Fig. 1(b) its persistence diagram. As the sublevel threshold $t$ increases, dark regions (intensity $\le t$) appear first and eventually merge or fill in. Two connected components are highlighted by light-blue dashed outlines, and two holes (enclosed bright regions) are marked by red dashed outlines. At $t = 1$, the first connected component (red dot in Fig. 1(b)) appears, and a hole (blue dot in Fig. 1(b)) appears when its surrounding boundary closes. A second connected component is born at $t = 3$ but cannot form a hole until the intensity-4 “barrier” pixel (circled in black) is added at $t = 4$. Once this barrier pixel activates, a new hole emerges. One connected component (the lower-right blob) merges into the first component at $t = 6$, causing its death in the $H_0$ diagram, while the two holes disappear at $t = 8$ and $t = 10$ as the last boundary pixels fill. Note that one component (the initial dark region) persists for all thresholds (an infinite-lifetime feature). Fig 1(b) shows the persistence diagram in (birth, death) coordinates: red and blue markers correspond to the $H_0$ and $H_1$ events described above. Features farther from the diagonal represent longer-lived topological structures.

We can endow the space of persistence diagrams with a metric that aims to quantify the topological differences.

<figure><img src="image_9.png" /><figcaption>FIGURE 2: The persistence diagrams $D_1$ and $D_2$ are matched through their optimal 1-Wasserstein matching.</figcaption></figure>

**Definition 1.** Given two persistence diagrams $D_1, D_2$ and $p \in \mathbb{N}$, the p-Wasserstein metric is

$$W_p(D_1, D_2) = \inf_{\gamma} \left( \sum_{x \in D_1} \|x - \gamma(x)\|_{\infty}^p \right)^{1/p}, \quad (1)$$

where $\gamma$ ranges over all bijections between $D_1$ and $D_2$.

We considered the 1-Wasserstein metric since it is particularly suitable for our analysis as it weights all features proportionally to their persistence, and maintains stability against small perturbations of input data. We show in Figure 2 the optimal matching between two persistence diagrams $D_1$ and $D_2$.

## B. DEEP LEARNING AND TOPOLOGICAL DATA ANALYSIS APPLIED TO SIGNAL RECONSTRUCTION

Traditional wireless signal reconstruction techniques often rely on spatial interpolation and model-based recovery. Classical methods such as inverse-distance weighting (IDW) or radial basis functions (RBF) interpolate radio maps from sparse samples, and geostatistical kriging has been widely used to fuse measurements into smooth coverage fields [59]. Compressive sensing and variational approaches have also been applied to channel reconstruction, exploiting sparsity or smoothness to recover signals from limited measurements [60]. For example, graph signal processing interprets the radio map as a signal on a graph and performs spectral interpolation; this can outperform local methods like IDW or RBF interpolation [61]. However, these approaches require strong prior assumptions (e.g. smoothness, known propagation models) and often oversmooth sharp features. In practice, traditional kriging or interpolation can be computationally expensive in high dimensions and may struggle to model complex multi-path or interference environments.

In recent years, data-driven neural methods have emerged as powerful alternatives. Convolutional neural networks (CNNs) have been used to predict radio propagation or coverage maps from environment inputs, essentially treating the problem as image-to-image regression [62]. Recurrent mod-
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els (e.g. long-short term memory (LSTM)) capture temporal correlations in dynamic scenarios: for instance, an LSTM-based architecture was shown to reconstruct time-varying road-side radio environment maps (REMs) for V2X vehicular networks with high accuracy [63]. Graph neural networks (GNNs) have been applied to REM completion by leveraging spatial connectivity: multi-source GNNs learn to fuse sparse spectra from monitoring stations into full reference signal received power (RSRP) or SINR maps, improving prediction in heterogeneous domains [64]. Self-supervised techniques have also been explored. For example, deep image prior treats the radio map as an image and fits an untrained neural network directly to the observed measurements, thereby reconstructing missing values without any external training data [65]. Neural architecture search (NAS) and semi-supervised learning can further improve performance with little data: a recent received signal strength (RSS)-mapping system uses NAS combined with self-training on unlabeled points, achieving lower error than traditional RBF, Kriging or manually designed nets under extreme sparsity [66]. These deep and graph-based approaches generally achieve higher fidelity than classical methods, especially with abundant data, but they tend to require large training sets, careful tuning, and may still produce unrealistic artifacts when data is very limited [67].

Complementary approaches model uncertainty and incorporate generative modeling. Bayesian and probabilistic deep networks have been used to quantify confidence in reconstructed maps, which is crucial for network optimization. Gaussian process regression, for example, provides uncertainty bounds in interpolating coverage maps, though it scales poorly with data size. More recently, conditional generative models treat map reconstruction as a data generation task: e.g. GUMBLE [68] introduces a Bayesian framework for uncertainty-aware generation of mobile data. GUMBLE can produce synthetic signal measurements conditioned on partial real samples, effectively augmenting sparse crowdsourced datasets. This line of work highlights the interplay between reconstruction and data augmentation: a generative model that fills in missing radio map data can be seen as performing reconstruction while also creating diverse samples for downstream learning. However, generative methods often lack hard constraints on physical consistency and may suffer from mode collapse or bias if the training data is unbalanced.

