Title: Probabilistic and Differentiable Wireless Simulation with Geometric Transformers
Venue: NeurIPS 2024
Accepted: no
Keywords: Wireless communication

electromagnetic signals

inverse problems

learning to simulate

inverse problems

geometric deep learning

equivariance

diffusion models
Abstract: Modelling the propagation of electromagnetic signals is critical for designing modern communication systems. While there are precise simulators based on ray tracing, they do not lend themselves to solving inverse problems or the integration in an automated design loop. We propose to address these challenges through differentiable neural surrogates that exploit the geometric aspects of the problem. We first introduce the Wireless Geometric Algebra Transformer (Wi-GATr), a generic backbone architecture for simulating wireless propagation in a 3D environment. It uses versatile representations based on geometric algebra and is equivariant with respect to E(3), the symmetry group of the underlying physics. Second, we study two algorithmic approaches to signal prediction and inverse problems based on differentiable predictive modelling and diffusion models. We show how these let us predict received power, localize transmitters, and reconstruct the 3D environment from the received signal. Finally, we introduce two large, geometry-focused datasets of wireless signal propagation in indoor scenes. In experiments, we show that our geometry-forward approach achieves higher-fidelity predictions with less
data than various baselines.

Abstract

Modelling the propagation of electromagnetic signals is critical for designing modern communication systems. While there are precise simulators based on ray tracing, they do not lend themselves to solving inverse problems or the integration in an automated design loop. We propose to address these challenges through differentiable neural surrogates that exploit the geometric aspects of the problem. We first introduce the Wireless Geometric Algebra Transformer (Wi-GATr), a generic backbone architecture for simulating wireless propagation in a 3D environment. It uses versatile representations based on geometric algebra and is equivariant with respect to E(3) , the symmetry group of the underlying physics. Second, we study two algorithmic approaches to signal prediction and inverse problems based on differentiable predictive modelling and diffusion models. We show how these let us predict received power, localize transmitters, and reconstruct the 3D environment from the received signal. Finally, we introduce two large, geometry-focused datasets of wireless signal propagation in indoor scenes. In experiments, we show that our geometry-forward approach achieves higher-fidelity predictions with less data than various baselines.

1 Introduction

Modern communication is wireless: more and more, we communicate via electromagnetic waves through the antennas of various devices, leading to progress in and adoption of mobile phones, automotive, AR/VR, and IoT technologies [12, 16]. All these innovations build upon electromagnetic wave propagation. Therefore, modelling and understanding wave propagation in space is a core research area in wireless communication, and remains crucial as we are moving toward new generations of more efficient and spatially-aware wireless technologies.

Wireless signal propagation follows Maxwell's equations of electromagnetism and is often accurately modelled by state-of-the-art ray-tracing simulation software. However, these simulators take substantial time to evaluate for each scene, cannot be fine-tuned on measurements, and are (usually [29]) not differentiable. This limits their usefulness for solving inverse problems.

In contrast, neural models of signal propagation can be evaluated cheaply, can be trained on real measurements in addition to simulation, and are differentiable and thus well-suited for solving inverse problems. Several such approaches have been proposed recently, often using image-based representations of the inputs and outputs and off-the-shelf vision architectures [6, 23, 34, 35, 44, 46, 51, 52]. However, wireless surrogate modelling faces various challenges. Realistic training data is often scarce, requiring surrogate models to be data efficient. Wireless environments can consist of complex meshes. Finally, input and output data consist of a variety of data types, including the shape of extended 3D objects, point coordinates and spatial orientation of antennas, and information associated with the transmitted signal.

