Physics-Informed Generative Modeling of Wireless Channels
Authors: Benedikt Böck
Andreas Oeldemann
Timo Mayer
Francesco Rossetto
Wolfgang Utschick
Venue: ICML 2025 poster (2025)
Keywords: sparse Bayesian generative modeling
wireless channel modeling
physics-informed
generative model
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We propose a physically interpretable and generalizable generative model for wireless channels that can learn from corrupted channel observations.

Abstract: Learning the site-specific distribution of the wireless channel within a particular environment of interest is essential to exploit the full potential of machine learning (ML) for wireless communications and radar applications. Generative modeling offers a promising framework to address this problem. However, existing approaches pose unresolved challenges, including the need for high-quality training data, limited generalizability, and a lack of physical interpretability. To address these issues, we combine the physics-related compressibility of wireless channels with generative modeling, in particular, sparse Bayesian generative modeling (SBGM), to learn the distribution of the underlying physical channel parameters. By leveraging the sparsity-inducing characteristics of SBGM, our methods can learn from compressed observations received by an access point (AP) during default online operation. Moreover, they are physically interpretable and generalize over system configurations without requiring retraining.

Physics-Informed Generative Modeling of Wireless Channels
Benedikt Böck
Andreas Oeldemann?
Timo Mayer? Francesco Rossetto Wolfgang Utschick
Abstract
(Meinilä et al., 2009) families are physically oriented and combine physics with statistical channel properties. These models use the laws of electromagnetic wave propagation to describe channels as function of the corresponding physical parameters (directions of arrival (DoAs), delays, etc.), which in turn are characterized statistically. These characteristics are calibrated and evaluated by real-world measurement campaigns different propagation scenarios such indoor, outdoor, urban, and rural (Yin & Cheng, 2016). While these models characterize channels in generic scenarios and are widely accepted in the communication community, they also exhibit considerable drawbacks.
Learning the site-specific distribution of the wire- less channel within a particular environment of interest is essential to exploit the full potential of machine learning (ML) for wireless communica- tions and radar applications. Generative model- ing offers a promising framework to address this problem. However, existing approaches pose un- resolved challenges, including the need for high- quality training data, limited generalizability, and a lack of physical interpretability. To address these issues, we combine the physics-related com- pressibility of wireless channels with generative modeling, in particular, sparse Bayesian genera- tive modeling (SBGM), learn the distribution of the underlying physical channel parameters. By leveraging the sparsity-indccing characteristics of SBGM, our methods can learn from compressed observations received by an access point (AP) dur- ing default online operation. Moreover, they are physically interpretable and generalize over sys- tem configurations without requiring retraining.
Due to their broad categorization into generic scenario types, such as indoor and outdoor, these channel models cannot represent the site-specific characteristics of particular envi- ronments. This limits their use in ML-aided wireless com- munications, where site-specific channel realizations are critical for training (Kim et al., 2023). The emerging variety of use cases in communications, such as millimeter wave, unmanned acrial vehicles, high-speed train, ultra- masssve (MIMO), joint communica- tions and sensing, and Internet of Things communications, all exhibit their own unique requirements for accurate chan- nel modeling and impose open challenges for measurement campaigns and channel sounders (Wang et al., 2020). Ad- ditionally, the improved accuracy of physically oriented channel models over time made more difficult to directly use them in real-time processing tasks, resulting in an unde- sired eecuracy-simplcciy trade-off (Imoize et al., 2021).
caltiple-inuut-muttileee
1. Introduction
The accurate modeling of wireless channels is critical for system design, network optimization, and performance eval- uation in wireless communications and radar (Wang al., 2018; 2020; Yin & Cheng, 2016). There are two approaches to wireless channel modeling: physically and analytically oriented (Almers et al., 2007). The former integrates the underlying physics in the description of wireless channels and focuses on their accuracy and realism. In contrast, the latter captures channel statistics, ignoring the underlying physics, and provides an easy-to-use framework for evalua- tion and system design schemes (Almers et al., 2007; Imoize al., 2)21). Standardized channel models such as the 3GPP (3GPP, 2024a), COST (Liu et al., 2012) WINNER
One physically oriented alternative is ray tracing. which models the channel purely deterministically. Ray tracing is technique to simulate the electromagnetic wave propa- gation in specific scenarios (Hofmann, 1990). Ray tracing for channel modeling requires a 3D digital replica of the scenario and an accurate characterization of the materials within the scene (Geok et al., 2018: McKown & Hamilton, 1991; Yun & Iskander, 2015). Commercial ray tracing tools like WirelessInside (Remcom) use databases to determine the material properties, requiring full environmental knowl- edge. To address this limitation, differentiable ray tracing is studied in (Gan et al., 2014; Hoydis et al., 2024; Orekondy et al., 2023), enabling the learning of material properties from data. On the downside, high-quality channel state information (CSI) must be acquired for training in each
'Technical University of Munich, Munich, Germany "Rohde & Schwarz, Munich, Germany. Correspondence to: Benedikt Böck bbenedikt.bocck @ @utm.de>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Physics-Informed Generative Modeling of Wireless Channels
scenario of interest, questioning its practicality. Addition- ally, modeling diffuse scattering and the complexity remain challenging objectives in ray tracing (Imoize et al., 2021).
receives during default online operation. Our method aligns with the idea of physics-informed MI., i.e., facilitating the learning from training data by incorporating underlying pre-known physics that the trained model must obey (Karni- adakis et al., 2021; Raissi et al., 2019). However, instead of integrating partial differential equations from physics into a loss function, we leverage the laws of ray opties, building upon electromagnetic wave propagation and relating the channel to its physical parameters (e.g., delays).
Another approach for channel modeling is based on genera- tive modeling, which aims to learn an unknown distribution from data (Bond-Taylor et al., 2022). Using the terminology from (Diggle & Gratton, 1984), generative models can be categorized into prescribed and implicit models. Prescribed models learn the parameters of statistical model (Girin et al., 2021). In contrast, implicil models directly learn to generate samples (Mohamed & Lakshminarayanan 2017). Examples of the former are variational autoencoders (VAEs) (Kingma& & Welling, 2014), while generative adversarial networks (GANs) (Goodfellow et al., 2014) represent the latter. Perhaps surprisingly, almost all current approaches to modeling wireless channels with generative models use GANs and, thus, rely on implicit modeling (Euchner et al., 2024; Hu et al., 2023; Li et al., 2018; Orekondy et al., 2022;
Main Contributions We combine the physics-related compressibility of wireless channels with both implicit and prescribed generative models. The resulting models over- come limitations of existing approaches, i.c., they do not need high-quality training data, generalize over system con- figurations without requiring retraining, and are physically interpretable. By leveraging pre-known structural knowl- edge about conditional channel moments, we additionally show that the prescribed approach, in particular SBGM, pro- vides advantages over the implicit approach by promoting sparsity and ensuring physical consistency. Moreover, we validate the performance on several datasets, showing the su- periority of SBGM for parameter and channel generation.'
Seyedsalehi et al., 2019; Tian et al., 2024; Xiao et al., 2022; Yang et al., 2019). Moreover, since this line of research applies GANs in a black box manner, these methods be- long to the analytically oriented branch of channel models. While these publications show some progress in channel modeling with implicir generative models, there are several open concerns with this approach (Euchner et al., 2024):
2. Related Work
High-quality datasets: The assumption of having access to (lots of) site-specific high-quality training data requires costly measurement campaigns in each scenario of interest.
Early work exploiting implicir GAN-based generative mod- els for channel modeling is given by (Li et al., 2018; Yang et al., 2019). Since then, several GAN variants have been proposed for time-varying (Seyedsalehi et al., 2019) or MIMO channel impulse responses (Orekondy et al., 2022; Xiao et al., 2022). For evaluation, (Orekondy et al., 2022; Seyedsalehi et al., 2019; Xiao et al., 2022) use cluster delay line, tap delay line, or pedestrian A-like 3GPP channel mod- els (3GPP, 2023; 2024a). These link-level models produce channels that all originate from the same cluster angles and delays and cannot represent proper communication scenar- ios where users experience different sets of delays, DoAs, and directions of depature (DoDs). The work in (Euchner et al., 2024; Hu et al., 2023) also studies GANs for MIMO channel impulse responses, but evaluates on the geometry- based stochastic channel model QuaDRiGa (Jaeckel et al., 4) or measurement data. Diffusion models (DMs) for MIMO channels are studied in (Lee et al., 2024; Sengupta et al., 2023). The work (Tian et al., 2024; Xia et al., 2022) considers parameter generation (e.g., delays) using a ground- truth training dataset of parameters. In (Baur et al., 2024b; Fesl et al., 2023), Gaussian mixture models (GMMs) and VAEs are used for channel estimation based on compressed training data. Moreover, (Baur et al., 2025) introduces tech- niques to evaluate generative models for wireless channels.
Generalizability The proposed GAN-based methods only generate channel realizations matching the system configu- ration (number of antennas, subcarrier spacing etc.) used for training, making it difficult to adapt to other configurations.
Physical interpretability & consistency: Their generation process cannot be interpreted physically and they cannot guarantee their samples to obey the underlying physics.
In this work, we demonstrate how prescribed generative models can effectively resolve all these limitations. Specif- ically, we show that the parameterized statistical models in prescribed generative modeling enable integrating the physics of electromagnetic wave propagation. This adapta- tion not only relaxes the requirements for the training data but also shifts the generative modeling approach from the an- alytically oriented to the physically oriented channel models and provides generalizability, interpretability, and physical consistency. Our approach shares features with standardized channel models, as it also describes the wireless channel as a function of the physical parameters. However, the sta- tistical model that characterizes these parameters does not need be calibrated through extensive and costly measure- ment campaigns but can be trained by a few compressed and noisy channel observations that an AP or base station (BS)
Source code is available at https: / http//gtttbbcco. beneboeck/phy- wwtp://hhyi--f-ifoo inniiin--g-nen goen-mod-uee wi reless.
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Physics-Informed Generative Modeling of Wireless Channels
3. Background and Problem Statement
{ M2
3. Ma
3.1. Physically Characterizing Wireless Channels
Receiver
Wireless signal transmission causes the signal to attenuate and undergo phase shifts. Moreover, as the signal prop- agates, it can also get reflected or scattered by obstacles. This results in multiple paths (or rays) between the transmit- ter and receiver, each characterized by its own attenuation and phase shift, which, in turn, are linked to the channel parameters (e.g., angles and delays) through geometrical optics. The wireless signal transmission can be modeled as a linear time-variant system (Tse & Viswanath, 2005) and, thus, these effects are captured by the wireless base channel (transfer function), which depends on the time i, the (baseband) frequency f. and the receiver and transmitter positions r'. and rT in local coordinate systems, i.e.,
Transmitter
Figure I. Path schematic when transmitting wireless signals.
to equidistantly sampling (1) in both the frequency and time domains, yielding the OFDM channel matrix
Me
H
Pt.m a (a10e,m)aayTTaae (TE,)
(3)
8= m=1
with a. (e c 21 114T and a (T(mm,)
e 2 (5-1)= For our following considerations, we define the set of physical channel parameters
P [pe,m: Di,ms Té,m: 2 (
LL t.m=
(4)
Me
We specify subsets containing the parameters associated with a specific domain by their subscript, c.g., PR
h(t, PT)
Pt.m el 22 e- 12rrend x
1=1 m=1
{pE,, L.m Emm=1 or Pt. (pe,m: De,m, T2,m 1.m=1-
J.
j ( Cam TTR (
Trr
(1) The channel parameters contain the number L of paths, the number Me of subpaths per path E, the complex path losses PE,m, the doppler shifts Dr,m, the delays Ti,m, the DoAs 22/30 and the DoDs 22.mm A schematic of the paths when transmitting wireless signals is given in Fig. 1. The wavevec- tor k-)) is defined as (-)) = (2T/ A) e--) with wavelength A and e(-) is the spherical unit vector. Transforming (1) via a Fourier transform (FT) with respect to any of its arguments results in their dual so-called dispersive domains (Yin & Cheng, 2016). Moreover, one is typically interested in sampled version of the wireless channel h(-). For instance, when the receiver is equipped with multiple antennas, receiv- ing signals can be viewed as sampling the wireless channel h(-) in (1) at the antennas' spatial positions. By exemplary ignoring the channel's time- and deeeedecy (i.e., t f = 0) and accounting for the spatial sampling via mul- tiple antennas at the receiver, (1) can be represented as
3.2. Sparse Representation of Wireless Channels
One possibility to beneficially represent channels is by ex- ploiting their compressibility (i.e., approximate sparseness) regarding a physically interpretable dictionary (Dai & So, 2021; Gaudio et al., 2022). For instance, by placing the re- ceiving antennas with X/2 spacing in a uniform linear array (ULA), aligning the local coordinate system's 's z-axis with the ULA, and assuming the ULA to be in the far-field, the DoAs (R) reduce to position-inddeendent angles wie. E [-x/2, m/2) with k(.)TrR, . 1) sin()) )) in (2). By additionally defining grid Gn = {gT SSn) Sn/2-1 cardinality SR. (2) can be represented by
Sa/2 of
Sn/2-1 Σ
y Sn.
h =
SR
QR
= DR
(5)
8R/2
with DR [an( -2 )...... ag (T
x )[, and SR Sn
(8n/2 1- If the DoAs wa, coincide [ (-Sn/2) Sn.
Me
h
Pe.m an ( R) (2
with gridpoints in GR. the entries sk fulfill SR = Pt.m (R)
8=1 m=
if w / and are zero else. In this case, 8R per- fectly determines the channel's parameters PR. Since this usually only approximately holds, &n is not exactly sparse but rather compressible. Equivalent compressible represen- tations exist in the frequency and temporal domains. By defining the grid G. x Gj with G. = 1./221 (- and
with ax(+))====)) = e- ((()))Tr...) and the position TR. of the ath receiving antenna. The vectors (n(()) are typically referred to as steering vectors. Equation (2) represents the generic iut-iiippuuuulipllouuuupuuu (SIMO) channel, which does not cover the time and frequency domains. As another ex- ample, the receiver and transmitter in -rtquennal-rrqquency- mulliplxiu (OFDM) are both equipped with sin- gle antennas, whose positions can be set to the coordinate systems' origins, i.e., FT/R,1 = 0. Thus, OFDM covers no spatial domains and is characterized by its subcarrier spac- ing Af and symbol duration AT. This setup corresponds
g, (JT/S,) $y-1 where 7 and 7 bound the maximally reachable doppler shift and delay, and S,, S₁ E N, we also find a compressible representation of the vectorized OFDM channel in (3) equivalent to (5), i.e., h = Dt with Dr, = D, & Dy. Appendix A.I provides a more detailed description.
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Physics-Informed Generative Modeling of Wireless Channels
3.3. Generative Learning through Inverse Problems
3.4. System Models and Problem Statement
In some cases, generative models should learn a data distri- bution that is only indirectly observable from the training data, c.g,, if only compressed training data is available.
We consider a scenario in which an AP receives channel observations y: from different locations within the environ- ment it serves. One example is outdoor communications, where users periodically send pilot symbols to their BS. The observations are noisy and potentially compressed, i.e.,
Implicit Approach The GAN variant AmbientGAN is an implicit generative model that addresses this objective (Bora
y = Ah + n
et al., 2018). GANs simultancously train two neural net- works (NNs), the generator Ga(-) and discriminator (.).), in a competitive setting (Goodfellow et al., 2014). The generator takes inputs 2, drawn from fixed distribution p(z) and aims to minimize a certain objective by producing samples Go(z,) that resemble the training samples y:- In ordinary GANs, the discriminator's goal is to differentiate directly between the generator's outputs and the training samples by maximizing the same objective. However, Am- bientGAN assumes a (stochastic) mapping f(.) that relates the training samples and the random variable of interest (i.e.. y/: f(:)) and applies f() (i..., fGe before forwarding the result to (-).) This trains the generator to generate samples s; while only having y: for training.