Another emerging trend is leveraging geometric and topological priors. TDA techniques have been applied to wireless signals to capture global structure: for instance, persistent homology and persistence images of radio frequency time-series have been used for modulation classification and anomaly detection [69]. These methods show that capturing the topology of signal features can improve interpretability. More broadly, group-equivariant learning has proven effective in vision and signal tasks: Cohen et al. [70] showed that neural networks designed to respect transformations (rotations, translations, scaling) can dramatically reduce sam-

ple complexity. Building on these ideas, GENEOS have been proposed as a theoretical framework for embedding symmetries and stability into learning [5], [71]. A GENEO is an operator commuting with a group action (e.g. shifting coordinates) and satisfying a 1-Lipschitz property. In practice, GENEO-based models (e.g. GENEOnet) have demonstrated that injecting equivariance and topological constraints yields compact, convex model spaces and good performance even with small training sets. In wireless signal mapping, equivariance could encode invariance of wireless signals under certain domain transformations (for example, translations of the coordinate frame) or known physical symmetries, while non-expansiveness enforces robustness to noise. By contrast, conventional CNNs or GNNs cannot incorporate by design diverse symmetries and therefore must rely on substantially larger datasets to achieve comparable generalization.

### C. THEORETICAL FOUNDATIONS OF GENEOS

We define an agent as a functional operator that transforms data preserving important invariances and symmetries and we formalize it as a GENEO. In our theoretical framework (see [5]), a data set is given by a set of bounded real-valued measurements on some suitable domain X:

$$ \Phi = \{ \varphi : X \to \mathbb{R} \} \subseteq \mathbb{R}_b^X, \quad (2) $$

where $\mathbb{R}_b^X$ is the set of all bounded real-valued functions on the set X. We can think of X as the space where one makes measurements, and of $\Phi$ as the set of admissible measurements. For example, a wireless signal can be represented as a function $\varphi$ from the real plane X to the real numbers.

The concept of symmetry is modeled as a group action. In our context, we consider the group action of planar rigid motions on signals. The consecutive application of two rigid motions is equivalent to applying their composition, and the action of the identity rigid motion leaves the signal unchanged. A data set $\Phi$ is naturally equipped with a group action:

$$ \rho: \Phi \times \mathrm{Aut}_{\Phi}(X) \to \Phi, \quad (\varphi, g) \mapsto \varphi g, \quad (3) $$

where $\varphi g$ is the usual function composition and $\mathrm{Aut}_{\Phi}(X)$ is the group of all bijections $g$ such that $\varphi g, \varphi g^{-1} \in \Phi$. Thus $\Phi$ is not just a set, but a set with an action of the group $\mathrm{Aut}_{\Phi}(X)$. To encode the symmetries of $\Phi$ induced by this action, we consider its perception pairs.

**Definition 2.** A perception pair is a pair $(\Phi, G)$ with $\Phi \subseteq \mathbb{R}_b^X$ and $G \subseteq \mathrm{Aut}_{\Phi}(X)$.

The choice of $G$ encodes certain symmetries of $\Phi$. GENEOS enable us to transform datasets (or perception pairs) while preserving symmetries and not increasing distances. In other words, we can say that in our framework data transformations are modelled as sets of GENEOS.

**Definition 3.** Consider two perception pairs $(\Phi, G)$, $(\Psi, H)$. A map $(F, T) : (\Phi, G) \to (\Psi, H)$ is said to be a GEO from $(\Phi, G)$ to $(\Psi, H)$ if $F$ is $T$-equivariant (i.e.,
$F(\varphi g) = F(\varphi)T(g)$ for every $\varphi \in \Phi$, $g \in G)$ and $T$ is a group homomorphism. Moreover, if $F$ is non-expansive (i.e., $\|F(\varphi_1) - F(\varphi_2)\|_{\infty} \le \|\varphi_1 - \varphi_2\|_{\infty}$ for every $\varphi_1, \varphi_2 \in \Phi$), then we say $(F, T)$ is a GENEO.

### III. GENEOS FOR SIGNAL RECONSTRUCTION

We describe the signal to be reconstructed as a function $\varphi: \mathbb{R}^2 \to [0, 1]$. The function $\varphi$ represents the intensity of the signal over a 2D grid, and is often referred to as ground truth or GT. We also consider the function $\psi: \mathbb{R}^2 \to [0, 1]$ taking each point $p$ to the reliability $\psi(p)$ of the value $\varphi(p)$ measured at the point $p^1$. We aim at reconstructing $\varphi$ by starting from a very poor sampling of $\varphi$, by using GENEOS. In literature, GENEOS accept as input only continuous functions that share the same domain, but samplings are functions that do not possess these properties. An analogous statement holds for the corresponding sampling of $\psi$. To obtain $\hat{\varphi}$ and $\hat{\psi}$, namely the continuous functional representations with the same domain extracted from $\varphi$ and $\psi$, let us assume that we know the intensity $\varphi(p_i)$ of the ground truth signal at $r$ points $p_1, \dots, p_r$ and the reliability $\psi(p_i)$ of each value $\varphi(p_i)$. We can now define the functions $\hat{\varphi}, \hat{\psi}: \mathbb{R}^2 \to [0, 1]$ by setting

$$ 
\begin{aligned}
\hat{\varphi}(p) &:= \sum_{i=1}^{r} \varphi(p_i) e^{-\frac{\|p-p_i\|^2}{2\sigma^2}}, \\
\hat{\psi}(p) &:= \sum_{i=1}^{r} \psi(p_i) e^{-\frac{\|p-p_i\|^2}{2\sigma^2}}
\end{aligned}
\qquad (4) 
$$

for a small value of $\sigma$. In our implementation, $\sigma$ is chosen sufficiently small such that the continuous functions $\hat{\varphi}$ and $\hat{\psi}$ can be approximated by two arrays consisting entirely of zero-valued pixels, except for a number less or equal to $r$ of pixels with nonzero values. We refer to Fig. 3 for an example of $\hat{\varphi}$ (Fig. 3(a)) and $\hat{\psi}$ (Fig. 3(b)). The color scheme for $\hat{\varphi}$ is as follows: the image takes the value $\varphi(p) \in [0, 1]$ if $\hat{\psi}(p) = 1$, and is white otherwise.

