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Graph neural networks for sampling-invariant embeddings of organized signal sets
Martin Bauw, Santiago Velasco-Forero, Jesus Angulo
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Summary
This paper investigates the use of Graph Neural Networks (GNNs), specifically Graph Convolutional Networks (GCNs), to create sampling-invariant embeddings for organized signal sets. The proposed method projects heterogeneously sampled signals (e.g., from radar or sensor networks) into a fixed-size latent space using contrastive learning (NT-Xent loss). Experiments on synthetic complex-valued radiofrequency signals demonstrate that the GCN encoder produces discriminative representations robust to sampling variations, outperforming autoencoders and traditional spectral/statistical baselines in waveform separation tasks, though it struggles with order-sensitive waveform identification.
Entities (7)
Relation Signals (5)
Graph Convolutional Network → uses → NT-Xent Loss
confidence 95% · The GCN is implemented with three layers... trained with the contrastive learning objective (2)... loss used to train the signal sets encoder is the Normalized Temperature-Scaled Cross-Entropy loss (NT-Xent)
Graph Convolutional Network → processes → Radiofrequency Signals
confidence 92% · The encoding and embeddings discrimination experiments conducted rely exclusively on synthetic complex-valued radiofrequency signals... The proposed encoder is a graph convolutional network (GCN)
Graph Convolutional Network → enables → Sampling-Invariant Embeddings
confidence 90% · This paper investigates organized signal sets neural network encoders... project heterogeneously sampled signal sets into an arbitrary fixed-size vectors space... designed to reduce the influence of heterogeneous sampling parameters
Graph Convolutional Network → outperforms → Autoencoder
confidence 88% · The GCN encoder... leads to the best signal sets templates classification performances... while the AE... struggles and ranks below spectral and statistical signal features.
Graph Convolutional Network → appliedto → Radar
confidence 85% · Encoding a graph of individual signals is particularly interesting in a radar context as it allows for adaptive radar cells processing.
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Abstract
Abstract:Sensor networks and radars can deliver signals as organized sets, e.g. ordered signals, signals describing range cells within a grid or signals perceived as graph nodes. Within such sets, individual signals may be characterized by distinct sampling parameters. This paper investigates organized signal sets neural network encoders. In the context of this work, the purpose of such encoders is to project heterogeneously sampled signal sets into an arbitrary fixed-size vectors space. This new representation space is designed so that signal sets can be processed as vectors rid of sampling differences to allow for arbitrary topology-aware processing with no signal processing constraints. Within this representation space designed to reduce the influence of heterogeneous sampling parameters, the relevance of signal sets representations is evaluated by considering signal sets discrimination potential with a focus on waveforms separation. The encoding and embeddings discrimination experiments conducted rely exclusively on synthetic complex-valued radiofrequency signals.
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- Source: https://arxiv.org/abs/2609.35934v1
- Canonical: https://arxiv.org/abs/2609.35934v1
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GRAPH NEURAL NETWORKS FOR SAMPLING-INVARIANT EMBEDDINGS OF ORGANIZED SIGNAL SETS Martin Bauw ⋆ Santiago Velasco-Forero † Jesus Gustavo Angulo ‡ ⋆ DEMR, ONERA, Universit ́ e Paris-Saclay, 91120 Palaiseau, France † Centre for Statistics and Images (STIM), Mines Paris, PSL University, Fontainebleau, France ‡ Centre for Applied Mathematics (CMA), Mines Paris, PSL University, Sophia Antipolis, France ABSTRACT Sensor networks and radars can deliver signals as organized sets, e.g.ordered signals, signals describing range cells within a grid or signals perceived as graph nodes. Within such sets, individual signals may be characterized by distinct sampling parameters. This paper investigates organized sig- nal sets neural network encoders. In the context of this work, the purpose of such encoders is to project heterogeneously sampled signal sets into an arbitrary fixed-size vectors space. This new representation space is designed so that signal sets can be processed as vectors rid of sampling differences to allow for arbitrary topology-aware processing with no sig- nal processing constraints. Within this representation space designed to reduce the influence of heterogeneous sampling parameters, the relevance of signal sets representations is evaluated by considering signal sets discrimination poten- tial with a focus on waveforms separation. The encoding and embeddings discrimination