where h represents the channel, A is the measurement ma- trix, and is AWGN. The AP eventually collects a dataset y = yy NN that captures site-specific information from the entire environment. In our work, we focus on the fol- lowing two systems.
Narrowband static SIMO: The receiver is equipped with multiple antennas, and the transmitter has single antenna. The channel h only covers the spatial receiver domain and equals (2). The matrix A in (9) is an identity, i.c., A = Mi
The channel h in (9) covers no spatial domain, and vectorizes (3), i.e., h = veo( H). The observation matrix A is a selection matrix, i.e., its rows are distinet unit vectors, masking out particular entries of h.
seletivselective OFDM:
Prescribed Approach SBGM is a prescribed generative modeling framework that also addresses the learning of hidden distributions (Böck et al., 2024b). can learn non- trivial parameterized distribution (s) for the compress- ible representation s of a signal h of interest given a dic- tionary D, i.e., h- = Ds. This learning can be done solely by noisy and compressed training samples y. = Ah, + ni with additive white Gaussian noise (AWGN) n, and known measurement matrix A. SBGM combines sparse Bayesian learning (SBL) (Wipf& & Rao, 2004) with a sub-class of prescribed models, namely, conditionally Gaussian latent models (CGLMs), i.e., models with a conditioned Gaussian on a latent variable z. After minorly extending it to complex numbers, the statistical model in SBGM is given by
As the channel h does not cover all four possible domains in these systems, only a subset of P in (4) determines h. Thus, depending on the employed system, the environment is characterized by a distribution over this subset of physical parameters, i.e., (PPR) in SIMO and p(Pr.)) in OFDM. This work aims develop physics-informed generative model that can learn these distributions solely using the imperfect training dataset V. While this work focuses on SIMO and OFDM, there is a straightfoward generalization to other systems, such as MIMO and MIMO-OFDM.
4. Physics-Informed Generative Approaches
Building on the sparse channel representation explained in Section 3.2 we observe that choosing a parameter grid GR or G. x Gy with sufficiently high resolution enables the com- pressible channel representation SR or st.f to uniquely deter- mine the channel's 's parameters PR or Pr. Thus, we can re- place the learning of p(PR) and p(Pr,)) by instead learning p( ((R) and p(st.)), ). with the grid resolution being the hyper- parameter that controls the approximation error. Moreover, by incorporating the sparse channel representation into the dataset in Section 3.4, we reformulate our problem to that of learning p(s) given y = {y: = ADs; + ) with 8; ~ p(s), and n. ~ Ne of 1).²
(|s~)(y)s) p(y)s) = Nw ADs, I), y|s ~
(6)
s|2 - pozs)) = NC (s: O,
(7)
~~ p=))
(8)
with known measurement matrix A, learnable parameters 0 and ö. variance o² of the measurement noise, latent vari- able 2, and s follows a conditionally zero-mean Gaussian (s|2) with z-dependent diagonal covariance matrix. The work in (Böck et al., 2024b) shows that this particular choice of (s(2) serves as a sparsity- promooinrooonggggooo regularization. It introduces the compressive sensing GMM (CSGMM) and compressive sensing VAE (CSVAE) as two particular imple- mentations of SBGM. In the latter, a NN decoder outputs the conditional variances (-) revealing its prescribed char- acteristic. After training, P5, (s)) is used for regularizing inverse problems and has been demonstrated to outperform standard priors in compressive sensing (Böck et al., 2024b).
FFoor readability, omit the subscripts of s. P. and & from now on when the argument applies to both OFDM and SIMO.
'While the actual noise variance mainly depends on the receiver hardware and, thus, does not vary between users at different loca- tions, the typical channel-wise pre-processing of normalizing by an estimated path loss effectively results in varying noise variances.
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Physics-Informed Generative Modeling of Wireless Channels
Channel Parameter Generator
Channel Parameter Generator
n2
(
8
D
A
Ygen
Enc 2a
Dec (a)
Yo
(-(-)
y
(2)-)
Po
A
D
Yreal
y
used for training (O.
1)
n
used for training
Figure 2. I Layout of the physics-informed implicit AmbientGAN.
Figure 3. Layout of the physics-informed prescribed CS CSVAE
Channel Parameter Generator
4.1. Incorporating Physics via Implicit Modeling
m
One possibility to address this problem is by utilizing the implicit AmbientGAN described in Section 3.3. By combin- ing (9) and the compressible representation of channels, cf.
h A
y + p(k)-)
k
D
D+y
with prior p(k)
(5), we identify the mapping f()) of AmbientGANs by
VTX
(,,)
n1
used for training-
y = fs.) ADs, + ni
(10)
Figure 4. Layout of the
prescribed CSGMM.
is assumed to be NN I). One forward operation during training consists of first inputing training samples y; to the encoder (Enc) to yield (z and ( We then forward the samples 2, drawn from ( (z(l)) to the de- coder (Dec) whose output represents Yo (2:) fully charac- terizing ( žx) (ef. (7)). A key aspect in SBGM is that po E y/) is Gaussian with closed-form mean and co- variance specified in Appendix A.3. Based on these mo- ments and (1a(())) (V:), Yo (ž:) derived during the for- ward operation, all terms in (12) can be calculated and dif- ferentiated with respect to (0, Φ). The closed forms and a more detailed explanation are given in Appendix A.4. Note that we solely require the processing chain in Fig. 3 up to v /Te for training. After training, we generate a new sample s by first drawing a ~ pa(z) and forwarding it through the decoder. This generates v (z), which allows us to draw a sample s ~ po 2) (cf. (7)). The necessity of this second sampling operation marks the key difference between the prescribed and implicit modeling approach.* +
with n; being AWGN exhibiting the same statistical char- acteristics as the noise present in the training dataset. A
schematic of the resulting AmbientGAN is given in Fig. 2. For one forward operation during training, we first gen- erate several z; drawn from a simplistic distribution (e.g.. (O, I)). These samples are then forwarded by the genera- tor G-(+) and the mapping f-). which in turn is character- ized by the pre-defined dictionary D, pre- pre-known measure- ment matrix A and noise n,, This procedure results in the generated @-differentiable observations ygen,
y/gnn. = ADG + n2.
(11)
In the final step of one forward operation, the discriminator (-)-) inputs ygen, as well as training samples Great, E y and approximates @-diiferrentable quantity measuring the distance between the desired and currently learned distribu- tion. The NN weights (ω, 0) are learned by solving mini- max optimization problem with Wasserstein distance-based objective (Gulrajani et al., 2017). A detailed explanation is given in Appendix A.2. After training, a new s can be generated by drawing a z and computing (z).
Similar to the schematic in Fig. 3, Fig. 4 shows schematic of the CSGMM when being applied to learn p(s). For the CSGMM, the distribution Pa Pa(z) (z) is categorial distribution, i.e., P5(2) = (k) = Pk (A: = 1,... K). Moreover, (z = Yk, i.e., s follows a GMM with zero means and di- agonal covariance matrices. The learning of (pk. Yt K k=1 (EM) algo-
4.2. Incorporating Physics via Prescribed Modeling
Building on prescribed generative modeling, we can also use SBGM, and in particular the CSVAE and CSGMM, explained in Section 3.3 to learn p(s) from y.
done by an extended
pntecttttioommammizzatte rithm considering the additional latent variable in (6)-(8)
(Böck et al., 2024b). In the E-step, the model's posterior p(s, klys) = p( [k, (k)p(k()))) ) has to be computed for each training sample y:. Equivalent to the CSVAE, p(s(k; y:) is Gaussian with closed-form mean and covariance (cf. Ap- pendix A.3) also resulting in a closed-form E-step. The M-step equals the one (Böck et al., 2024b). Both are sum- marized in Appendix A.5. Similar to CSVAES, we generate new sample after training by first drawing a le ~ ps(k) and then another sample s ~ Po k) (cf. (7)).
Fig. 3 shows schematic of the CSVAE when being ap- plied to learn p(s). Equivalent to VAEs, the training of the CSVAE is based on lower bound of the log-evidence (Böck et al., 2024b), i.c.,
E Do [ogp(,,(s)]
(12)
(200(22))))l)l - (po (s|2,, y. (p) (s).))
with &, drawn from a variational distribution (2 y:) =
Note that VAEs also he used as implicit models by having the decoder output only means, cf. (Dai & Wipf, 2019).
N(2; p diag(o? (v))). The distribution pa( (z) in (8)
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Physics-Informed Generative Modeling of Wireless Channels
4.3. Why SBGM Is Capable of Parameter Generation
5. Discussion
In (Böck et al., 2024b), SBGM is introduced to directly solve inverse problems, and not to generate new samples. Indeed, (Böck et al., 2024b) discusses that SBGM is not suited for generation due to its restriction on $ to have universal conditional zero mean, i.e., E[s] x] O for all z (cf. (7)). This assumption strictly limits its capability of generating realistic samples as it is typically not possible to decompose the true unknown distribution p(s) in this way (Böck al., 2024b). This raises the question whether this limitation also hinders SBGM from properly generat- ing wireless channels. However, as the following discussion shows, wireless channels exhibit unique property that elim- inates this limitation in its entirety when incorporating the common assumption of stationarity. Moreover, we show that the conditional zero mean and diagonal covariance prop- erty in (7) perfectly aligns with model- insights about the structure of conditional channel moments in general.
Generalizability Section addresses how the physics- informed generative approaches circumvent the need for high-quality training data and provide physical interpretabil- ity. One more claim from Section 1 is generalizability, i.e., the adaptability to other system configurations, such as different numbers of antennas or subcarrier spacings af- ter training. The link between the compressible channel representation s and the physical parameters P is solely de- termined by the choice of the parameter grid 9 (cf. Section 3.2). On the other hand, the system configuration is solely encoded in the columns of the dictionary D, but the pa- rameter grid G is system configuration independent. Thus, the learned channel parameter generators in Section 4 (cf. Fig. 2-4) do not depend on the dictionary used for training. In consequence, when sampling a new s from the genera- tive model after training and computing the corresponding channel realization h, we can use new dictionary D (new), whose domain must match the dictionary used for training, but whose range can be easily adapted to newly desired system configuration without any retraining.
The work in (Böck et al., 2024a) explores whether the struc- tural properties of the first and second channel moments (i.e., E(h] and E(hh")) DD) remain intact when conditioning the channel on some side information z. This publication considers the channel representation which is based on the sum of steering vectors as it is done in (2) and (3). In par- ticular, it shows that under some mild conditions, such stationarity, the channel is guaranteed to have a conditional zero mean and (block-) Toeplitz-struuttred covariance, i.e..'
Providing Training Data for Wireless Communication Most ML-baaed methods for the physical layer in wire- less communications either require lots of ground-truth and site-specific channel realizations for training or at least site- specific pairs of input and output signals linked by the wire- less channels. However, finding efficient and scalable ways to obtain this data in each scenario of interest is open challenge (Kim al., 2023). With the models from Section 4, we present a solution to this problem. These methods can learn from corrupted data that a wireless receiver obtains, e.g., during default online operation. It then can generate a desired amount of site-specific clean channel realizations for, e.g., training ML-based methods.
E hb) xz| = 0, Eihh" [[]] (ooepi-TToeplitz
(13)
if does not contain information about the phases of the path losses Pe,m in (1). Noting that these phases cannot represent site-specific channel features, (Böck et al., 2024a) argues and empirically demonstrates that the latent variable z in CGLMs satisfies this requirement. As a result, when restricting any CGLM (e.g., VAEs or GMMs) to satisfy (13), we enforce the model to be physically consistent.
and E|ss to be diagonal (cf. (7)). Moreover, the channel h and are related linearly by a dictionary D (cf. (5)). Thus, SBGM enforces the channel h to exhibit
SBGM models such that E[s|z]
Implicit versus Preseribed Modeling The implicit and prescribed modeling approaches in Section share the prop- erties of not requiring ground-truie -trth data for training, being physically interpretable, and generalizable over system con- figurations. However, there are also two key properties in which these approaches differ from each other, rendering the prescribed approach to be generally superior. As the com- pressible channel representation s typically exhibits a much larger dimension than the channel observation y, learning p(s) from a dataset y = {3, ) is an ill-posed problem. Therefore, to properly learn p(s). some regularization re- quired. On the one hand, when applying SBGM for learning regulariza-
E[h|z] = D [[s]2]] = 0, E E/hh = (x))
(14) Latter is Dolock-YToeplizz when inserting the dictionaries from Section 3.2, and, thus, SBGM perfectly aligns with (13). In consequence, the conditional zero mean and diago- nal covariance property in SBGM (cf. (7)) is no limitation for generating wireless channels, but rather the opposite. It regularizes the search space of the statistical models to those that are physically consistent. 6
p(s). we impose the general sparsity-promotie tion of SBGM in (cf. Section 3.3). On the other hand, as discussed in Section 4.3, this regularization extends beyond sparsity and additionally enforces the learning process to yield a physically consistent model. In comparison, the im-
A block of Toeplitz matrices arises when considering multiple OFDM.
domains at once, e.g., time and frequency in 'A more detailed explanation is given in Appendix A.6.
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Physics-Informec Generative Modeling of Wireless Channels
Table 1. Comparison between different channel modeling schemes.
Requires No High Quality
Site- Specific
No Accuracy Simplicity Trade-Off
Generalizable other System
Physically Interpretable
Guuranteed
Sparsity- Inducing
Channel Model
Training Data/
Physical
Measurement Campaigns
Contigurations
Consistency
Standardized Channel Model (e.g. 3GPP))
Black-Box Gen. Modeling (Leaming From la and Generating h))
Physics-Informse Implicit Gen. Modeling (Section 4.1)
V
Preacribed SBGM (e.g. CSVAE, CSGMM) (Section 4.2)
plicit-based learning process in Section 4.1 allows to learn p(s) from V, but imposes neither sparsity nor physical con- sistence during training and, thus, keeps the overall problem ill-posed. Table provides a comprehensive comparison between the implicit and prescribed modeling approaches well as standardized channel models and black-box bos-beed generative modeling, both discussed in Section 1.
represents an urban macro-cell, in which users can be in line- of-sight (LOS), non-LOS (NLOS), as well as indoor and outdoor. The other (5G-Ruraa (EG-Rural) represents a rural macro- cell, in which all users are in LOS. We also use the ray tracing database DeepMIMO (Alkhateeb, 2019) for SIMO in Appendix F. For the channel observations in OFDM, we generate one random selection matrix A extracting M entries from h and apply it to every training channel (cf. (9)). In all simulations and each training sample, we draw signal-to-nn gosiie ratios (SNRs) uniformly distributed between and 20dB, defining the noise variance of (cf. Section 4). A detailed description of the datasets, chosen configurations, and pre-processing is given in Appendix D.
Foundation Modeling Perspective While SBGM can generate training data, can also be directly used for real- time processing tasks. Recently, VAES and GMMs have been used for channel estimation (Baur et al., 2024a; Böck et al., 2023; Fesl et al., 2024; Koller et al., 2022), channel prediction (Turan et al., 2024a), and precoder- and pilot design (Turan et al., 2024b; 2025). Thus, the CSVAE and CSGMM not only generalize to arbitrary system configura- tions but also provide information that can be directly used for these downstream tasks without the need for specific training. This aligns with the idea of foundation modeling, ie., training generalized ML model and applying it to various downstream tasks without retraining.
Evaluation metrics performance by the power angular profile
We evaluate the parameter generation
Nus
(4) 12
((R) (q)
R
(15)
Niest
Su 12/22-1
(g)
g= Sp/2
R
as well as the channel-wise angular spread
S (sn) = g S2/2 (wg
(n) Σ SR/2-1
(4)
(g)
MR
(16)
Both proposed approaches in Section 4 have limitations related to non-stationary channel generation, off- grid mismatches, and pilot patterns, ef. Appendix
Limitations
Σ y== Sn/2
Sn/2-1
1012
(g)
(g) with *R.; being the gth entry in the ith newly gen-
erated sample $R,6- Sa/2-1 and (10)) (I y== y===Sa/? wa
Moreover, wg (R)
gT gT/Sp
Model Extensions Using ray tracing for wireless channel modeling allows customizing the number of paths for each channel (Alkhateeb, 2019). SBGM offers the same flexibil- ity, cf. Appendix C.1. Moreover, when considering multiple domains at once (e.g., OFDM), one can reduce the number of learnable parameters by further constraining the learned variances SBGM, as detailed in Appendix C.2.
(R) | ( gg=-5//2 |
Sa/211
(Zhang al., 2017). For the channel generation perfor- mance, map newly generated sR (or $t,f) to h using
dictionary DR (or D..,) (ef. Section 3.2) and evaluate the channel generation with the cross-validation method from (Baur et al., 2025; Xiao et al., 2022). Specifically, we first train each generative model using y (cf. Sec- tion 4) and generate N(gen) channels with each model to train an autoencoder for reconstruction by minimizing the mean squared error (MSE) for each generative model separately. We then compress and reconstruct ground- truth (i.e., QuaDRiGa) channels using these trained au- toencoders and evaluate the normalized MSE (NMSE) nMSE = 1/Nies MMan (11b; - hal3/N and the cosine sim- ilarity Pe 1/Nes MMr((