Now we present how we built GENEOS for the reconstruction of $\varphi$. The main idea is to look for meaningful patterns in the signal. In our model, a pattern is a pair of functions $P = (h, \chi_h)$, where:

* $h: \mathbb{R}^2 \to [0, 1]$ represents a shape we are looking for in the signals;

* $\chi_h: \mathbb{R}^2 \to \{0, 1\}$ establishes where the values expressed by $h$ are reliable.

Furthermore, we assume that the support $D_h$ of $\chi_h$ is compact, i.e., closed and bounded. This assumption implies that the effect of each pattern is local.

Let us now consider a pair $S = (f, \psi_f)$ of functions from $\mathbb{R}^2$ to $[0, 1]$. In this paper, $f = \hat{\varphi}$ and $\psi_f = \hat{\psi}$, but we want to emphasize that our model is applicable to broader contexts. After choosing the pair $S$ representing the data and the pair

<figure><img src="image_13.png" /><figcaption>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.</figcaption></figure>

$P = (h, \chi_h)$ representing a pattern, we can define:

$$ 
\begin{aligned}
S_{S,P}(x, y) &= \int_{\mathbb{R}^2} |f(x + \xi, y + \eta) - h(\xi, \eta)| \\
&\qquad \cdot \psi_f(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\mu(\xi, \eta),
\end{aligned}
\qquad (5) 
$$

which tells us how much $S$ differs from $P$ in a neighborhood of the point $(x, y)$; and

$$ A_{S,P}(x, y) = \int_{\mathbb{R}^2} \psi_f(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\mu(\xi, \eta), \quad (6) $$

which quantifies the reliability of our data in the region where we can perform the comparison between $S$ and $P$. In Eq.(5) and Eq.(6), defining $\bar{\mu}$ as the standard Lebesgue measure on $\mathbb{R}^2$, we set $\mu(A) = \bar{\mu}(A)/\bar{\mu}(D_h)$ for each measurable $A \subseteq \mathbb{R}^2$, where $D_h$ is assumed to have finite positive measure. Hence, we can consider the following GENEO $F$ that takes the function $f$ to the function $\hat{c}_{S,P}$ defined by setting:

$$ \hat{c}_{S,P}(x, y) = A_{S,P}(x, y) - S_{S,P}(x, y). \qquad (7) $$

The value $\hat{c}_{S,P}(x, y)$ measures the similarity between the pattern described by $h$ and the structure of $f$ in the neighborhood of $(x, y)$.

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

**Proof:**

The proof is provided in the Appendix. $\square$

**Proposition 2.** Assume that two signals $S_1 = (f_1, \psi_{f_1})$, $S_2 = (f_2, \psi_{f_2})$ and a pattern $P = (h, \chi_h)$ are given. If $\psi_{f_1} = \psi_{f_2}$, it holds that $A := A_{S_1,P} = A_{S_2,P}$ and

$$ \|\hat{c}_{S_1,P} - \hat{c}_{S_2,P}\|_{\infty} \le \|A\|_{\infty} \|f_1 - f_2\|_{\infty}. \qquad (8) $$

**Proof:**

<footnote n="1">¹In our work, we define a point as $p = (x, y) \in \mathbb{R}^2$.</footnote>
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>FIGURE 4: Illustrative toy example, where we present (a) a ground truth $\varphi$, (b) its sampled version $\hat{\varphi}$, and (c)-(f) four patterns $h_1\chi, \dots, h_4\chi$.</figcaption></figure>

The proof is provided in the Appendix. □

We will assume that the ground truth $\varphi$ is reliable everywhere (i.e., $\psi \equiv 1$). Hence, the resulting sampling reliability $\hat{\psi}$ takes value in $\{0, 1\}$, where $\hat{\psi}(x, y) = 1$ if $(x, y)$ is a known sample, otherwise $\hat{\psi}(x, y) = 0$. In addition, $\chi_h \equiv 1$ for every pattern $P$.

**Proposition 3.** The map $F$ taking $S = (f, \psi_f)$ to the function $\hat{c}_{S,P}$ is a GENEO with respect to the group of planar translations.

*Proof:* From Proposition 2 and the translation invariance of the Lebesgue measure, the statement immediately follows. □

**Remark 1.** Depending on the geometry of $D_h$ and the choice of patterns, we can allow $F$ to be equivariant w.r.t. a larger group, i.e. the group of isometries of the plane.