experiments conducted rely exclusively on synthetic complex-valued radiofrequency sig- nals. Index Terms— signal encoding, organized signal set, graph neural network, sampling-invariant embeddings, repre- sentation learning 1. INTRODUCTION This paper compares encoding methods producing fixed-size vector representations for organized signal sets. The orga- nized signal sets may represent signals produced by sensor networks, signals contained over a radar cells grid, or any graph whose nodes are defined by signals. Encoding a graph of individual signals is particularly interesting in a radar con- text as it allows for adaptive radar cells processing. This relates to range-Doppler processing, where the content of a neighborhood of range-Doppler cells can be used to detect and discriminate targets. Replacing CFAR detectors which rely on limited context with neural networks that adaptively process radar echoes in larger contexts is an active research question [1] [2]. Corresponding author: martin.bauw@onera.fr. The fixed-size vector representations motivating this pa- per aim at transposing a classification or anomaly detection (AD) problem from a signal processing representation space to a data processing representation space in which no sam- pling parameters heterogeneity hinders the signal sets dis- crimination task. The encoder this work aims at building is therefore expected to produce sampling-invariant, waveform- aware embeddings, this while remaining sensitive to wave- form ordering within sets. This waveforms order awareness led the experiments to include graphs. Related works in- clude [3] which proposed heterogeneously sampled signals encoding for downstream sampling-agnostic discrimination in a radar context. In [4], the authors propose training an encoder with a contrastive learning objective to encode and subsequently classify individual signals. Creating generic embeddings for radiofrequency (RF) signal recognition has also been investigated in [5], which considers both raw 1D IQs and 2D time-frequency input representations. Both in- put types have been considered in the literature [6] [7] [8] [9], the 2D time-frequency input making it easy to leverage image processing-inspired pipelines. The originality of this work stems from the combination of the following elements: the signals being processed as an ordered set and encoded with a specific graph, the signals intra and inter-sets sam- pling heterogeneity, the choice of raw IQs as input, and the radiofrequency waveforms selected. 2. SIGNAL SETS ENCODING FORMALIZATION A graph neural network (GNN) [10] Φ will take the signal set composed of P input M -dimensional signals Z ∈ C P×M along with an adjacency matrix defining the graph topology A P×P (where A[i,j] equals 1 if the i-th signal shares an edge with the j-th signal) as input to produce a real-valued latent space representation also called signals set embedding E ∈ R N . In this setup, input signals carry at most M samples with varying sampling frequencies, and end up represented by latent vectors of size N . Individual signals within signal sets have an actual length M ′ such that M ′ ≤ M , M defining a zero-padding length where needed. Zero-padding ensures arXiv:2609.35934v1 [cs.LG] 28 Sep 2026 uniform input dimensions for signals with varying lengths. This setup is illustrated for a set of five signals on Fig. 1. The encoding, which amounts to an inference, of a batch tensor G of B signal sets thus corresponds to the following equation: Φ(G,A) = E ∈ R B×N (1) with G∈ C B×P×M . The adjacency matrix A is required for graph neural network operations: it specifies which neigh- boring nodes’ features are aggregated to update the node rep- resentations in the next layer [11]. The aggregation of neigh- boring nodes features amounts here to an aggregation of sig- nals (e.g. radar cells in a grid). To ensure similar signal sets end up similarly encoded and distinct sets correspond to eas- ily separable embeddings, the loss used to train the signal sets encoder is the Normalized Temperature-Scaled Cross- Entropy loss (NT-Xent) [12]. This loss is computed thanks to positive pairs of signal sets (e i ,e j ), i.e. two signal sets sharing the same label: ℓ(i,j) =− log exp sim(e i ,e j ) τ P B k=1 1 [k̸=i] exp sim(e i ,e k ) τ (2) In the previous expression, τ is the temperature parameter and sim(e i ,e j ) = e T i e j ∥e i ∥e j ∥ the cosine similarity. The denom- inator includes both positive and negative pairs. The NT-Xent loss is a contrastive learning loss pulling the embeddings of pairs sharing labels together while pulling apart pairs of rep- resentations corresponding to distinct labels. Fig. 1. The encoding presented processes sets of signals with different sampling parameters. Individual signals within sets are organized as graphs 3. SIGNAL SETS ENCODING IMPLEMENTATION AND EVALUATION Organized signal sets were represented with and without a neural network encoder to produce a comparison revealing the relevance of neural