6. Experiments
6.1. Experimental Setup
Datasets For evaluation, we use four datasets. We use a modified standardized 3GPP spatial channel model for SIMO, which we adapted to better illustrate our method. For simulations with OFDM, we use two different QuaDRiGa- based datasets (Jaeckel et al., 2014). One (5G-Urban)
[|]]///|2||bbb|aa with h; and h, being the ground-trui and reconstructed channel.
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Physics-Informee Generative Modeling of Wireless Channels
a)
Ground Truth
CSGMM
CSVAE
AmbientGAN for
M-SBL
0.06
Counts
1500
1000
0.03
500
90
60
30
0
30
60
90
0
10
20
30
40
50
60
70
(R)
S
(8R)
wi
b) Exemplary training
Newly generated samples |s, after training (8, E C256)
samples yi C
Rc
Ground Truth
CSGMM
CSVAR
AmbientGAN for
M-SBL.
antennas
90
0
90
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-90
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Figure 5. a) Power angular profile Pdd (g) and a histogram of the angular spread 5(x) (sn) from 10 000 generated samples by CSVAE, CSGMM, AmbientGAN for and M-SBL, compared ground truth, b) eight exemplary training samples, c) squared absolute value of four exemplary generated samples from all models, respectively.
Baselines & architectures For parameter generation in SIMO, we compare the prescribed CSVAE, the prescribed CSGMM (cf. Section 4.2), the implicit Ambien(GAN trained to output s (cf. Section 4.1), and M-SBL (Wipf & Rao, 2007). While M-SBL. was not originally designed for generation, it is equivalent to the CSGMM from (Böck et al., 2024b) with K component, effectively fitting a Gaussian s that can be used for sampling. For generat- ing channel realizations, we also evaluate AmbientGAN (Bora et al., 2018) that learns to directly output h. A de- tailed overview of all architectures, hyperparameters, and the ground-truth baseline is given in Appendix E.
outliers with large angular spread, both closely resemble ground truth. In contrast, the implicit AmbientGAN sig- nificantly overestimates the angular spread. This is due to the missing promotion of sparsity and guarantee of physical consistency during training (cf. Section 5). Since M-SBL fits a Gaussian to 8, can not assign different directions to different samples, resulting in largely overestimating the angular spread. Since M-SBL equals CSGMM with K 1 component, this histogram also illustrates the advantage of CSGMM having more than one component. The different characteristics are also illustrated by the generated samples in Fig. 5 c). CSGMM and CSVAE generate samples that resemble ground truth, whereas AmbientGAN produces samples with a broader angular spread, and M-SBL encodes all directions of the power angular profile into every sample.
6.2. Results
Modified 3GPP In Fig. 5 a). the power angular pro- file p! (q) well as a histogram of the angular spread (.()) (sR) is given. The number of antennas N = .M is set to 16, and the number of gridpoints S is set to 256. The number N; of training samples is 10 000. In Fig. 5 b) and c), exemplary training samples and newly generated sam- ples shown. In general, all power angular profiles are consistent with ground truth by, e.g., assigning power to directions absent in the ground-truth profile. CSGMM and M-SBL produce sharper power angular profiles with more peaks than the ground truth. CSVAE yields a power angular profile that best matches the ground truth. When consider- ing the histogram of angular spreads, the methods exhibit very different characteristics. While CSGMM slightly over- estimates the angular spread, and CSVAE exhibits few
QuaDRiGa In Fig. 6 a) and b), the nMSE and Pa of the autoencoder reconstruction are shown over the num- ber N, of training samples with fixed M = 30 for the QuaDRiGa dataset 5G-Urbad (cf. Appendix D.2). We choose S, = S5 = 40, 7 6js and & = 0.25kHz (cf. Section 3.2). all simulations, (( = 30 000. Ambient- GAN for h performs significantly worse than all other meth- ods. AmbientGAN for s improves over N,. but does not out- perform M-SBL. Overall, CSVAE and CSGMM achieve the best results. Notably, CSGMM closely approaches ground truth performance even for small Nr. for which we train the autoencoder by means of 30 000 ground-truth channels. Given that M-SBL fits a Gaussian to s and performs con- sistently well, we infer that the true p(s) in the 5G-Urban
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Physics-Informed Generative Modeling of Wireless Channels
Ground Truth
CSVAE -E- CSGMM --
AmbientGAN for -
M-SBL
AmbientGAN for h
110
b)
-10-2
-10
-10
3
3
nMSE.
HMSE
1
0.3
2
0.3
0.1
0
10a
10
10a
10a
10
10
0.7/15 1.4/30 2.1 2.1/45 2.8/60 0 in kHz T in uss
0.7/15 1.4/30 2.1 2.1/45 2.8/60 9 in kHz / in SUS
N₂
N₂
e)
Exemplary training samples y E C
Newly generated samples |s.
(s: E C 100x40 of CSGMM after
Re
Im
iterations
600 itcrations
1200 itcrations
1800 iterations
360
Ojts
14.63
kHz
0.7kHl Okllz
0.66kl
0 ms
1
frequency
delay
duppler
time
6
0.05 0
0.03 0
0.1
Figure 6. a) - d) nMSE and Pe for reconstructing ground-truth channels by an autoencoder trained on channels h = Ds produced by the trained models, respectively. In a) and b). we vary N, for the generative models with fixed M
and dataset 5G-Urban, and in c) and d), we vary 9 and 7 with fixed N₂ = 3 000 and dataset 5C-Rural e) four exemplary training samples, f) illustration of the CSGMM training by the squared absolute value of two generated samples after 0, 600, 12000 and 1800 iterations.
dataset exhibits Gaussian characteristics. This is in con- trast to the results in Fig. 6 c) and d), where we used the 5G-Rural dataset (cf. Appendix D.2). Here, we set N₂ = 3 000, but vary the maximally resolvable Doppler 7 and delay 7. We choose S. = S = 40, M = 30. Overall, M-SBL performs the worst. In general, 7 and F are regular- izing hyperparameters. When choosing both just encom- pass all occurring ground-truth delays and Dopplers without causing ambiguities, the implicis AmbientGAN shows good performance. However, increasing & and F weakens this reg- ularization, leading to a significant drop in AmbientGAN's performance. CSGMM and CSVAE additionally regularize their training by inducing sparsity and ensuring physical consistency. Thus, they maintain strong performance even when 7 and F are large. Fig. 6 e) shows four exemplary training samples. To illustrate the OFDM masking with pilot symbols, we plot the real (Re) and imaginary (Im) part of the 30-dimensional observations within the OFDM grid representing the underlying channel E 144x14 with = vec H). Fig. 6 f) illustrates the training of CSGMM when being trained on 5G- Rural. Specifically, we plot the squared absolute value of two samples in the delay-Dopple domain after O, 600, 1200, and 1800 training iterations, demonstrating the increase in sparsity during training.
Additional Results In Appendix F.1, analyze the gen- eralizability of SBGM to other system configurations (cf. Section 5). Moreover, in Appendix F.2 and F.3, we evaluate controling the number of paths per sample, and the reduction of learnable parameters by enforcing additional structure. In Appendix F.4, we also test the parameter generation for the DeepMIMO dataset. In Appendix G, we provide pseu- docode and some implementation specifications.
7. Conclusion
In this work, we combined the physics-related compress- ibility of wireless channels with prescribed and implicit generative models to yield prysics-informed generative modeling framework for wireless channels. We also estab- lished that due to the promotion of sparsity and guarantee of physical consistency, SBGM as prescribed model is su- perior to implicit alternatives. These methods can generate the physical channel parameters and channel realizations themselves. We validated that the parameter generation consistent with ground truth and that the introduced models outperform GAN-based black-box baselines for generat- ing channel realizations. Limitations, such as generating non-stationary channel trajectories, are part of future work.
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Physics-Informed Generative Modeling of Wireless Channels
Impact Statement
wireless PHY layer. JEEE Transactions on Machine Learning in Communications and Networking, pp. 1-1,
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
2025. doi: 10.1099TMLLCN.2225333577 1022.
Bello, P. Characterization of randomly time-variant linear
channels. IEEE Trans. on Communn Sysz. 11(4):360-393, 1963. doi: 10.1 1099TCOMM.193.1088777777
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Physics-Informed Generative Modeling of Wireless Channels
Yang, Y., Li, Y., Zhang, W., Qin, F., Zhu, P., and Wang, C.-X. wireless channel modeling: Challenges and opportunities. IEEE Commun. Mag., 57(3):22-2 27, 2019.
Yin, X. and Cheng, X. Propagation Channel Characteriza- fion, Parameter Extimation, and Modeling for Wireless Communication. Wiley-IEEE Press, November 2016.
Yun, and Iskander, M. F. Ray tracing for radio propaga- tion modeling: Principles and applications. IEEE Access, 3:1089-11000 1100, 2015.
Zhang, R., Lu, X., Zhao, J., Cai, L.,, and Wang, Mea- surement and modeling of angular spreads of three- dimensional urban street radio channels. IEEE Transac- tions on Vehicular Technology, 6665):3555-3570, 2017.
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Physics-Informed Generative Modeling of Wireless Channels
A. Additional Explanations
A.1. Channel Sparsity in OFDM Systems
The OFDM channel matrix is given by (3), ie.,
Mr
H
Pt.m Or ar (a
(17)
(=1 mm
with ar(Ve.)), = (f-1)_ 11-1) and a (Tt,m) = e j2 ( (1-1))) We assume that we have access to (not necessarily tight) upper bounds F and for all delays and doppler shifts, i.e.,
max Tt,m t. [0 S 8
(18)
max E,
(19)
By deciding for number of grid points Sy and S; in the delay and doppler domain, respectively, we define the grids g, = (jT / S as well as G. = S,/2 In consequence, we can represent the OFDM channel matrix in (17) using 9 x Ge as
5./2-1 -
i20 JT )) = Dt.f8t.f aas
h h= Σ
(1.)) f
vec (arl
(20)
St
i=- 51/2 f=0
with = =
St,lks
ttf and Di, Dt ? Dy. Moreover,
(1.))
27
D. = [a((-)) 7) (T
(21)
S
Dy = (as (0). as(T
(22)
S
and we know $t, to be compressible in general.
A.2. Detailed Explanation of Training the Implicit AmbientGAN
The implementation of the implicit AmbientGAN is based on the Wasserstein GAN with gradient penalty (Gulrajani et al., 2017). In particular, the min-max objective function is given by
min max D.D (AD + m)] B(Do(y)) + A[[([ - 1)
(23)
Φ
with y - +1
n) and (40 1). The inner maximization realizes the maximization included in the Wasserstein distance according to the Seattorovich-Rubintter duality, and with the outer minimization, aim minimize this distance. The third term regularizes Do towards being Lipschitz-1 continuous, required for the proper definition of the Wasserstein distance (Arjovsky et al., 2017). The second expectation can be estimated batch-wise using the training dataset, while the first one can be estimated by Monte-Carlo sampling z and n from their known distributions p(z) and p(n). The parameter À is hyperparameter.
A.3. Detailed Description of the Closed-Form Moments of po p0(s)2,1 y)
The distribution pa(s|z, z, y) forms a conditioned conjugate prior of yls (conditioned on 2). Thus, we can compute the moments pe (s(s) c y:) as (see Appendix in (Böck et al., 2024b))
H
(ž.)
C v.
(2,)
(C Crul (2:)
y/1
(24)
C colurn (ž.) diag(ro (ε.))
C5.. 2n (2:)
(CHilz (z.))()))
(
(25)
"The term an (12 (8) )a (12 (T) is also sometimes represented by its equivalent tensor product an an (R) @ (12 ef. (Böck
al., 2024a).
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Physics-Informed Generative Modeling of Wireless Channels
with
C² Ei (2) ADdiag(ro(E)) AH +
(26)
and C (2:) diag(Tr((;)) D# Minor differences the setup in (Böck et al., 2024b) the sample-dependent noise variance of in (26) as well taking the Hermitian instead of the transpose.
One efficient and differentiable implementation of computing the inverse in (24) and (25) is by first calculating the Cholesky decomposition L (Z (. (a.) H of C3 & (ž:), then computing Lg (8 (2.) by solving M linear systems of equations with triangular matrix L |& (ž,), and finally computing Lg (2.)- L 8 (2,)- H, This can be done for the CSGMM and the CSVAE equivalently. Also note that C (ž;) in (25) does not have to computed explicitly for training the VAE (cf. Appendix A.4). Moreover, for training the CSGMM, we solely require its diagonal entries (cf. (34)).
A.4. Detailed Description of Closed Forms for the CSVAE Objective
The objective (12) for CSVAEs consists of three terms, i.e., 2p) flog (lgg p(Bss(() DKL( (xL(go))))))l((())) and DxL. (po E. (s) 2.). The second is given by (Kingma & Welling, 2014)
1 Nr. 2
(x.(gg(z\uu)llo<z) =
Σ (1 + log (B))) - (U: - (3
(27)
with (=/1/))) = N(2; (Bi), diag( (3 (v:))) and (v,) and (1:)) being the jth entry of (*(U)) and o (vr), respectively. Moreover, Nr. is the CSVAE's latent dimension. Separate closed-form solutions for the first and third term in the real-valued case with constant noise variance are given in (Böck et al., 2024b). When adjusting the derivation to the complex-valued case with varying noise covariances o² both terms equal
1 s)] --((A) (M log((T??)
elv.... (2,)11 + LADDC ll.. (A + o? (Iv. AD 1
E
(28)
and
log det ((:))() leg (Colun:B(E)))
DKL (s E. yi vi)llpo
(29)
+ir [daag ((70(2))~1) coe . (E:) + P4
elv. R: (ž,) diag (ro(2:) (2:)) 11 alui. (2:)
where 140 1igh 8. (2)) and C ( are the mean and covariance matrix of ( y:) (cf. (24) and (25)). Moreover, (Böck et al., 2024b) reformulates both terms and shows that they partially cancel out, leading to a more efficient computation, i...
Ep [og - (pe 2 v.)l po(() = - (M lMM(TTTT + (-M log of + det det Cyi (Z,) Note that due to having complex-valued Gaussians, (30) differs slightly from the terms in (Böck et al., 2024b).
1 (lv: AD al (2) o?
(30)
o| ui (ž.) diag (2)(
140 a ely. (()))
110
A.5. Detailed Explanation of the E-step and M-step for the CSGMM
E-step Generally, the E-step corresponds computing the model's posterior p(s, kly;) p(s[k:, y:)p(k(y;)) for each training sample ys Equivalent to the CSVAE, p(s k, y:) is Gaussian whose mean and covariance can be computed via (24) and (25). Moreover, ((k)y)) can be calculated using Bayes, i.c.,
k(pkk
p(kl)))) Σn P(y (k))
(31)
Due to modeling sik to be zero-mean Gaussian and y being linearly dependent on s (cf. Section 3.3), the distribution p(y:kk) [k) is zero-mean Gaussian with covariance matrix (cf. (26))
k
ADdiag 7h (>)) ) A of
(32)
Thus, (31) computable closed form.
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Physics-Informed Generative Modeling of Wireless Channels
M-step
The M-step for the (u 1)th iteration for CSGMMs solves
(Px,(u+)), (k,(+1))
argmax
Σ
(oo(a.ll 8. s.t. Pk Pk==1 =1
(33)
{pa:7e) } N.
with (s, klys)) being the posterior for the ith training sample computed in the uth iteration of the EM algorithm. The closed-form solution is given by (Böck al., 2024b)
(k,(++)) = Evey Pa (klp)) MR
A2 + diag( RR, )
(34)
Σ Eney Pu (Alps))
Pa(u+1)) Eneey Cikeypulie (klps))
(35)
Note that for improving training stability, one can clip (k,(++)) from below at, e.g., 10-?
A.6. Detailed Explanation for the Guarantee of Physical Consistency by SBGMs
We require three lines of argumentation to fully explain why the statistical model in (6)-(8) aligns with the inherent properties of wireless channels and, thus, is suited for generating wireless channels.
The complex path losses Pt.m in (1) model several physical as well as technical effects contributing to the wireless channel, such as the antennas' radiation patterns or changes in polarization due to reflections (Jaeckel et al., 2023). Additionally, their phases also contain the phase shift due to the center frequency modulation, i.e., 2n Te. with fc heing the center frequency (Tse& & Viswanath, 2005). We reformulate Te m de,m Axe with de,m. being the (sub-)path length between the transmitter and receiver (cf. (1)), and be being the corresponding wavelength (i.e., C Ae with the speed of light e). Typical center frequencies fc for the wireless transmission of signals range from a few GHz to tens or even hundreds of GHz. Thus, the corresponding wavelength Aa takes values of at most a few centimeters or smaller. Consequently, by just changing the ppat))path length by d (i.e., a few centimeters or less), the path loss phase arg(pe, takes a full turn of 2T. other words, by just slight movements users, the path loss phases arg( rapidly change. Since we want statistically describe the wireless channel over a whole scenario whose dimensions are much larger than the center wavelength and, thus, lead d >> Ae, the path loss phases arg (pe,)) are generally modeled to be uniformly distributed (Tse & Viswanath, 2005).
When choosing the dimension of the latent variable in CGLMs (e.g., VAEs) to be smaller than the dimension of the data whose distribution the CGLM is supposed to be learned, the training of the CGLM aims to be a statistically meaningful (lower-dimensional) representation of the data of interest (Bishop & Bishop, 2023).
By building probabilistic graph representation of wireless channels, the work in (Böck al., 2024a) proves that if variable does not contain any information about the complex path loss phases arg(Pt,m). the conditioning this variable preserves the structural properties of channel moments, i.e., the channel's zero mean and Toeplitz covariance structure.
By combining these three arguments, (Böck et al., 2024a) reasons and experimentally validates that the latent variable of CGLMs is trained to not capture arg arg(Pe..)-related -rellate information as these phases do not contain distinct statistically characteristic channel features. In other words, (Böck et al., 2024a) shows that the statistical model of CGLMs, i.e.,
po,s(x)- - f pe (z)2))p(()) = -f Nc(a:: (z) C=()))pp(z)dz
(36)
is learned to fulfill (=) 0 and Cz(z) being (block-)Toeplitz structured when being properly trained on channel realizations.
In SBGM, we use (7) and (8) as the statistical model for the compressible representation su (or st.)). By that, we implicitly model h be distributed according CGLM with
(h) = / po(h)2))ss2)dd
/Ne(h: DRdia((o
Nc(h:;
(37)
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---