In the following, we will assume that our samplings and our patterns are reliable everywhere, i.e., $\psi \equiv \psi_{f_1} \equiv \psi_{f_2} \equiv 1$ for every $f_1, f_2$. Given a sampled signal $\hat{S} = (\hat{\varphi}, \hat{\psi})$, we have to consider a library of $N$ patterns to reconstruct $\varphi$, that is

$$ \{P_1 = (h_1, \chi_{h_1}), \dots, P_N = (h_N, \chi_{h_N})\}. \quad (9) $$

Therefore, we can compute $\{\hat{c}_{\hat{S},P_i}\}_{i=1}^N$. For any index $i$ we define the similarity coefficient of the pattern $P_i$ at the point $p$, when $P_i$ is centered at another point $q \in \mathbb{R}^2$, as

$$ \mathrm{sim}(p, P_i, q) = \hat{c}_{\hat{S}, P_i}(q)\chi_{h_i}(p-q). \quad (10) $$

In plain words, the value $\mathrm{sim}(p, P_i, q)$ quantifies how plausible it is that at the point $p$ the dominant pattern is $P_i$, centered at the point $q$. The term $\chi_{h_i}(p-q)$ is needed because we are interested only in the points of $P_i$ at which the pattern is reliable (i.e., $\chi_{h_i}(p-q) \neq 0$). Fig. 4 shows (from left to right) a toy example of $\varphi$, its sampled version $\hat{\varphi} = \varphi\psi$ with a 10% sampling rate and four patterns $h_1, \dots, h_4$, displayed as $h_1\chi, \dots, h_4\chi$, where $\chi$ is the characteristic function of the unit disk. Fig. 5 displays the corresponding $\hat{c}_{S,P_i}$, where $S = (\hat{\varphi}, \hat{\psi})$ and $P_i = (h_i, \chi)$ for $i = 1, 2, 3, 4$.

The last step in our procedure consists of using the functions $\mathrm{sim}(\cdot, P_i, q)$, varying $P_i$ and $q$, to produce the reconstructed signal $\varphi_{\mathrm{rec}}: \mathbb{R}^2 \to [0, 1]$. To do this, considering

$$ (\bar{q}(p), \bar{\iota}(p)) = \underset{\substack{i \in \{1, \dots, N\} \\ q \in \mathbb{R}^2}}{\operatorname{argmax}} \mathrm{sim}(p, P_i, q), \quad (11) $$

<figure><img src="image_18.png" /><figcaption>FIGURE 5: Heatmaps of the coefficients $\hat{c}_{S,P_i}$ for $i = 1, 2, 3, 4$, where $S = (\hat{\varphi}, \hat{\psi})$ and $P_i = (h_i, \chi)$, with $\hat{\varphi}, h_i, \chi$ as defined in Fig. 4. Sub-figures (a)–(d) correspond to $i = 1, 2, 3, 4$, respectively.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIGURE 6: (a) Heatmap of the maximum confidence $\max_i \hat{c}_{S,P_i}(p)$ at each point $p$, indicating how well the best-matching pattern $P_i$ locally agrees with the original sample $\varphi(p)$ displayed in Fig. 4(a). (b) Map of indices $\bar{\iota}(p) = \underset{i}{\operatorname{argmax}} \hat{c}_{S,P_i}(x, y)$, showing which pattern $P_i$ achieves the highest confidence at each point. (c) Reconstructed output $\varphi_{\mathrm{rec}}(p)$ produced by the GENEO-based pattern matching.</figcaption></figure>

we reconstruct the point $p$ using the $\bar{\iota}(p)$-th pattern with a shift equal to $\bar{q}(p)$ as

$$ \varphi_{\mathrm{rec}}(p) := h_{\bar{\iota}(p)}(p - \bar{q}(p)). \quad (12) $$

Continuing the toy example, Fig. 6 (a) displays the maximum among the $\hat{c}_{S,P_i}$ values for each point. This image essentially represents how confident the GENEO is about having found a pattern effectively explaining $\varphi$'s sampling at each point. Fig. 6 (b) is colored according to the index of the pattern that achieves the optimal $\hat{c}_{S,P_i}$. Finally, Fig. 6 (c) shows the corresponding reconstruction $\varphi_{\mathrm{rec}}$ according to Eq. (12). We observe that the reconstruction operator that associates $N$-tuple $(\hat{c}_{S,P_1}, \dots, \hat{c}_{S,P_N})$ with $\varphi_{\mathrm{rec}}$ is a GEO.

# IV. NUMERICAL RESULTS

## A. SIGNAL RECONSTRUCTION

In our numerical experiments, we focus on reconstructing the SINR over a two-dimensional area. We generate SINR measurements with the Sionna RT ray-tracing simulator [44], using its built-in outdoor urban scenarios for Munich and Paris. Each scenario is discretized into a $L \times L$ grid of $1 \text{ m}^2$ pixels $\{p_j\}_{j=1}^L$.

The network contains $N_{\mathrm{tx}} = 100$ transmitters, each operating at carrier frequency $f_c = 10 \text{ GHz}$ with transmit power $P_{\mathrm{tx}} = 1 \text{ dBm}$ and bandwidth $B_w = 50 \text{ MHz}$. Every
transmitter is equipped with a planar array of 128 isotropic elements (16 rows of 8). Similarly, at each receiver pixel j, we place a planar array of 16 isotropic elements (2 rows of 8). Denote by $P_{\text{rx}}^{j,i}$ the received power at receiver j from transmitter i.

We define the instantaneous SINR at receiver j as

$$ \gamma_j = \frac{\max_{1 \le i \le N_{\text{tx}}} P_{\text{rx}}^{j,i}}{\sigma^2 + \sum_{i=1}^{N_{\text{tx}}} P_{\text{rx}}^{j,i} - \max_{1 \le i \le N_{\text{tx}}} P_{\text{rx}}^{j,i}}, \quad (13) $$

where the noise power is

$$ \sigma^2 = k_B T_K B_w F_\sigma, \quad (14) $$

with $k_B = 1.38 \times 10^{-23}$ J/K (Boltzmann's constant), $T_K = 290$ K (system temperature), and $F_\sigma = 5$ dB (noise figure), yielding $\sigma^2 = 10^{-12}$ W. To obtain a bounded target to reconstruct, we normalize the SINR values across the grid:

$$ \bar{\gamma}_j = \frac{\max_{1 \le k \le L^2} \gamma_k - \gamma_j}{\max_{1 \le k \le L^2} \gamma_k - \min_{1 \le k \le L^2} \gamma_k}. \quad (15) $$

We then pose the reconstruction problem as reconstructing the mapping

$$ \hat{\varphi} : \{p_j\}_{j=1}^{L^2} \mapsto \{\bar{\gamma}_j\}_{j=1}^{L^2}, \quad (16) $$

i.e., from each pixel to its normalized SINR value.