networks and graphs for the discrim- ination task considered. The following section details which representations and encoding approaches were investigated. The downstream discrimination evaluating the embeddings relevance is a signal sets templates (cf. Table 1) classifica- tion. 3.1. Graph neural network implementation The main motivation of this work being the production of discriminative representations of organized signal sets, graph neural networks [10] appeared as the most natural deep learn- ing approach. The proposed encoder is a graph convolutional network (GCN) [11] trained with the contrastive learning ob- jective (2). The GCN is implemented with three layers and produces a 128-dim signals set embedding thanks to mean- pooling. This architecture carries 2.33M parameters and can be said to be a graph2vec [13] [14] encoding. Inspired by radar test cell-centric processing, the graph topology eventu- ally selected as the method put forward is a chain-like graph with directed edges oriented towards the central node, as il- lustrated on Fig. 1. 3.2. Neural network baselines Several deep learning baselines were considered in our initial experiments. A graph autoencoder (GAE) [15] [16] leverag- ing the topology defined by the adjacency matrix and a regular autoencoder (AE) ignoring the latter were trained along with the proposed contrastive GCN. The graph autoencoder imple- mented failed to achieve any substantial signal sets separabil- ity and thus will be absent from the presented results. Evalu- ating regular autoencoders using stacked, zero-padded signals as 2D inputs aims at including a non-graph neural network en- coder to our baselines. The 1D AE eventually selected relies on 1D convolutions spanning the temporal axis of individual signals within a signals set, and carries 1.28M parameters. It processes the five complex-valued signals of a set as 10 inputs channels through a real and imaginary parts separation. Train- ing is conducted with a length-masked mean squared error (MSE) reconstruction loss, and the signal sets representations are extracted from the 128-dim bottleneck. The order of the set remains available to the AE through the signals stacking order leading to the complex-valued 2D input representation. 3.3. Features engineering baselines Machine learning-free representations were defined to evalu- ate the actual contribution of approaches relying on a trained neural network to produce a representation with discrimina- tive power. The two retained baselines rely on spectral (base- line name rawspectrum) and statistical signal (baseline name raw dsp) features [17]. The spectral baseline extracts 320 features from Fourier bins computed over individual signals within sets, whereas the statistical signal baseline combines temporal and spectral representations attributes to produce 44 features as a signals set representation. 3.4. Evaluating the encoding quality without classifica- tion/AD In the case of neural network encoders, the silhouette score [18] and k-nearest neighbors (KNN) were used to quantita- tively evaluate the relevance of the latent space distribution before classification. The training signal sets templates sep- arability was thus evaluated with two methods relying on Euclidean distances in the representation space. These two scores motivated the choice of both the GCN and the AE encoders among the different neural networks of each type trained. The separability of distinct signal sets templates could additionally be qualitatively evaluated through 2D la- tent space distribution visualization thanks to PCA, t-SNE and UMAP-provided dimensionality reductions. One such latent space visualization is depicted on Fig. 4. 4. SYNTHETIC SIGNAL SETS GENERATION The signal sets used for training and evaluating the proposed approaches are synthetic, i.e. they were generated from con- trolled parameters. Each signals set includes five signals, each with possibly unshared sampling parameters within the set. Each set corresponds to a multiple waveforms template, all of which are defined in Table 1. A positive pair in (2) is de- fined by a common template. Each of the 19 templates fixes an ordered five-node sequence, and node order is preserved at generation time so the graph depicted on Fig. 1 reads the set as a line with a centre. The individual signals generation parameters, presented in 2, are selected to ensure training, validation and test sets are mutually exclusive with respect to sampling frequencies, signal lengths and SNR (Signal-to-Noise Ratio). Each indi- vidual signal within a set independently draws its length M ′ and sampling rate f s from the dataset valid values pool, the SNR is drawn once per set and shared by its five nodes. Valid signal parameters pools are regular grids with the held-out values excised, so Train + , Val and Test are pairwise mutu- ally exclusive for all three parameters. Train + is an enriched training set with increased sampling and noise parameters cre- ated to investigate the