Physics-Informed Generative Modeling of Wireless Channels
and (=))a)(())))))))))))))) DR being Toeplitz (or block-Toeplitz when using D.,)), which perfectly aligns with the findings from (Böck et al., 2024a).
B. Limitations
Our proposed methods exhibit some limitations, which are discussed below.
Non-Stationary Channel Generation When modeling the wireless channel using its compressible representation (5) and (20), one assumption is that each dictionary column only depends on one single grid point. More precisely, in SIMO, each column ( SR
gm From a physical perspective, this means that at each antenna, @ corresponds to only one single angle SR the impinging waveform comes from the exact same direction. This is called the far-field approximation and is a common assumption in radar and wireless communication exploiting multiple antennas (Yin & Cheng, 2016). From statistical perspective, the far-field approximation leads to a spatially stationary process. Equivalently, in OFDM, each dictionary column in (20) only corresponds to a single delay-doppler tuple (j7/Sz, /.) Thus, each timestamp and subcarrier in the OFDM grid is modeled to experience the same delays and doppler shifts. This results in a stationary process in the (WSSUS) assumption (Bello,
time and frequency domain, referred to as the wide-sense-stationnayy wawrryyooooollaeedscaaateeeeeeee 1963). While the stationary assumptions are common in radar and wireless communication, some applications require non-stationary channel characteristics, such as the generation of long channel trajectories, that our proposed method, in its current form, cannot model.
Off-Grid Mismatches Another limitation that our method requires a discretization of the physical parameter space (cf. Section 3.2). In consequence, when channel in the training dataset exhibits physical parameters that do not exactly match any grid point, our proposed method cannot distinguish between having a single path with parameters between two grid points and having two paths on neighboring grid points. This mismatch is controlled by the grid resolution.
OFDM Pilot Pattern A further limitation is specific for OFDM. It well known that compressive sensing (CS) methods experience a decrease in performance when using selection matrices that correspond to regular pilot pattern compared using (oseudo-)random selection matrices (Gaudio et al., 2022). As the training in SBGM as well as the physics-informed implicit AmbientGAN builds on CS, this limitation also holds for our proposed methods and constrains the set of suitable OFDM pilot patterns.
C. Model Extensions
C.1. Customizing the Number of Paths per Channel Realizations
When modeling wireless channels using ray tracing, one possibility to customize the number of paths per channel realization (Alkhateeb, 2019). This can be beneficial to, e.g., reduce computational complexity or only capture the most relevant effects in the channel realization of interest. Equivalent to ray tracing, our proposed method also allows the adjustment of the number of paths. More precisely, our method generates new complex-valued vectors st.f (or 8R) where each entry represents the complex-valus path loss corresponding to one gridpoint, i.c., one delay-doppler tuple or one angle. Thus, we can interpret each non-zero entry in 8t as a single path, where the corresponding index in &t, together with the complex-valued entry determines the path's 'ss's physical parameters. Thus, when we want to restrict our generated channel realizations to only possess Pmux paths, we can set all entries in the newly generated 8t,f (or 8R) to zero that do not belong to the Pmax strongest entries in a squared absolute value sense.
C.2. Reducing the Number Learnable Parameters when Considering Multiple Domains
In case of considering multiple domains (e.g., the time and frequency domain in OFDM, cf. Section 3.4), one possibility to reduce the number of learnable parameters is to constrain the conditional covariance matrix E[hh"|> | to not only be blopk-Toeplitz (ef. Section 4.3) but rather Kronecker of Toeplitz matrices forming subset of all block-Toeplitz matrices. In particular, by constraining (0(2) = 7/0 (z) @ 2 (z) Dr, di..diag(70(z)) in OFDM is given by
(2) the resulting conditional channel covariance matrix
(De @ Dy) diag(n
(2) 2 70 (f) (D) (D, 8 D,)
(38)
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Physics-Informed Generative Modeling of Wireless Channels
and, thus, can be written Kronecker product of Toeplitz covariance matrices. This approximation reduces the number of variance parameters learned from S to St Sy- For the CSVAE, this constraint be enforced by letting the NN output Sy- and S,-dimensional vector and subsequently building the Kronecker product instead of outputting St -iidenmnnsio vector. For CSGMM, incorporating this constraint requires a new M-step. Specifically, the M-step with
(*) ( exhibits no closed form and must solved iteratively. In the following, we derive closed the constraint 7 = Y & 7
forms for the global solution the update steps when applying coordinate search.
We aim solve (33) with the constraint 7/s = 7 considering that we have complex-vallee -valued Gaussians, we reformulate the optimization problem as
( & 7 Following the derivation in Appendix F of (Böck et al., 2024b) and
K
(- (-(S)og
* yi 19. Pkk
C b K.J.J -
))
argmax
Σ
Σ
Pu (kllys)
S log +
Σ
log Thij +
+ log Ps:
(pE.Te)
Vk,j
M.EY 1=1
/=1
(39)
s.t. Σ Pk 1,
7h 7k
(8)
2 7k
for all k
k
alvi.k & denote the jth entry of Pk, with S = St y and . and k.j.)
and the diagonal of C Chlliick Bick in the uth iteration,
Mk.j
respectively, By defining p(g.p) = (q 1)5, + we can rewrite
(1) (f)
Tkrr(g.)) = -khi '
(40)
and, thus, (39) can be rewritten as
K
St
S
Sr Σ
2 + Calyi vi.k
argmax
ΣΣ
Σ
Pa
(k)
-S log π - S
log 7i.0
(4)
Σ
log TR.P
(
Mk.r(g.))
k.r(g.)).r(a.p))
log Pk
(t) (f)
(P V.EY k=1
k=1
9=1
p=1
9.p=1
7k.q
kk.p
Σ Pk =1
(41)
The corresponding Lagrangian C is given by
K
S.
S St
S
11
elus (krlla.))
12 Cellick
Pu Pulklyu) (-Sooui
log ग - S
2
log kk.g
Σ Σ pm
log kkp
krr(g.))r +
V:EY
q=
9.pm1
7k1 kk.p
(42)
log pk) +v[1 Σ pk)
with Lagrangian multiplier v. In the following, we consider coordinate search, i.e., we keep {7k }k= fixed and solely
(f) (f)
(t) optimize over {7k 1k=1 (and vice versa). By taking the derivative of L with respect to FT.g
(() and setting zero, we end up
with
S₁
| Alvekk k.((g.))
12 + Coll..
C = Σ (Klp))
S
+
T.((.)))r(...)
0
(43)
0,10
(8)
(1) (1)__(0
M.y
R.q
F.g EFP
and, thus,
8,
Σ No
i= Pro (Fllya)
11.00 C Al (f)
Crapp)itam))
(0
(44)
k
Sy kop Ei p(klps))
p=1
for all 7 1,..,,,, k 1, 1,... K. Due to the symmetry of (41) with respect to x and 7 we find an equivalent update for 7
(f) i.e.
S.
N Pa (Flz:)
Aly..
+
( F
(45)
S
No p(klys))
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Physics-Informed Generative Modeling of Wireless Channels
0.04
p(w
0.02
-90
0 R
90
Figure 7. Distribution p(w") of the path angle in the modified 3GPP dataset.
for p= L..... Sy. ,,..., of the weights {pk } k= equal those in (Böck et al., 2024b) and are given by
K. We suppressed the index for the coordinate search due to readability. The update steps
E EmEy Pa (k(g)))
P
(46)
= |
(47)
D. Dataset Description and Pre-Processing
D.1. Modified 3GPP Dataset Description and Pre- Pre-Processing
Our modified 3GPP dataset is based the conditionally normal channels described in (Neumann et al., 2018). More precisely, each channel realization is generated following two steps. First, we draw one random angle (R) from the distribution illustrated in Fig. 7. Subsequently, we draw the channel h from (h:; O, Cum)) where
CWiR) [" " (O)))
(48)
and (R): Laplacian with mean (R) and standard deviation of 2 degree. Moreover, aR()),; = e- ()
corresponding ULA with equidistant antenna spacing (ef. Section 3.2). The angle distribution in Fig. 7 artificially represents scenario with, e.g., four street canyons where the users' positions are mainly distributed in four different angular regions. The 2-degree standard deviation is in line with the 3GPP standard and referred to as per-path angle spread (3GPP, 2024b). It simulates small spread of the main angle (R) due to scatterers close to the users. We apply no other pre-processing. We define the SNR in dB to be SNR = 10 log [/]] number of antennas, o² the noise variance, and [|[||/ is estimated using 10000 samples generated in the discussed manner. For producing the training dataset, we draw SNR; uniformly between OdB and 20dB for each training sample and compute the corresponding noise variance of2 Subsequently, we generate the training dataset
where the dimension of h. i.e., the
y = {3: h, + +
(49)
with n, Ne of I).
D.2. QuaDRiGa Dataset Description and Pre-Processing
QuaDRiGa a freely accessible geometry-based stochastic simulation platform for wireless channels (Jaeckel et al., 2023). The two QuaDRiGa datasets considered our work build on the "3GPP 38.901 UMa" and the "3GPP 38.901 URa" scenario, which simulate an urban and rural macrocell environment with parameters consistent with the 3GPP standard, respectively (Jaeckel et al., 2014). For the former, outdoor. The scenario contains three streets in | 120-degree sector of outdoor cellular network. An illustration of the scenario is given in Fig. 8. The specific scenario parameters are given in Table 2. For the latter, we use setup with users all in LOS and outdoor. An illustration of the scenario is given in Fig. 9. The specific scenario parameters are given in Table 3.
setup with users in LOS as well NLOS, as well indoor and
For both scenarios, we simulate OFDM channels with system configuration specified in Table 4. We denote these datasets with 5G-Urban and AG-Rurall. 1, respectively. For evaluating the generalization performance of our proposed models, we
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Physics-Informec Generative Modeling of Wireless Channels
X Access Point
500
O NLOS Users
Access Point
LOS Users
3000
Users
Sector
Indeor
250
Sector
2000
Pgp
y-coord in [m]
3 3-c00ru in ju]]
1000
4204
0x
Boo Bola CO OEO doe
-1000
-250
-2000
-3000
-500
0
100
200
300
400
500
0
1000
2000
3000
4000
x-coord in [m]
X-coord in [m]
Figure 8. Urban QuaDRiGa scenario with three streets and regions with in- and outdoor as well as LOS and NLOS users.
Figure 9. Rural QuaDRiGa scenario with four streets and only
LOS users.
Table Specific scenario parameters for the urban scenario.
Table 3. Specific scenario parameters for the rural scenario.
Distance Range
Velocity Range
Number Streets
In- to Outdoor User Ratio
Distance Range
Velocity Range
Number
In- to Outdoor User Ratio
User to BS
User to BS
Streets
20-500m
0-50km/h
3
1/4
100-4000m
70-100km/h
Table Specifie parameters for the 5G system configuration.
Table 5. Specific parameters for the La rgg system config.
Subcarrier Spacing
Slot
Symbol
Subcarrier
Slot
Symbol
Bandwidth
Bandwidth
Duration
Duration
Spacing
Duration
Duration
360 kHz
15 kHz
ms
1/14 ms
1200 kHz
60 kHz
5.14 ms
1/3.5 ms
also simulate OFDM channels in the urban scenario with changed system configuration, specified in Table 5, The resulting dataset is denoted as Large-Urbar
In all cases, we pre-process the data before training the generative model. More specifically, we apply the common scaling of each channel realization by its effective path gain (PG), which models the attenuation due to the distance of the user and the BS as well as shadowing effects (Jaeckel et al., 2023). Subsequently, we scale the training dataset such that the estimated signal energy [[] equals the number of resource elements in the corresponding OFDM grid (i.e., 336 for the 5G system configuration and 360 for the Large system configuration). For each tuple of number M of pilots and system configuration (e.g., (30, 5G)), we draw one random selection matrix as measurement matrix E R x
(cf. (9)) and keep this matrix fixed for all training and validation samples. This selection matrix extracts M random elements from the N-dimensional
channel h. Equivalent the modified 3GPP dataset, we draw SNR, uniformly between 5dB and 20dB for each training sample and compute the corresponding noise variance o3 |]]]AA/1))/MM 100.ISNR., Then, the datasets are given by
y {yi y: Ah; + n
(50)
with n, Ne(0, of I).
D.3. DeepMIMO Dataset Description and Pre-Processing
The modified 3GPP dataset (cf. Appendix D.1), as well as the QuaDRiGa dataset (cf. Appendix D.2), involve statistics in their channel generation process, due to. e.g., sampling underlying angles (modified 3GPP) or randomly placing scatterers in simulated environment (QuaDRiGa). In addition these channel models, we evaluate our method for SIMO based on purely deterministic channel model, i.e., ray tracing. Next to being purely deterministic, another key difference between ray tracing and standardized channel models (and QuaDRiGa) is the absence of subpaths. More concretely. Mr in (1) is set to I for all e (Alkhateeb, 2019). Subpaths typically originate from minor effects, such as the spread of an impinging waveform at a rough surface into several paths with similar properties. Since these phenomena are difficult to simulate deterministically. they are left out in ray tracing. As described in Section 1, ray tracing requires a 3D replica of the environment of interest and the material properties within the scene. The DeepMIMO dataset proposed in (Alkhateeb, 2019) is a benchmark dataset for
20