## B. SCENARIOS

We generate target signals $\varphi$, samplings $\hat{\varphi}$, and reliability masks $\hat{\psi}$ from two outdoor ray-tracing scenarios (Munich and Paris), each discretized into a set of $L \times L$ grids with $L = 270$ (i.e., $270 \times 270$ m² per grid). For each scenario:

* We sample only $M\%$ of the SINR values, selected uniformly at random, and mark those as “known.” We define

$$ \hat{\psi} : \{p_j\}_{j=1}^{L^2} \rightarrow \{0, 1\}, \\ \sum_j \hat{\psi}(p_j) = \frac{M}{100} L^2, \quad (17) $$

where $\hat{\psi}(p_j) = 1$ if the measurement $\hat{\varphi}(p_j)$ is retained, and 0 otherwise. We consider $M \in \{1, 2, 3\}$.

* To simulate noisy or corrupted measurements, we replace a further $Q\%$ of those retained pixels with uniform random noise in $[0, 1]$. Concretely, out of the $\frac{M}{100} L^2$ locations with $\hat{\psi} = 1$, we choose $Q\%$ uniformly at random and for each such pixel $\bar{p}_j$ set

$$ \hat{\varphi}(\bar{p}_j) \leftarrow u \sim \mathcal{U}(0, 1), \quad (18) $$

with $Q \in \{15, 30\}$.

This process yields incomplete and noisy observations of the true normalized SINR map, reflecting realistic measurement limitations. Although having substantially more data from multiple cities would be ideal, our experiments are based on the only two default urban scenarios provided by Sionna RT. While target signals $\varphi$ are fixed given a scenario, many different realizations of samplings $\hat{\varphi}$ and reliability masks $\hat{\psi}$ can be generated through the random selection process.

### 1) Munich

For the Munich scenario, we cover an overall area of $810 \times 810$ m², partitioned into nine non-overlapping square subregions of $270 \times 270$ m² each (see Fig. 7(a)). We evaluate GENEO and the baselines via leave-one-out cross-validation over these nine areas: in each of nine trials, eight subregions provide the known/corrupted measurements (for pattern preparation and training, where applicable), while the ninth subregion's SINR values (masked by $\hat{\psi}$) form the reconstruction target $\varphi$.

### 2) Paris

For the Paris scenario, we span $540 \times 540$ m², divided into four square subregions of $270 \times 270$ m² each (see Fig. 8(a)). We assess generalization by training solely on all nine Munich subregions and then reconstructing each of the four Paris subregions in turn: the entire Munich collection provides known/corrupted measurements for GENEO and baseline training, and each Paris subregion's masked SINR values constitute the test targets. This experimental setup challenges the generalization capabilities of tested approaches, it requires training exclusively on data from one urban environment (Munich) and testing to a distinct and unseen city layout (Paris) without any retraining.

## C. GENEO IMPLEMENTATION

Each $L \times L$ SINR map $\varphi$ (as defined in Sec. A) is partitioned using circular tiling, covering the full $270 \times 270$ m² area and keeping minimal overlap between circles. This process yields 67 patterns per area. In this paper, each $\chi_i$ is modelled as a fixed, rotation-invariant mask with a support of diameter of 45 m:

$$ \chi(x, y) = \begin{cases} 1 & \text{if } x^2 + y^2 \le 22^2, \\ 0 & \text{otherwise.} \end{cases} \quad (19) $$

Each pattern is therefore a pair $P_i = (h_i, \chi)$, with $i = 1, \dots, 67$. To enrich the library, we generate 24 rotated variants of every pattern by using $15^\circ$ increments, such that pattern $h_{i\frac{\theta}{15}+i}$ is equal to pattern $h_i$ rotated by $\theta$ degrees. Thus each area contributes $67 \times 24 = 1608$ patterns.

The construction of the pattern library differs between the two scenarios:

* **Munich:** In each leave-one-out trial, patterns are drawn from the eight “training” subregions (i.e., $8 \times 1608$ patterns) to set up the GENEO operator; the remaining subregion's masked SINR map - given only the known entries indicated by $\hat{\psi}$ - is reconstructed via GENEO operator. An example of patterns extracted is presented in Fig. 9.

* **Paris:** All nine Munich subregions supply patterns (i.e., $9 \times 1608$ in total) for GENEO set up, and the four Paris subregions are subsequently reconstructed, one at a time, from their partial observations $\hat{\varphi}$.
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /><figcaption>FIGURE 7: Comparison of reconstruction methods on the Munich scenario, with only $M = 3\%$ of SINRs known, whose $Q = 15\%$ of them is featuring errors. (a) ground truth, (b) GENEO, (c) 1-KNN, and (d) U-Net.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>FIGURE 8: Comparison of reconstruction methods on the Paris scenario, with only $M = 3\%$ of SINRs known, whose $Q = 15\%$ of them is featuring errors. (a) ground truth, (b) GENEO, (c) 1-KNN, and (d) U-Net.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>FIGURE 9: Example of patterns extracted from areas of Munich.</figcaption></figure>

This setup ensures that GENEO leverages spatially equivariant operators from rich, minimally overlapping, local signal structures and applies them to reconstruct the full normalized SINR map $\varphi$ under varying availability and noise conditions.