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In this work, we present a new approach to modelling wireless signal propagation. It is grounded in 37

the observation that wireless propagation is inherently a geometric problem: a directional signal is 38

transmitted by an oriented transmitting antenna, the signal interacts with surfaces in the environment, 39

and the signal eventually impinges an oriented receiving antenna. We argue that it is critical for 40

neural surrogates to model and flexibly represent geometric aspects (e. g. orientations, shapes) in 41

the propagation environment. We therefore develop surrogate models based on flexible geometric 42

representation and strong geometric inductive biases. 43

We first propose the Wireless Geometric Algebra Transformer (Wi-GATr), a backbone architecture for 44

wireless signal propagation problems. A key component is a new tokenizer for the diverse, geometric 45

data of wireless scenes. The tokens are processed with a Geometric Algebra Transformer (GATr) 46

network [9]. This architecture is equivariant with respect to the symmetries of wireless channel 47

modelling, but maintains the scalability of a transformer architecture. 48

Second, we study Wi-GATr models as differentiable, predictive surrogates for the simulator (see 49

Fig. 1a). Here the network predicts observables such as the received power as a function of transmitter 50

position, receiver position, and 3D environment. We show how this enables forward modelling, and 51

in addition, inverse problem solving due to Wi-GATr's differentiability. 52

Next, we propose an alternative, more versatile probabilistic approach to prediction and inference 53

tasks: training Wi-GATr diffusion models (Fig. 1b) on the joint distribution of transmitter, receiver, 54

channel information, and 3D environment. At test time, the model can be flexibly conditioned on any 55

available information to predict the received power, localize a transmitter or receiver (Fig. 1c), or 56

even reconstruct the (full or partial) 3D geometry from the wireless signal (Fig. 1d). 57

To enable machine learning development for wireless problems, we finally introduce two new datasets, 58

Wi3R and WiPTR . Each dataset consists of thousands of indoor scenes of varying complexity and 59

include all the geometric information that characterizes a wireless scene. 60

Finally, we demonstrate the predictive and the probabilistic models on these datasets. Our experi61

ments show that the Wi-GATr approach gives us a higher-fidelity predictions than various baselines, 62

generalizes robustly to unseen settings, and requires up to 20 times less data for the same perfor63

mance than a transformer baseline. 64

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2 Background and related work

Wireless signal propagation. How do wireless signals propagate from a transmitting antenna 66

(Tx) to a receiver antenna (Rx) in a (static) 3D environment? While the system is fundamentally 67

described by Maxwell's equations, for many realistic problems the ray approximation of geometric 68

optics suffices [31]. It approximates the solution to Maxwell's equations as a sum of planar waves 69

propagating in all directions from Tx. Each planar wave is represented as a ray, characterized by 70

various attributes (e. g., power, phase, delay) since transmission. As a ray reaches an object-that is, 71

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it intersects with its mesh-the interaction is modelled as reflection, refraction, or diffraction. During 72 such interactions, the power, phase, polarization, and propagation direction of the wave can change in 73 complex, material-dependent ways. In addition, new rays can emanate from the point of interaction. 74 After multiple interactions, the rays eventually reach the receiving antenna. The Tx and Rx are then 75 linked by a connected path p of multiple rays. The effects on the received signal are described by the 76 channel impulse response (CIR) h ( τ ) = ∑ p a p δ ( τ -τ p ) , where a p ∈ C is the complex gain and τ p 77 the delay of the incoming rays [53]. 78

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Maxwell's equations and in extension ray propagation are highly symmetric. The received signal does not change under rotations, translations, and reflections of the whole scene, as well as the exchange of transmitter and receiver. The latter property is known as reciprocity [37].

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Wireless simulators. Wireless propagation models play a key role in design and evaluation of communication systems, for instance by characterizing the gain of competitive designs in realistic settings or by optimizing systems performance as in base station placement for maximal coverage. Statistical approaches [2] represent propagation as a generative model where the parameters of a probabilistic model are fitted to measurements. On the other hand, wireless ray-tracing approaches [1, 5, 29] are increasingly popular due to their high accuracy and because they do not require expensive field measurement collection campaigns.

Neural wireless simulations. Both statistical and ray-tracing simulation techniques are accompanied by their own shortcomings, subsequently mitigated by their neural counterparts. Neural surrogates for statistical models [19, 40, 42, 56] reduce the amount and cost of measurements required. Neural ray tracers [29, 41, 58] address the non-differentiability of simulators using a NeRF-like strategy [38] by parameterizing the scene using a spatial MLP and rendering wireless signals using classic ray-tracing or volumetric techniques. While these techniques are faster than professional ray tracers, they are similarly bottlenecked by expensive bookkeeping and rendering steps (involving thousands of forward passes). In contrast, we propose a framework to simulate wireless signals with a single forward pass through a geometric transformer that is both sample-efficient and generalizes to novel scenes.