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Generative Modeling of Wireless Channels
User Ond
Use Gnd
Figure 10. Ray tracing Boston scenario used in our simulation (cf. (Alkhateeb, 2019)).
wireless channel modeling, building on the ray tracing tool Remcom (Remcom) and offering several predefined scenarios. In our work, we use the Boston 5G scenario with 3.5GHz center frequency. In Fig. 10, we plot a 2D perspective of this scenario (Alkhateeb, 2019). We only consider users in "User Grid 2" since most of the users "User Grid 1" are blocked and have no connection to the BS. DeepMIMO offers the possibility simulate channels gridpoints over the whole scenario in Fig. 10, where the spacing between two adjacent users is 37em. In a first step towards the used dataset in our work, exclude all users in "User Grid 2", which are blocked and have no connection to the BS, i... we only consider non-zero channels (either LOS or NLOS) in our dataset. Furthermore, we consider single antenna users and the BS is equipped with a ULA with N antennas and 2/2 antenna spacing. This ULA is first placed along the x-axis and then rotated by 45 degrees around the z-axis. Moreover, we allow up to paths to be simulated per channel realization.
We pre-process the data before training the generative models. Since the simulated channels in ray tracing involve large-scale fading (e.g., path losses due the distance between the user and BS), their norms range over several orders of magnitude, making them ill-suited to directly train MI. models. DeepMIMO provides the absolute values of each path's 's path loss (denoted as DeepMIMO.dataset (ie. path.params. . power in DcepMIMO Version v2, and denoted as Pt DM in the following). Similar to how QuaDRiGa distinguishes between large-scale- and small-scale wwalllssaae--aaee fading (cf. (Jaeckel
al., 2023)), scale each channel individually by /1/(E, P.M). 8.8 Subsequently, we scale the training dataset such that the estimated signal energy E []][k]]]] equals the number of antennas, i.e., N. We then apply the same method to compute noise variances of for each training channel realization as in Appendix D.1 to construct the used dataset. We consider the case N = 32 (and N 256 for computing ground truth, cf. Appendix E), and denote the dataset as 2/
E. Architectures, Baselines, and Hyperparameters
For the CSVAE encoder, we use simple fully coly-cooneected NN with ReLU activation function (cf. Table 6). We tested different architectures for the decoder, from which deep decoder-motivatee architecture performed the best (cf. (Heckel & Hand, 2019)). Specifically, the decoder contains two fully connected layers with ReLU activation followed by several blocks consisting of a convolutional layer with kernel size of 1, a ReLU activation, and a (bi-)linear upsampling operation. For the simulations on OFDM, we incorporated of these blocks (cf. Table 7), while for the modified 3GPP and DeepMIMO dataset (cf. Table 8), we used of them. The final layer is convolutional layer with kernel size of 1. We applied hyperparameter tuning for each simulation setup to adjust the width and the depth d for the encoder well as the linear layer width, the number of convolutional channels in the decoder, and the learning rate. For that, we took the model resulting in the largest evidence lower bound (ELBO) over a validation set of 5000 samples. For the optimization, we used the Adam optimizer (Kingma & Ba, 2015).
CSVAE
*Note that this normalization of each individual channel requires an estimation of the attenuation of the channel based on the distance between the user and the BS, which typically changes very slowly. The alternative normalization for all channels to have norm 1, respectively, requires genie-knowledge of each channel realization for offline training as well online operation, which is why we do not normalize in this manner.
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Physics-Informed Generative Modeling of Wireless Channels
Table 6. CSVAE.
Table 7. CSVAE (OFDM).
Table CSVAR (3GPP & DeepMIMO))
Encoder
Decoder
Decoder
dxLineer
2inai
2xnerr
ReLU
ReLU
ReLU
1 LxLina
1xUnflatten
1 Unnnlattenn
ConvDD (kernel size=1)
2 2xConvID (kernel size=1) 1)
ReLU
ReLU
Upsample (scale= 2, bilinear)
Upsample (scale=2, linear)
x Conv2D (kernel size=1) 1)
1 x xXCovvlD ID (kernel size=1) 1)
Table 9, Autoencoder.
Table 10. Autoencoder.
Table 11. GAN (OFDM).
Table 12. GAN (OFDM).
Encoder
Decoder
Generator
Discriminator
dxConv2D ReI.U
IxLinear
1xLincar
3xConv2D
ReLU
IxUnflatten
LeakyRel.U(0.2)
LxLinear
IxUnflatten
Dropout(p 0.25)
dx DCConnTranspos:2D
1x Tanh
ReLU
2 Uppsmple (scale=2,nearest)
1xFlatten
Conv2D
IxConv2D (kernel size= 1)
1xLinc
Batch Normalization
ReLU
IxConv2D
Note that we need to enforce the variances o (1) the output of the encoder as well as the variances (z) the output of the decoder to be positive. To do so, we interpret the corresponding outputs of the final layer to represent the logarithm of these variances and apply an exponential function to the output before any further processing. For increased stability, we also recommend implementing a lower bound for the decoder output representing log yo (e.g., 10).
CSGMM
The only hyperparameter for CSGMM is the number K of components. We tune this parameter by choosing K, leading to the largest likelihodd over the training dataset.
Autoencoder We use an autoencoder to apply the tross-validation method from (Baur et al., 2025; Xiao et al., 2022). The chosen architecture resembles the proposed architecture in (Rizzello Utschick, 2021) and is given in Table and 10. The kernel size, the stride, and the padding of the 2D convolutional layers in the encoder are set 3, 2, and 1, respectively. The stride and the padding in the 2D transposed convolutional layers of the decoder are set to 2 and 1, and the kernel size is set the output dimension be matched. The final 2D convolutional layer has kernel size, stride, and padding of 1, 1, and O. We used the same architecture for the simulations on DeepMIMO but with 1D convolutional layers instead of 2D ones. All autoencoders in all simulations exhibit a latent dimension of 16. We applied hyperparameter tuning to adjust the width of the linear layer, the number of convolutional blocks d, the channel size of the convolutional layers, as well as the learning rate based on a ground truth validation dataset. For QuaDRiGa, we used 10000 validation channel realizations. For the optimization, we used the Adam optimizer (Kingma & Ba, 2015).
to either 3 4, depending
AmbientGAN for and h For both GAN variants, we utilize the architecture from (Doshi et al., 2022), i.e., the generator consists of a single fully connected layer with ReLU activation, an unflatten operation and, subsequently, two blocks of a nearest-neighbor upsampling operation, convolutional layer, batch normalization and a ReLU activation (cf. Table 11). The final layer is convolutional layer. The kernel size, the stride, and the padding of all convolutional layers are set to 3, 1, and 1, respectively. The discriminator contains three blocks of a convolutional layer, a LeakyReLU activation, and a dropout layer. Subsequently, it contains a flatten operation and a final linear layer. For the simulations on DeepMIMO, we use the same architecture but with ID convolutional layers instead of 2D ones. The width of the linear layers and the channel number of the convolutional layers is determined by hyperparameter tuning. For the simulations on QuaDRiGa, we
22