## D. BASELINES

### 1) 1-KNN

As a simple, non-parametric reference, we implement a one-nearest-neighbor (1-KNN) interpolation over the spatial grid. The following steps break down the procedure:

1) Let $\mathcal{K} = \{p_k | \hat{\psi}(p_k) = 1\}$ be the set of pixels with available SINR measurements, and $\mathcal{P} = \{p_j | \hat{\psi}(p_j) = 0\}$ the set of pixels to reconstruct.

2) For any target $p_j \in \mathcal{P}$, we compute the 2-norm to each known point $p_k \in \mathcal{K}$ to identify the closest known pixel:

$$p^* = \underset{p_k \in \mathcal{K}}{\operatorname{argmin}} \|p_j - p_k\|_2. \qquad (20)$$

3) Finally, we assign

$$\varphi_{\text{rec}}(p_j) = \hat{\varphi}(p^*), \qquad (21)$$

i.e., we recover the SINR value of the nearest sampled pixel for reconstruction.

This 1-KNN baseline exploits the spatial locality of SINR: each missing value is simply replaced by its nearest neighbor's measurement in the two-dimensional plane, providing a simple yet effective benchmark.

### 2) U-Net

We employ a fully supervised U-Net model [72] to recover the complete SINR map $\varphi$ from its partially observed version $\hat{\varphi}$. The network produces an output $\varphi_{\text{rec}} \in \mathbb{R}^{L \times L}$, an estimate of the full reconstructed SINR.
<figure><img src="image_25.png" /><figcaption>FIGURE 10: U-Net architecture.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIGURE 11: MSE achieved by GENEO, U-Net, and 1-KNN for normalized SINR reconstruction in Munich scenario, where $M \in \{1, 2, 3\}$ and $Q = 15$.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>FIGURE 13: 1-Wasserstein achieved by GENEO, U-Net, and 1-KNN for normalized SINR reconstruction in Munich scenario, where $M \in \{1, 2, 3\}$ and $Q = 15$.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>FIGURE 12: MSE achieved by GENEO, U-Net, and 1-KNN for normalized SINR reconstruction in Munich scenario, where $M \in \{1, 2, 3\}$ and $Q = 30$.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>FIGURE 14: 1-Wasserstein achieved by GENEO, U-Net, and 1-KNN for normalized SINR reconstruction in Munich scenario, where $M \in \{1, 2, 3\}$ and $Q = 30$.</figcaption></figure>

Similarly to [72], we consider a U-Net architecture comprising a four-level encoder, a central bottleneck, and a sym-

metric four-level decoder, connected via skip-connections. The considered U-Net architecture is presented in Fig. 10.
<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /><figcaption>FIGURE 15: MSE achieved by GENEO, U-Net, and 1-KNN for normalized SINR reconstruction in Paris scenario, where $M \in \{1, 2, 3\}$ and $Q = 15$.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>FIGURE 17: 1-Wasserstein achieved by GENEO, U-Net, and 1-KNN for normalized SINR reconstruction in Paris scenario, where $M \in \{1, 2, 3\}$ and $Q = 15$.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>FIGURE 16: MSE achieved by GENEO, U-Net, and 1-KNN for normalized SINR reconstruction in Paris scenario, where $M \in \{1, 2, 3\}$ and $Q = 30$.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>FIGURE 18: 1-Wasserstein achieved by GENEO, U-Net, and 1-KNN for normalized SINR reconstruction in Paris scenario, where $M \in \{1, 2, 3\}$ and $Q = 30$.</figcaption></figure>

The 4-layers encoder features ReLU-activated $3 \times 3$ convolutional layers and $2 \times 2$ max-pooling, where the number of channels doubles at each depth. The bottleneck leverages two additional $3 \times 3$ convolutions with ReLU activation. The decoder successively performs upsampling by $2 \times 2$ transposed convolutions that halve the channel count. After each upsampling, the result is concatenated with the corresponding encoder feature map, and two $3 \times 3$ ReLU-activated convolutions restore spatial detail. A final $1 \times 1$ convolution produces the reconstructed map $\varphi_{\text{rec}}$.

As supervised training loss, we train U-Net to minimize the mean squared error between $\varphi_{\text{rec}}$ and $\varphi$ using the Adam optimizer with learning rate $10^{-4}$. The ground truth $\varphi$ is assumed fully available for supervision at training time, enabling U-Net to both reconstruct missing information and correct noisy measurements.

## E. RESULTS ON MUNICH SCENARIO

We report leave-one-out cross-validation results on the Munich areas for both MSE and 1-Wasserstein distance, under two noise levels, with $Q \in \{15, 30\}$, and three sampling

ratios, with $M \in \{1, 2, 3\}$. Each plot summarizes nine trials, where in each trial eight subregions provide the known (and corrupted) SINR measurements and the remaining subregion is reconstructed. An example of reconstructions using the three approaches is presented in Fig. 7(b)-(d).