performances improvements allowed by increased training diversity while keeping the same num- ber of training signal sets. Train and Train + are thus alterna- tive training sets and are never used together. Table 1. Signal set templates. Abbreviations: C↑/C↓ linear up/down chirp, Q↑/Q↓ quadratic up/down chirp, η complex Gaussian noise. ClassNode sequence cwCW CW CW CW CW amAM AM AM AM AM fskFSK FSK FSK FSK FSK bpskBPSK BPSK BPSK BPSK BPSK chirp upC↑ C↑ C↑ C↑ C↑ chirp downC↓ C↓ C↓ C↓ C↓ chirpquadupQ↑ Q↑ Q↑ Q↑ Q↑ chirpquaddownQ↓ Q↓ Q↓ Q↓ Q↓ chirp3fsk2C↑ C↑ C↑ FSK FSK fsk3 chirpup2FSK FSK FSK C↑ C↑ chirpup2fsk3C↑ C↑ FSK FSK FSK chirp3noise2C↑ C↑ C↑ η η noise2 chirp3η η C↑ C↑ C↑ noisechirpnoise η C↑ C↑ C↑ η am fskamAM FSK AM FSK AM chirpupdownC↑ C↓ C↑ C↓ C↑ chirpupdownquadQ↑ Q↓ Q↑ Q↓ Q↑ cwchirpcwAM C↑ C↓ C↑ AM mixedC↑ η FSK CW AM Table 2. Dataset generation parameters. All signals are com- plex baseband, corrupted by additive white Gaussian noise, and zero-padded to M = 4096 samples. TrainTrain + ValTest Sets100 000100 00010 00010 000 Seed4242530971319468 N (samples)2048–40961920–40962150–40302304–4007 pool size136544 f s (MHz)1.70–3.101.60–3.201.95–3.051.90–3.00 pool size115744 SNR (dB)7–305–3210–299–28 pool size44744 5. EXPERIMENTS AND RESULTS The proposed GCN encoder and the selected baselines lead to the performances presented in Fig. 2. These performances correspond to a decision tree, one of the classification heads considered for latent space classification. The GCN encoder trained on the Train + dataset over 60 epochs leads to the best signal sets templates classification performances in the latent representation space, closely followed by the GCN baseline trained on the Train dataset (cf. Table 2), while the AE trained over 40 epochs struggles and ranks below spectral and sta- tistical signal features. The poor AE performances are not surprising as the training objective is generative and not con- trastive for this baseline. The reconstruction objective fails at producing a competitive representation. The comparison between the AE baseline and the GCN encoders amounts to comparing a general-purpose encoder with encoders specifi- cally trained for the discrimination implemented by the down- stream classification task. For the three neural network en- coders, the validation set was used to adjust training hyper- parameters and avoid using an overfitted encoder to produce signal sets embeddings for the test set. The test set confusion matrix associated with the decision tree classification head on top of the trained GCN encoder is depicted on Fig. 3. It reveals a limited confusion be- tween linear (template chirp down) and quadratic (template chirp quaddown) chirps with decreasing frequencies. More interestingly, it shows the best and proposed approach still struggles to identify distinct waveforms orders within the signal sets : chirp3 noise2 is substantially confused with noise2chirp3, and a similar confusion can be observed for chirpup2fsk3 and fsk3chirpup2. In the experiments con- ducted, the raw spectrum and rawdsp baselines were unex- pectedly hard to beat, and in some cases with different signal sets templates not presented here, systematically performed better in downstream classification with respect to trained neural encoders. Fig. 2. Performance of a decision tree classifier on embed- dings from: GCN candidate, GCN baseline, AE (1D autoen- coder), raw spectrum (spectral features), and raw dsp (statisti- cal features). Neural networks metrics are averaged over three seeds and the error bar depicts± std. Fig. 3. Confusion matrix for the latent space classification task. The performances reported stem from a classification head implemented with a decision tree. Fig. 4. Latent space 2D distribution of test signal sets, com- puted with t-SNE. Top: colormap describes organized signal sets templates that should be separable. Bottom: colormap describes a signal sampling parameter that should not impact the latent space distribution. 6. CONCLUSION This work put forward the relevance of graph neural networks to encode organized signal sets for downstream discrimina- tion. The proposed encoder produces discriminative repre- sentations that exhibit robustness to heterogeneous sampling parameters under the synthetic conditions investigated. Tak- ing into account a graph structure allows for the representa- tions to be order-sensitive within the signals set, making the presented method relevant for sensor networks and radar cells neighborhoods processing. The experiments conducted ad- ditionally reaffirmed the relative competitiveness of spectral and statistical signal features as embeddings for the classi- fication task studied. Future work will consider larger and more complex signal sets, will include real data and more di- verse waveforms, and will evaluate the relevance of the pro- duced embeddings with both classification and anomaly de- tection as the latent space discrimination tasks. 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