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Physics-Informed Generative Modeling of Wireless Channels
Table 13, Generalization Performance for GG-Urban to Large-Urban and vice versa (M = 30, N. 10000).
CSGMM trained
CSGMM trained
5G-Urban
on Large-Urbas
nMSE for the 5G-Urban config.
0.00109
0,00096
Pe for 5G-Urban contig.
0.99756
0.99783
nMSE for the Large-Urban config. Pr for the Large-Urbar config.
0.00178
0,00155
0.99675
0.99701
Ground Truth B CSGMM (Pmxx
1600)
CSGMM (variable prmx)
UMSE
0.0011
0.001
0.0009
32
64
96
128
Peax
Figure 11. nMSE over the number of considered paths Pmax for the 5G-Urban dataset with M 30 and Nr 10 000.
take the autoencoder's reconstruction performance on a ground-truth validation set as the objective for the hyperparameter tuning. For parameter generation, we take the AmbientGAN leading the smallest average generated angular spread and, thus, the best fitting histogram of spreads. In line with (Doshi et al., 2022), we used the RMSProp optimizer, adjust the learning rate via hyperparameter tuning, and the GAN variants are optimized using the Wasserstein GAN objective with gradient penalty explained in Appendix A.2.
Ground-Truth Baseline for the Parameter Generation Performance In Section 6.2, we evaluate the parameter generation performance for the 3GPP dataset and compare our method to ground truth. The same is done for the DeepMIMO dataset in Appendix 6.2. In general, for the simulation, we decide computing the results for the ground-truth baseline, we artificially assume that we have many antennas gridpoints, i.e., 256. This results in Dn in (5) to be squared and invertible. In consequence, we can compute sn for each channel realization in the test set and estimate the power angular profile (15) and the angular spread (16) using these calculated vectors.
number of angle grid points S, set to 256. For
F. Additional Results
F.. Experiments for Validating the Generalization Performance
We analyze SBGM's capability to generalize to other system configurations without being retrained. More specifically, we train CSGMM using 5G-Urban and N;- = 10000. After training, we use the dictionary Dt.f for the Large-Urban configuration, i.e., Aʃ = 60kHz, A7 3.5 ms with Dr.f E ((20-18)x(10-100 (cf. Appendix A.1) to map generated 8t, to new channel realizations that are different to the ones involved in the training dataset 5G- Urban. Subsequently, we apply the same wross-validation method as explained in Section 6.2, i.e., we use these newly generated channel realizations to train an autoencoder whose performance is then evaluated on ground- truth QuaDRiGa channels in the Large-Urban configuration. We also do the same procedure vice versa, i.e., we train CSGMM using Laa and then adapt the dictionary to the SG-Urban configuration.
and N, = 10000,
The results are given in Table 13. The underlined numbers indicate the performance where CSGMM has been trained on different system configuration compared to the one used for channel generation. We can see that there is almost no difference between whether CSGMM has been trained on the system configuration that matches with the generated channel realizations afterward or different one.
E2. Experiments for Controlling the Number of Paths Considered
In Fig. 11, we analyze the effect on the generated samples when varying Pmax (cf. Appendix C.I), More precisely. in Fig. 11, the nMSE is shown for varying Pmax where we used QuaDRiGa channels from 5G-Urban (cf. Section D.2), M = 30 and N, = 10 000. Thus, we first train CSGMM with 5G-Urban. We then generate 10000 new samples st,f- We filter out all entries for each of these samples with a smaller squared absolute value than the Paux strongest entries. We then map the
23