### a: MSE Performance

Fig. 11 shows the MSE for $Q = 15$: GENEO (green) consistently attains better performance for $M \in \{2, 3\}$, compared to 1-KNN (red) and U-Net (blue), while showing similar MSE to 1-KNN for $M = 1$. For the higher corruption level $Q = 30$, (Fig. 12), GENEO again outperforms both baselines, with its advantage being most pronounced across all values of $M$. U-Net exhibits the largest errors in all settings.

### b: 1-Wasserstein Performance

Fig. 13 reports the 1-Wasserstein distance for $Q = 15$, GENEO delivers dramatically lower topological discrepancy than 1-KNN and U-Net across all $M$. At $Q = 30$ (Fig. 14), GENEO maintains its superiority, achieving the smallest 1-
Wasserstein distances in every case. The baselines remain clustered at substantially higher values, with U-Net and 1-KNN showing minimal relative improvement as sampling density increases.

## F. RESULTS ON PARIS SCENARIO

We evaluate zero-shot generalization on the Paris areas using models trained exclusively on the nine Munich subregions, under two noise levels, with $Q \in \{15, 30\}$, and sampling ratios, with $M \in \{1, 2, 3\}$. Each boxplot aggregates four trials, one per Paris subregion. Results are collected for both MSE and 1-Wasserstein distance. An example of reconstructions using the three approaches is presented in Fig. 8(b)-(d).

### a: MSE Performance

Fig. 15 ($Q = 15$) reveals that GENEO (green) achieves the lowest reconstruction error in terms of MSE across all sampling ratios $M$. Under heavier corruption $Q = 30$ (Fig. 16), GENEO's advantage persists: it consistently outperforms 1-KNN and U-Net, with the largest margin at $M = 1$.

### b: 1-Wasserstein Performance

Fig. 17 shows the 1-Wasserstein distance for $Q = 15$: GENEO again yields substantially lower topological error, improving over 1-KNN and U-Net across all $M$. For $Q = 30$ (Fig. 18), GENEO maintains its lead in topological fidelity, while both baselines exhibit worse performance in terms of signal reconstruction.

## V. CONCLUSION

In this work, we introduced a novel framework for reconstructing SINR maps in urban mobile radio scenarios from extremely sparse measurements, grounded in the theory of GENEOS. First, we develop a GENEO-driven reconstruction method that embeds domain symmetries directly into the operator, yielding compact models with strong inductive biases. Second, we couple traditional statistical metrics (MSE) with a TDA metric (1-Wasserstein distance) to capture not only intensity errors but also the preservation of geometric and spatial signal characteristics. Third, we deliver a thorough empirical evaluation - using realistic Munich and Paris scenarios generated via the Sionna RT ray-tracing tool - demonstrating that our approach consistently outperforms state-of-the-art ML baselines in both statistical fidelity and topological accuracy. These results validate our approach as an effective solution for reconstructing sparse urban SINR maps with high topological fidelity and compact, symmetry-aware models.

## Appendix

Here we provide the proof of the theoretical results described in the previous sections. First we present the proof of Proposition 1:

### Proof:

Given a signal $S = (f, \psi_f)$ and a pattern $P = (h, \chi_h)$, we know that $A_{S,P}$ and $S_{S,P}$ are nonnegative since the integrating functions are nonnegative. In order to conclude the proof, it will suffice to show that $S_{S,P} \le A_{S,P} \le 1$. We recall that in our model we assume that $\psi_f \le 1$, $f \le 1$, and $\int_{\mathbb{R}^2} \chi_h d\mu \le 1$. Since $\bar{f}(\xi, \eta) := |f(x+\xi, y+\eta)-h(\xi, \eta)| \le 1$, we have that

$$ 
\begin{aligned}
S_{S,P} &= \int_{\mathbb{R}^2} \bar{f}(\xi, \eta) \psi(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\mu(\xi, \eta) \\
&\le \int_{\mathbb{R}^2} \psi_f(x + \xi, y + \eta) \chi_h \, d\mu \\
&= A_{S,P}(x, y) \\
&\le \int_{\mathbb{R}^2} \chi_h \, d\mu \le 1.
\end{aligned}
$$

Then, we can prove Proposition 2:

### Proof:

Fixing a point $(x, y) \in \mathbb{R}^2$, we have that

$$ 
\begin{aligned}
A_{S_1,P}(x, y) &= \int_{\mathbb{R}^2} \psi_{f_1}(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\xi \, d\eta \\
&= \int_{\mathbb{R}^2} \psi_{f_2}(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\xi \, d\eta \\
&= A_{S_2,P}(x, y).
\end{aligned}
$$

Before proceeding, we set $\mathcal{A} := A_{S_1,P} = A_{S_2,P}$. Denoting $\bar{f}_i(\xi, \eta) := |f_i(x + \xi, y + \eta) - h(\xi, \eta)|$ for $i = 1, 2$, using the reverse triangle inequality we obtain that

$$ 
\begin{aligned}
|\bar{f}_1(\xi, \eta) - \bar{f}_2(\xi, \eta)| &= \left| |f_1(x + \xi, y + \eta) - h(\xi, \eta)| - |f_2(x + \xi, y + \eta) - h(\xi, \eta)| \right| \\
&\le |f_1(x + \xi, y + \eta) - f_2(x + \xi, y + \eta)| \\
&\le \|f_1 - f_2\|_{\infty}.
\end{aligned}
$$