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Physics-Informec Generative Modeling of Wireless Channels
Ground Truth
AmbientGAN for s - CSGMM -E-
CSVAE
CSGMM kron
CSVAE kron
M-SBL - AmbientGAN for
a) -10-2
b)
-10
2
10
d) -10
6
4
nMSE
aMSE
& 1
0.3
2
0.3
0.1
0
10a
10ah
10
10a
10a
10
0.7/15 1.4/30 2.1/45 2.8/60
0.7/15 1.4/30 2.1/45 2.8/60
No
N
9 in kHz / in uss
1 in kHz / T in μs
Figure 12. a) - d) nSSE and Pe for reconstructing ground-truth channels by an autoencoder trained on channels h = Da produced by the trained models, respectively. In a) and b), we vary N, for the generative models with fixed M 30 and dataset 5G-Urban, and c) and d), we vary 1 and 7 with fixed N₂- = 3 000 and dataset 5G-Rural.
a)
Ground Truth
CSGMM
CSVAE
AmbientGAN for
M-SBL
0.03
800
Counts
P.R)
0.015
400
0
0
90
60
-30
30
60
90
0
10
20
30
40
50
60
70
80
Wi (R)
SW
(SR)
b) Exemplary training
samples Yi E C32
Newly generated samples |s:|² after training (8, E C256)
Ground Truth
CSGMM
CSVAR
AmbientGAN for 8
M-SBL.
90
90
90
0
90
90
90
antennas
90
90
90
90
90
0
90
90
90
90
900
0
90
-90
90
90
90
900
90
90
90
-90
90
Figure 13. a) Power angular profile (2( (g) and histogram of the angular spread ((R)(an) CSGMM, AmbientGAN for a and M-SBI for DeepMIMO, compared to ground truth, b) eight exemplary training samples, c) squared absolute value of four exemplary generated samples from all models, respectively.
(R)
(sn) from 10000 generated samples by CSVAF,
resulting vectors to channel realizations using the dictionary Dr,f and train the autoencoder. We see that by incorporating the prior knowledge that most channels only exhibit few relevant paths, improve our method even further and reach almost ground-truth performance.
E33. Experiments for the Kronecker Constraint on the Covariances (cf. Appendix C.2)
In Fig. 12, plot the same results as in Fig. 6, but, additionally, we also plot the performance of the CSGMM and CSVAE with Kronecker-based covariance, explained in Appendix C.2. The Kronecker-based CSVAE does perform similarly or even outperforms the ordinary CSVAE. While the Kronecker-basse CSGMM performs well in some configurations, it also exhibits outliers with significant performance drop. As the M-step of the Kronecker- based CSGMM in Appendix C.2 not the guarenteed global solution of the corresponding optimization problem (39), these performance drops come from the Kronecker-base CSGMM training converging to bad local optima.
24

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Physics-Informed Generative Modeling of Wireless Channels
F.4. Experiments for the Parameter Generation the DeepMIMO Dataset
For the DeepMIMO dataset on SIMO, we evaluate the parameter generation. Equivalent to the results on the modified 3GPP dataset in Fig. 5, plot the power angular profile and histogram of the angular spread in Fig. 13 a), exemplary training samples in Fig. 13 b), and the absolute squared value for exemplary newly generated samples of CSVAE, CSGMM, AmbientGAN for and M-SBL compared to Ground Truth in Fig. 13 c). The results are in line with Fig. 5 with CSGMM exhibiting less peaks and matches the underlying power angular profile more closely. In addition, CSVAE underestimates the amount of LOS channels, which can seen at the missing peak in the histogram with very small angular spreads. AmbientGAN provides less sparsity than SBGM by not generating parameters with small angular spread.
G. Pseudocode for Parameter and Channel Generation and Implementation Specifications
Algorithms 1-3 summarize the generation process of parameters, channels, and channels with constraining the path number, respectively. All models and experiments have been implemented in python 2.1.2 2.2.2 using pytorch 3.10. 13, and pytorch-cuda 12. 1. All simulations have been carried out on a NVIDIA A40 GPU.
Algorithm Parameter Generation with the CSGMM (CSVAE) Input:
Output: complex-valued vectors { (depending on the grid used for training) for
Nen whose entries together with their indices correspond to the physical parameters
N(gen) do draw k; p(k) (or draw z; p(=)) (ef. Section 4.2)
2) draw 8; p(s(k;) (s diag(r:) (or draw s, p(s|z;) (,, (ef. Section 4.2)
end for
Algorithm 2 Channel Generation with the CSGMM (CSVAE)
Input: dictionary D (new) (can be chosen according to the desired system configuration, not necessarily the one used for training) Output: complex-valued channel realizations [h. Nge that are scenario-specific and match the desired system configuration of interest for i = 1 to N(gm) do
1) draw k; p((k) (or draw 2, p(z)) (cf. Section 4.2)
2) draw 8; p(s(k:) Ns(s; O, diag(y:)) (or draw 8; ~ p(s)zz) -Ne(s; diag((ro (cf. Section 4.2)
3) compute h, D (now)
end for
Algorithm 3 Channel Generation with the CSGMM (CSVAE) when only considering Pmax paths
Input: dictionary (new) (can be chosen according to the desired system configuration, not necessarily the one used for training)
Output: complex-valued channel realizations [h, y that are scenario-specific and match the desired system configuration of interest for = to Nig=s) do
1) draw k; p(k) (or draw z; p(z)) (cf. Section 4.2)
2) draw s; ~ p(s[k:)
Ne(s; O, diag(?:)) (or draw s; ~ p(s[z:) = (s( 0, (iig((7 (cf. Section 4.2)
3) compute |s; 2 and identify the indices Z; of the Pmux strongest entries in |s;
4) compute 8, which equals 8, for the indices in I, and is zero elsewhere
3) compute h, (now) 8;
end for
25