Then, setting $\psi := \psi_{f_1} = \psi_{f_2}$, we get

$$ 
\begin{aligned}
|\hat{c}_{S_1,P}(x, y) - \hat{c}_{S_2,P}(x, y)| &= |S_{S_1,P}(x, y) - S_{S_2,P}(x, y)| \\
&= \left| \int_{\mathbb{R}^2} \bar{f}_1(\xi, \eta) \psi(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\mu(\xi, \eta) - \int_{\mathbb{R}^2} \bar{f}_2(\xi, \eta) \psi(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\mu(\xi, \eta) \right| \\
&\le \int_{\mathbb{R}^2} |\bar{f}_1(\xi, \eta) - \bar{f}_2(\xi, \eta)| \psi(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\mu(\xi, \eta) \\
&\le \int_{\mathbb{R}^2} \|f_1 - f_2\|_{\infty} \psi(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\mu(\xi, \eta) \\
&= \|f_1 - f_2\|_{\infty} \int_{\mathbb{R}^2} \psi(x + \xi, y + \eta) \chi_h(\xi, \eta) \, d\xi \, d\eta \\
&\le \|\mathcal{A}\|_{\infty} \|f_1 - f_2\|_{\infty}.
\end{aligned}
$$
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LORENZO MARIO AMOROSA (Member, IEEE) received the B.S. degree (summa cum laude) in Computer Engineering and the M.S. degree (summa cum laude) in Artificial Intelligence from the University of Bologna, Italy, in 2019 and 2021, respectively. In 2025, he received the Ph.D. degree in Electronics, Telecommunications and Information Technologies Engineering from the University of Bologna. Currently he is with the Department of Electronic, Information and Electrical Engineering

“Guglielmo Marconi” as a Postdoctoral Research Fellow at University of Bologna. He is Research Associate at the National Laboratory of Wireless Communications (WiLab) of CNIT (the National, Inter-University Consortium for Telecommunications). His main research interests include Decentralized Artificial Intelligence, Cooperative Multi-Agent Systems, Machine Learning for Industrial IoT.

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FRANCESCO CONTI is a PostDoc Researcher at Inria, Sophia-Antipolis, specializing in topological data analysis and geometric machine learning. He holds a PhD in Mathematics from the University of Pisa (2024), where he established theoretical connections between persistent homology and group equivariant non-expansive operators (GENEOs). His current postdoctoral research applies topology-infused generative models to galaxy mapping, while his broader work focuses on the topological and geometrical properties of GENEOS

and TDA. Francesco has published in high-impact journals including Nature Scientific Reports and received the Under 35 Best Paper Award from AITA (2023). He has expertise in Python and specialized topological tools.
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**NICOLA QUERCIOLI** is a researcher working at the intersection of mathematics and machine learning, with a focus on symmetry, topology, and data analysis. His work explores how structural properties of data can be formalized and integrated into AI models through advanced mathematical frameworks. He received his Ph.D. in Mathematics from the University of Bologna, where he studied the theory of Group Equivariant Non-Expansive Operators (GENEOs). He has collaborated with WiLab/CNIT on research projects involving topology-driven approaches to machine learning, with applications ranging from signal processing to high-performance computing environments.

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**PATRIZIO FROSINI** received his Ph.D. in Mathematics from the University of Florence in 1991. He taught for many years at the University of Bologna and is currently an Associate Professor in the Department of Computer Science at the University of Pisa. His research focuses on the topological and geometric properties of manifolds and topological spaces equipped with filtering functions taking values in $R^n$, particularly in the presence of symmetry groups. In the early 1990s, he introduced and began studying the theoretical foundations that would later evolve into topological data analysis and persistent homology. Together with his collaborators, he has also introduced and begun developing the topological theory of group-equivariant non-expansive operators (GENEOs), as well as their applications in topological data analysis, machine learning, and artificial intelligence.

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**FLAVIO ZABINI** (Member, IEEE) received the Laurea (summa cum laude) in telecommunications engineering and the Ph.D. in electronic engineering and computer science from the University of Bologna, Italy, in 2004 and 2010, respectively. In 2004, he developed his master's thesis at the University of California San Diego, La Jolla. In 2008, he was a Visiting Student with the DoCoMo Eurolabs of Munich, Germany. From 2013 to 2014, he was a Post-Doctoral Fellow with the German Aerospace Center, Cologne, Germany. He was with the IEIIT-Bo of CNR. He is currently an Associate Professor at the University of Bologna. His current research interests include stochastic sampling, molecular communications, and joint sensing and communications. He served as an associate Editor for the IEEE Communication Letters and the KSII Transactions on Internet and Information Systems.

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**TAYEBEH LOTFI MAHYARI** is a Senior Research Engineer at Huawei Technologies in Ottawa, Canada, where she focuses on applying artificial intelligence to wireless communication systems and standardization, with a particular emphasis on massive MIMO technologies for future 6G networks. She holds B.Sc. and M.Sc. degrees in Computer Engineering from Isfahan University of Technology and Sharif University of Technology, respectively, and earned a second M.Sc. and a Ph.D. in Computer Science from Simon Fraser University and Carleton University. Prior to joining Huawei, she taught part-time at Isfahan University of Technology and Islamic Azad University. Her broader research interests include machine learning, statistical modeling, and advanced wireless system design.

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**YIQUN GE** is a Distinguished Research Engineer and 6G scientist at Huawei Technologies Co., Ltd. His research focuses on wireless communication systems and standardization, with particular interests in low-power chip design for wireless applications, channel coding, AI, and massive MIMO technologies for future 6G networks. He has over two decades of R&D experience in wireless systems and has been actively contributing to cutting-edge innovations that bridge theory and industry standards. He received his B.Eng. degree from Shanghai Jiao Tong University in 2000 and his M.Sc. degree from ENST-Bretagne, France, in 2003.