Reviews:
[{"review_id": "KuykxjoXLq", "reviewer": "Reviewer_9aFT", "summary": "The paper proposes a physics‐informed generative modeling framework for wireless channels that integrates physical channel knowledge with sparse Bayesian generative modeling (SBGM). The method consists of several key components:\n\nChannel Representation via Compressibility:\nThe authors begin by exploiting the fact that wireless channels are compressible when expressed in a physically interpretable dictionary. \nIn an SIMO channel, the representation is a sparse vector that captures the physical parameters (such as delays and angles). A similar representation is used for OFDM channels by defining a grid in the delay–Doppler domain.\n\nSparse Bayesian Generative Modeling (SBGM):\nBuilding on SBGM, the method learns a parameterized distribution for the compressible representation of the channel. \n\nIncorporating Physical Structure:\nThe model leverages known physical properties of wireless channels. Because path loss phases are uniformly distributed, channels tend to be zero-mean and, under stationarity assumptions, have structured (e.g., Toeplitz) covariance matrices. By choosing the dictionary  appropriately, the conditional covariance naturally inherits a Toeplitz (or block-Toeplitz) structure. This aligns the learned model with established physical channel models.\n\nModel Variants – CP-VAE and CP-GMM:\nTwo concrete implementations are proposed: \nChannel Parameter-VAE (CP-VAE): A variational autoencoder variant that learns the latent distribution and uses neural networks to parameterize $\\gamma_\\theta (z)$.\nChannel Parameter-GMM (CP-GMM): A Gaussian mixture model version where the latent variable is discrete and the model learns mixture weights and associated diagonal covariance matrices.\nBoth variants are designed to capture the distribution of the compressible representations, which in turn determine the physical channel parameters.\n\nKronecker Approximation for High-Dimensional Channels:\nIn scenarios such as OFDM where the channel spans multiple domains (e.g., time and frequency), the model incorporates a Kronecker structure to the conditional covariance. This reduces the number of parameters by expressing the covariance as a Kronecker product of two smaller Toeplitz matrices, making the method more scalable.\n\nGeneralizability and Data Efficiency:\nA notable aspect of the approach is its ability to learn from compressed and noisy observations (e.g., those collected during standard online operation) rather than requiring extensive high-quality training data. Furthermore, the learned generative model is shown to generalize to different system configurations (such as varying numbers of antennas or subcarrier spacings) by adapting the dictionary without retraining.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer 9aFT for the thorough summary and review. In the following, we address the concerns and questions raised.\n\n**To Methods and Evaluation Criteria (2. paragraph) & Relation to Broader Scientific Literature & Questions for Authors (1. paragraph):** We agree with the reviewer that Section 4 contains a mix of our contributions and already existing ideas and techniques, such as channel sparsity, sparse Bayesian generative modeling, and knowledge of conditioned channel moments. We decided for this structure since our main contribution lies in recognizing that these topics can be combined to yield a generative model for wireless channels that is very different from existing approaches and overcomes the main concerns in concurrent methods. However, we understand that this might make extracting our contribution difficult.\n\nFrom our perspective, our work has considerable application-driven contributions: It addresses and overcomes the major concerns of state-of-the-art generative models for wireless channels by combining three topics that are non-trivial to combine, i.e., channel sparsity, sparse Bayesian generative modeling (which is originally not meant to generate samples), and prior knowledge of conditioned channel moments. By doing so, we derive a model that can train on corrupted data, is physically interpretable, and requires no retraining for changing system configurations. In addition, our work has a methodological contribution by deriving a new EM algorithm in Appendix F that exploits the conditioned Toeplitz structure.\n\nWe will rewrite the final version so that our contribution is better separated from previous work. We thank the reviewer for making us aware of this potentially misleading structure.\n\n**To Methods and Evaluation Criteria (3. paragraph) & Questions for Authors (2. paragraph):** We would like to refer to Fig. 2 a) and b), where we investigate the effect of the training size. In the considered QuaDRiGa scenario, 100 training samples are sufficient for our method to perform well.\n\n**To Experimental Designs or Analyses (1. paragraph):** Generative modeling for wireless channels is rather new, and there are basically no proposed methods that learn from corrupted data. Almost all approaches are based on GAN-variants and assume lots of ground-truth training data. Although the methods in Baur et al. and Fesl et al. can learn from compressed data, they are not derived to generate samples but directly apply to estimating wireless channels from observations.\n\nHaving said that, we included two more baselines now. In our submission, we compared to AmbientGAN, which can output channels h. However, AmbientGAN can also be trained to output the compressible channel representation s by including the dictionary D between generator and discriminator. This also allows us to compare to AmbientGAN for parameter generation. Moreover, we included M-SBL [1]. M-SBL is an SBL extension that assumes several observations instead of one. Indeed, M-SBL is equivalent to the CSGMM from [2] with only one component. In Fig. 1 in the anonymous repository https://github.com/anonymousicml2529/rebuttal/, we plot preliminary results for the 3GPP dataset. While AmbientGAN for s and M-SBL perform similarly for the power angular profile, both significantly overestimate the channel-wise angular spread. For AmbientGAN, this is due to the lack of sparsity prior knowledge. For M-SBL, it is because it fits a Gaussian to s, which is not expressible enough to capture different directions individually. This is also seen in the plotted samples (c).\n\n**To Experimental Designs or Analyses (2. paragraph) & Questions For Authors (3. paragraph):** There might be a misunderstanding regarding our evaluation method. We do **not** evaluate the reconstruction performance of our method in any experiments. An autoencoder reconstructs generated data as a means to evaluate the **generation** performance of our model. Specifically, the autoencoder is first trained on data generated by our model. It then reconstructs ground-truth channels. As our model generates new channels that are supposed to resemble ground truth, we do not think that training the autoencoder on compressed data is a good fit for evaluation. Applying our model to compressed sensing, as in [2], evaluates the performance for reconstruction. However, we consider the generation performance, so we evaluate the quality of newly generated samples. If the reviewer is referring to something different and we misinterpret the statement, we would appreciate it if the reviewer could comment on this point again.\n\nWe would like to thank the reviewer again as we think that restructuring our work together with the newly added baselines will improve our work's quality.\n\n[1] Wipf et al., ”An Empirical Bayesian Strategy for Solving the Simultaneous Sparse Approximation Problem,” IEEE TSP., vol. 55\n\n[2] Böck et al., ”Sparse Bayesian generative modeling for compressive sensing,” NeurIPS, 24", "claims_and_evidence": "The evidence for the claims are reasonable. The authors validate their generative model by studying the power angular profile and channel-wise angle spread. They also use the channels generated by their methods and baselines to train autoencoders. The reconstruction error of these autoencoders are then evaluated, and the proposed method performs better.", "methods_and_evaluation_criteria": "The method and evaluation datasets make sense, although there is confusion about what the contributions of this work are in comparison to prior work. \n\nThe paper is written in an extremely confusing manner. The authors start section 4.1 with sparse modelling of wireless networks (which seems fairly standard and I don't understand what the innovation is here), then move to sparse generative modelling (again, seems like prior work from Wipf and Rao directly transfer), then move to section 4.3 where they talk about how wireless channels don't suffer the same problems suffered by other sparse dictionaries, and the techniques from Bock et al directly transfer with some modifications. There's too much unnecessary text that blurs the actual method proposed by the authors and what their contributions are. The introduction and background material take up almost 4.5 pages, which is far too much IMO.\n\nThere's also no clear reasoning as to why this algorithm needs fewer training samples than existing work. There are no experiments that study the effect of training size on the proposed algorithm and baselines.", "theoretical_claims": "The paper does not report theoretical results.", "experimental_designs_or_analyses": "The benchmark algorithms seem weak. AmbientGAN is work from 2018 and WGAN-rbf seems like a method the authors made up (it does not contain a citation). Surely there must be more recent algorithms for learning wireless channels (I'm not an expert on wireless channel generation, so I cannot provide feedback on appropriate algorithms). Why not compare to the algorithms by Baur et al and Fesl et al from the related works section?\n\nAnother reasonable evaluation would be on the performance of these models for compressed sensing estimation-- the current work is evaluate on reconstruction of fully observed channels, and I did not find experiments for estimation from compressed measurements.", "relation_to_broader_scientific_literature": "The paper seems to be a significant contribution towards the generative modelling of parameter-based wireless channels. However, there are some concerns regarding the novelty over prior work by Bock et al. My reading of the paper is that most of the techniques for sparse generative modelling transfer to this case, with some modifications for wireless channels.", "essential_references_not_discussed": "Seems reasonable.", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "My main concerns are regarding the writing/description of the algorithm. A more detailed description of why your method is not a direction extension over the prior works by Bock et al, is appreciated. \n\nThere's also no clear reasoning as to why this algorithm needs fewer training samples than existing work. There are no experiments that study the effect of training size on the proposed algorithm and baselines.\n\nAdditionally, the performance of these models on compressed sensing estimation seems like the most relevant metric for judging the quality of these models.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GqlLB7I2N6", "reviewer": "Reviewer_9VKv", "summary": "This paper introduces a physics-informed generative modeling approach for wireless channels. By using sparse Bayesian generative modeling and knowledge about the conditional channel moments, this paper addresses limitations in existing methods such as the need for high-quality data, lack of generalizability, and poor physical interpretability.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer 9VKv for the review and the reviewer’s assessment. In the following, we address the concerns and questions raised.\n\n**To Relation To Broader Scientific Literature & Other Strengths and Weaknesses (bullet point 1):** We agree that our proposed model can be utilized in the context of communications and signal processing. Having said that, we also strongly believe that our work fits well with the application-driven machine-learning track of ICML. Our model builds a machine learning-based framework to simulate the propagation of electromagnetic waves in indoor and outdoor scenarios. For that, we enhance a machine learning-based framework (sparse Bayesian generative modeling) by incorporating specific insights from geometric optics and electromagnetic waves (i.e., the channel sparsity and the structural knowledge about conditional channel moments). In this sense, we derive a new application-driven machine learning-based tool that is not meant to solve one particular problem from communications or signal processing but can be used generally. We hope that the reviewer agrees with us on this.\n\n**To Other Strengths And Weaknesses (bullet point 2):** We now included two more baselines for physical parameter generation: an AmbientGAN that does not output channel realizations $\\mathbf{h}$ but their compressible representation $\\mathbf{s}$, and M-SBL from [1]. In Fig. 1 in the anonymous repository https://github.com/anonymousicml2529/rebuttal/, we plot preliminary results for the 3GPP dataset. It can be seen that AmbientGAN, as well as M-SBL, significantly overestimate the channel-wise angular spread. We would also like to refer to our response *To Experimental Designs or Analyses (1. paragraph)* to Reviewer 1 (9aFT), where we discuss the new baselines and the results in more detail. \n\n**To Questions For Authors (1)):** In our simulations for OFDM, $\\mathbf{A}$ is chosen as a selection/masking matrix, i.e., it masks out a certain amount of entries in the corresponding channel. This is in line with the OFDM standard in wireless communications. The indices that correspond to the entries that are masked out are drawn randomly. Moreover, this masking operation is illustrated in, e.g., Fig. 2 e), where we plot the corresponding compressed training samples.\n\n**To Questions For Authors (2)):** Changing the observation matrix for different training samples is possible and has also been considered in [2]. Generally, changing the observation matrix either has no impact or even improves the overall performance as the model experiences varying ”subspace information” from the training samples during training. In our work, we only considered fixed measurement matrices as this is arguably more relevant for wireless channels. However, we will consider including simulations regarding varying the observation matrix in the final version.\n\nWe would like to thank the reviewer 9VKv again for the valuable comments, and we think that the added baselines will improve our work's quality.\n\n[1] Wipf et al., ”An Empirical Bayesian Strategy for Solving the Simultaneous Sparse Approximation Problem,” IEEE TSP., vol. 55\n\n[2] Böck et al., ”Sparse Bayesian generative modeling for compressive sensing,” NeurIPS, 24", "claims_and_evidence": "The claims are well supported.", "methods_and_evaluation_criteria": "This paper figures out a way to solve the drawback of sparse Bayesian generative modeling by using the conditional channel moments.", "theoretical_claims": "Not appliable.", "experimental_designs_or_analyses": "This paper made experiments for both SIMO and OFDM, and used open datasets to evaluate the proposed method. The experimental designs are sound and the evaluation metrics are well chosen.", "relation_to_broader_scientific_literature": "The contributions of this paper is useful for designing algorithms for communication and signal processing, but I don't realize the contribution to machine learning.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths: The paper provides a data-knowledge dual-driven way to model the wireless channel, which is important for the simulations in communication and signal processing. According to the experiments, the proposed methods present a superior performance over the other methods.\n\nWeakness: \n- The paper seems more suitable for communication/signal processing in my opinion.\n- It would be better if the paper could compare with more other methods.", "other_comments_or_suggestions": "Not appliable.", "questions_for_authors": "1. How do you choose the selection matrix \"A\"? Do you choose a good one by cross-correction?\n2. What if the matrix “A” changes for different training channels?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IYa62kowey", "reviewer": "Reviewer_dNNa", "summary": "The paper presents a new way to model wireless channels by combining basic physical principles with modern generative modeling techniques. Instead of using complex black-box methods like GANs—which require lots of clean data—the authors introduce more structured approaches based on simplified versions of VAEs and GMMs (called CP‑VAE and CP‑GMM). These models learn the key physical characteristics of wireless channels (such as the angles, delays, and Doppler shifts) even when the data is noisy or incomplete. By incorporating known physics rules into the learning process, the method produces realistic and interpretable channel models. Additionally, the approach uses clever algorithmic tricks (like a closed-form update using Kronecker factorization) to reduce computational effort and can adapt to various wireless setups without needing to start over with training. Tests on several datasets show that this new method generates accurate channel representations and outperforms traditional GAN-based models.\n\n## Update After Rebuttal:\nThe authors sufficiently address my review and I vote to keep the score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We want to thank reviewer dNNa for the thorough review and the reviewer's assessment. In the following, we address the reviewer's additional questions.\n\n**To Experimental Designs Or Analyses (1. paragraph) & Weaknesses & Questions For Authors (1):** We agree with the reviewer that experimental results on real measurement data would strengthen our paper's contribution. We indeed have preliminary results of our models on a real-world measurement campaign that was conducted on the Nokia campus in Stuttgart, Germany, in 2017. In Fig. 4 in the anonymous repository https://github.com/anonymousicml2529/rebuttal, you can see the scenario in which measurements have been conducted. Moreover, the green regions indicate that users experienced a LoS path to the base station, while the red regions have solely NLoS paths. The base station was equipped with a 16x4 antenna array. In our preliminary results, we only consider the first row of the antenna array. In Fig. 5 (https://github.com/anonymousicml2529/rebuttal), we tested the CP-VAE on this data. The data consisted of 16-dimensional noisy channel measurements in the spatial domain. In a), you can see the overall power angular profile of newly generated samples (256-dimensional). In Fig. b), we trained the CP-VAE solely on LoS channels from the \"LoS\"-street canyon, as well as solely on NLoS channels respectively. It can be seen that the generated channel parameters from the LoS CP-VAE exhibit smaller angular spreads, which is expected. Moreover, in c), you can see exemplary plotted samples validating that the CP-VAE can easily distinguish between the LoS and NLoS channels. We will consider including these results in our work.\n\n**To Experimental Designs Or Analyses (2. paragraph) & Questions For Authors (2):** We agree with the reviewer that a further analysis regarding the hyperparameters would improve our work's contribution. In Fig. 2 and 3 in https://github.com/anonymousicml2529/rebuttal, we have preliminary results for these experiments. \n\nMore precisely, in Fig. 2, we investigate the effect of the CP-GMM's number of components on the angular spread for the modified 3GPP data. For K=1, the CP-GMM coincides with M-SBL [1]. It can be seen that for K<4, the angular spread is significantly overestimated, which is to be expected as the underlying data is coming from a scenario with power from four different angular regions (cf. Fig. 3 in Appendix H). \n\nIn Fig. 3, we investigate different grid resolutions for the QuaDRiGa 5G OFDM dataset. In these simulations, we vary the resolution of the delay-doppler grid with $S^2$ gridpoints from $S=16$ to $S=64$, while keeping the maximally resolvable delay at $8\\mu\\text{s}$ and the maximally resolvable doppler at $100\\text{Hz}$. It can be seen that the grid resolution has only marginal effects on the performance of the autoencoder in a) and b). In c), we plot exemplary samples from the trained CP-GMMs, respectively. \n\nIn our final version, we will consider including a more thorough discussion and experimental validation regarding various hyperparameters, such as the number K of CP-GMM components, the grid-resolution, and the maximally resolvable delay and doppler. Moreover, we will consider including some results for the measurement campaign. We would like to thank the reviewer dNNa again as we think that both remarks will improve our work's quality.\n\n[1] Wipf et al., ”An Empirical Bayesian Strategy for Solving the Simultaneous Sparse Approximation Problem,” IEEE TSP., vol. 55", "claims_and_evidence": "Most of the paper’s claims are supported by a mix of theoretical derivations and experimental results on simulated datasets.", "methods_and_evaluation_criteria": "## Methods\n\n- **CP-VAE and CP-GMM Models:**\n  - Special versions of common generative models (variational autoencoders and Gaussian mixture models).\n  - Leverage physical properties of wireless channels (like compressibility and regular covariance patterns) to learn key parameters (angles, delays, Doppler shifts) even from noisy or incomplete data.\n  - Designed for common wireless setups such as SIMO and OFDM.\n  - Can adapt to different system configurations without needing to retrain.\n\n## Evaluation and Datasets\n\n- **Datasets Used:**\n  - A modified 3GPP model for SIMO channels.\n  - QuaDRiGa for OFDM channels.\n  - The DeepMIMO ray tracing database.\n\n- **Evaluation Metrics:**\n  - **Power Angular Profile**\n  - **Angular Spread**\n  - **Normalized Mean Squared Error (NMSE)**\n  - **Cosine Similarity**\n\nOverall, these methods and evaluation criteria are well-suited for capturing and validating the real-world behavior of wireless channels.", "theoretical_claims": "There are no theoretical proofs in the first 9 pages, and I did not read them.", "experimental_designs_or_analyses": "The paper runs experiments on simulated data. While simulated datasets are standard, the experimental validation does not include real-world measurement data. Testing on real-world channels would further confirm the practical viability of the approach.\n\n Details on hyperparameter selection and sensitivity analysis are limited. More thorough reporting on these aspects could help validate the robustness of the methods.", "relation_to_broader_scientific_literature": "The paper builds on ideas from sparse Bayesian learning and compressive sensing (as developed by Tipping (2001) and Wipf & Rao (2004)), adapting them to the wireless channel context. It leverages SBGM, previously explored in works like Böck et al. (2024b), to derive models (CP‑VAE and CP‑GMM) that are both data-efficient and physically interpretable. The use of standard datasets like QuaDRiGa and DeepMIMO situates the work within the broader literature, allowing direct comparison with existing channel models.", "essential_references_not_discussed": "The related work in this paper appears to be appropriate for the paper.", "other_strengths_and_weaknesses": "## Strengths\n- **Originality:**\n  - Integrates physical channel properties with sparse Bayesian generative modeling.\n  - Introduces novel prescribed models (CP-VAE and CP-GMM) that reduce the need for high-quality data.\n- **Significance:**\n  - Addresses key challenges in wireless channel modeling, enhancing interpretability and adaptability.\n  - Demonstrates potential for use across various system configurations without retraining.\n- **Clarity:**\n  - Overall presentation is clear with well-defined equations and detailed explanations.\n\n## Weaknesses\n- **Experimental Validation:**\n  - Evaluation is primarily based on simulated datasets; real-world datasets might further strengthen the claims.", "other_comments_or_suggestions": "None", "questions_for_authors": "1. **Real-World Data Validation:**  \n   Could you clarify whether you have tested your method on real-world channel measurements, and if not, what are your expectations or plans for such validation?  \n   *If the authors provide evidence or detailed plans that indicate strong performance on real-world data, it would significantly reinforce the practical impact of the work.*\n\n2. **Hyperparameter Sensitivity and Robustness:**  \n   How sensitive is your approach to the choice of hyperparameters (e.g., grid resolution, noise variance estimates) and what guidelines can you offer for setting them in different scenarios?  \n   *A detailed sensitivity analysis or robust guidelines could increase confidence in the method’s reproducibility and general applicability.*", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GGme6W7JF4", "reviewer": "Reviewer_BGie", "summary": "This paper introduces a physics-informed generative model for wireless channel modeling, addressing challenges in data efficiency, generalizability, and physical interpretability. Unlike traditional stochastic models or recent GAN-based methods, the authors propose a Sparse Bayesian Generative Modeling (SBGM) framework that leverages the compressibility of wireless channels to learn distributions of physical channel parameters (e.g., delays, angles of arrival/departure). The results demonstrate that the proposed approach produces physically interpretable channel models that maintain statistical and structural consistency with real-world wireless propagation environments.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer BGie for the reviewer's comments. In the following, we address the reviewer's questions and concerns. \n\n**To Other Strengths And Weaknesses (2. paragraph):** We agree with the reviewer that further ablations studies regarding various hyperparameters would improve our work's quality. Therefore, we conducted preliminary experiments regarding the number K of CP-GMM components as well as our models' grid resolution. More specifically, in Fig. 2 in the anonymous repository https://github.com/anonymousicml2529/rebuttal, we plot the angular spread for different numbers K of CP-GMM components for the modified 3GPP dataset. For K = 1, the CP-GMM coincides with M-SBL [1]. It can be seen that for K<4, the angular spread is significantly overestimated, which is to be expected as the underlying data is coming from a scenario with power from four different angular regions (cf. Fig. 3 in Appendix H). In Fig. 3 (https://github.com/anonymousicml2529/rebuttal), we investigate different grid resolutions for the QuaDRiGa 5G OFDM dataset. In these simulations, we vary the resolution of the delay-doppler grid with $S^2$ gridpoints from $S=16$ to $S=64$, while keeping the maximally resolvable delay at $8\\mu\\text{s}$ and the maximally resolvable doppler at $100\\text{Hz}$. It can be seen that the grid resolution has only marginal effects on the performance of the autoencoder in a) and b). In c), we plot exemplary samples from the trained CP-GMMs, respectively. \n\n**To Questions for Authors (1):** There are various effects influencing the channel when considering a different frequency band. Interestingly, for spatial channels, we can adapt to different frequency bands by considering a new dictionary $\\mathbf{D}$, that takes the newly considered frequency band into account. More precisely, the so-called beam squint (cf. [2]) is the major cause for changed spatial channel characteristics when varying frequency bands. The beam squint describes that the spatial steering vectors are frequency-dependent as they depend on the ratio between the antenna spacing and the considered wavelength. However, as the steering vectors constitute the dictionary $\\mathbf{D}$ in our model, we can adapt this ratio in a newly used dictionary after training. The remaining frequency-dependent effects regarding, e.g., the path loss are rather slowly changing, which is why we expect our model to be valid for a reasonably broad frequency range. We will consider including experiments regarding this in our work. \n\nWe would like to mention that our model is supposed to capture the scenario-specific channel characteristics from one concrete environment. Only then, it can generate new samples that are better suited for this environment in comparison to generic standardized channel models (e.g., 3GPP). A generalization for a large variety of environments is what the already existing standardized channel models are designed for. This is why we do **not** want our model to generalize very well to unseen environments, as this would likely lead to a considerable drop in performance for the environment it has been trained for. Of course, there is a certain trade-off that can be optimized. However, we do not think that this is within the scope of our work, and we hope that the reviewer agrees with us on this point. \n\n**To Questions for Authors (2):** In our simulations, we added noise to our training samples whose noise variance is drawn such that the corresponding SNR is uniformly distributed between 0 and 20 dB. Moreover, SBGM is specifically designed to deal with noisy and corrupted data and considers the noise level during training. Therefore, SBGM is robust with respect to different noise levels. In our OFDM simulations, we consider solely training samples that contain missing data, and our model performs very well. As a consequence, SBGM also works well for missing data. We hope that this is what the reviewer is referring to and we did not misunderstand the reviewer's question. If not, we would appreciate it if the reviewer could comment again on this point. \n\nWe hope that we addressed all the reviewer's concerns and would like to thank the reviewer again, as we think that the additional experiments regarding the hyperparameters will improve our work's quality. \n\n[1] Wipf et al., ”An Empirical Bayesian Strategy for Solving the Simultaneous Sparse Approximation Problem,” IEEE TSP., vol. 55\n\n[2] B. Wang, M. Jian, F. Gao, G. Y. Li and H. Lin, \"Beam Squint and Channel Estimation for Wideband mmWave Massive MIMO-OFDM Systems,\" in IEEE Transactions on Signal Processing, vol. 67", "claims_and_evidence": "Good to me.", "methods_and_evaluation_criteria": "The authors evaluate their methods on three different dataset: a modified standardized 3GPP spatial channel model for SIMO for a better illustration; QuaDRiGa for simulations with OFDM; and a further evaluation on the ray tracing database DeepMIMO in Appendix L. It justify the physical parameter distribution accuracy by the comparison of generated vs. real distributions for angular spread, power angular profile, and delay-Doppler characteristics. The channel generation performance is evaluated using cross-validation autoencoder reconstruction error (normalized MSE and cosine similarity). It demonstrates it outperformance by the comparisons with GAN-based baselines (AmbientGAN, GAN-rbf). Overall, these metrics align well with the goals of wireless generative modeling and provide strong empirical validation.", "theoretical_claims": "Good to me.", "experimental_designs_or_analyses": "The experimental setup is comprehensive, covering a range of scenarios and datasets to validate the proposed model. The authors conduct experiments on standardized 3GPP spatial channel models, QuaDRiGa simulations, and DeepMIMO ray-tracing datasets, ensuring that their results are applicable to both empirical and simulated wireless environments. The evaluation spans different system configurations, including single-input-multiple-output (SIMO) and orthogonal frequency-division multiplexing (OFDM), demonstrating the flexibility of the approach.\n\nA notable aspect of the experimental design is the inclusion of cross-validation using an autoencoder-based reconstruction method, which provides a robust metric for assessing the realism of generated channels. The authors also perform sensitivity analyses, varying the number of training samples and pilot symbols to examine the model’s robustness. One limitation of the experimental design is the lack of ablation studies to isolate the contributions of different components, such as the impact of sparsity constraints or the Kronecker factorization. Additionally, a discussion on training stability and sensitivity to hyperparameters would further improve the evaluation.", "relation_to_broader_scientific_literature": "The paper bridges physics-based wireless models and ML-driven generative methods, ensuring interpretability and generalizability. Unlike GANs, it leverages sparse Bayesian learning and structured priors, aligning with trends in physics-informed ML.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "One of the major strengths of this paper is its ability to generate physically consistent channel models while maintaining computational efficiency. The approach is novel in that it shifts from implicit GAN-based modeling to a prescribed statistical framework that ensures generalizability across system configurations. The integration of structured priors further enhances sample efficiency and interpretability.\n\nHowever, the paper lacks ablation studies that isolate the effects of different components, such as the impact of sparsity constraints, Kronecker factorization, and dictionary-based representations. Additionally, while the method generalizes across system configurations, there is limited discussion on generalization to different propagation environments or frequency bands.", "other_comments_or_suggestions": "Good to me.", "questions_for_authors": "1. How well does the model generalize to unseen frequency bands or new environments? \n2. How sensitive is SBGM to different levels of noise or missing data?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This submission presents a novel integration of physical channel properties with sparse Bayesian generative modeling (SBGM) to address key challenges in wireless channel modeling. The proposed CP-VAE and CP-GMM frameworks aim to improve data efficiency, generalizability, and physical interpretability compared to GAN-based approaches. 

Based on the reviews, I would like to recommend acceptance due to the following strengths of this submission:

1. The paper addresses a timely challenge in wireless communications with a technically sound, physics-informed approach.

2. Author revisions substantially improved clarity, baseline comparisons, and empirical validation (including real-world data).


Nevertheless, as also pointed out by reviewers, some improvement can be made as follows: 

1. Explicitly highlight methodological novelties (e.g., EM algorithm for Toeplitz structure) in the introduction.

2. Include real-world measurement results (Nokia dataset) in the main text.

3. Discuss limitations regarding frequency band generalization.
Decision: Accept (poster)