Geometric deep learning. The growing field of geometric deep learning [11] aims to incorporate structural properties of a problem into neural network architectures and algorithms. A central concept is equivariance to symmetry groups [15]: a network f ( x ) is equivariant with respect to a group G if its outputs transform consistently with any symmetry transformation g ∈ G of the inputs, f ( g · x ) = g · f ( x ) , where · denotes the group action. Of particular interest to us is the Euclidean group E(3) of isometries of 3D space, that is, transformations that leave Euclidean distances invariant. This group includes spatial translations, rotations, reflections, and their combinations. As we argued above, the physics of wireless signal propagation are invariant under this group.

GATr. The Geometric Algebra Transformer (GATr) [9] is an E(3) -equivariant architecture for geometric problems. Among equivariant architectures, it stands out in two ways. First, it uses geometric (or Clifford) algebras [14, 22] as representations. For a rigorous introduction to these algebras, we refer the reader to Dorst [20]. From a practical machine learning perspective, these algebras define embeddings for various geometric primitices like 3D points, planes, or E(3) transformations. We will show that this representation is particularly well-suited for wireless channel modelling. Second, GATr is a transformer architecture [54]. It computes the interactions between multiple tokens through scaled dot-product attention. With efficient backends like FlashAttention [17], the architecture is scalable to large systems, without any restrictions on the sparsity of interactions like in message-passing networks.

Diffusion models. Diffusion models [25, 48, 50] are a class of generative models that iteratively invert a noising process. They have become the de-facto standard in image and video generation [26, 45]. Recently, they have also shown to yield promising results in the generation of spatial and sequential data, such as in planning [30] and puzzle solving [28]. Aside from their generative modelling capabilities, diffusion models provide a flexible way for solving inverse problems [13, 36] through multiplication with an appropriate likelihood term [48]. Furthermore, by combining an invariant prior distribution with an equivariant denoising network, one obtains equivariant diffusion models [33]. These yield a sampling distribution that assigns equal probability to all symmetry transformations of an object, which can improve performance and data efficiency in symmetry problems like molecule generation [27] and planning [10]. We will demonstrate similar benefits in modelling wireless signal propagation.

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3 The Wireless Geometric Algebra Transformer (Wi-GATr)

3.1 Problem formulation

Our goal is to model the interplay between 3D environments, transmitting and receiving antennas, and the resulting transmitted wireless signals. More precisely, we consider wireless scenes consisting of:

The 3D geometry F of the environment. We specify it through a triangular mesh with a discrete material class associated with each mesh face.

A set of transmitting antennas t i for i = 1 , . . . , n t . Each t i is characterized by a 3D position, an orientation, and any antenna characteristics. We will often focus on the case of a single Tx and then omit the index i .

Analogously, a set of receiving antennas r i for i = 1 , . . . , n r .

The channel or signal h ij between each transmitter i and each receiver j , which can be any observable function of the CIR.

In this setting, we consider various downstream tasks:

Signal prediction is about predicting the signal received at a single antenna from a single receiver, p ( h | F, t, r ) with n t = n r = 1 . This is exactly the task that ray-tracing simulators solve. Often, the signal is modelled deterministically as a function h ( F, t, r ) .

Receiver localization : inferring the position and properties of a receiving antenna from one or multiple transmitters, r ∼ p ( r | F, { t i } , { h i } ) , with n r = 1 .

Geometry reconstruction or sensing: reconstructing a 3D environment partially, inferring p ( F u | F k , t, r, h ) , where F u and F k are the unknown and known subsets of F , respectively.

The latter two problems are examples of inverse problems , as they invert the graphical model that simulators are designed for. They are not straightforward to solve with the simulators directly, but we will show how neural surrogates trained on simulator data can solve them.

3.2 Backbone

Core to our approach to this family of inference problems is the Wireless Geometric Algebra Transformer (Wi-GATr) backbone. It consists of a novel tokenizer and a network architecture.

Wireless GA tokenizer. The tokenizer takes as input some subset of the information characterizing a wireless scene and outputs a sequence of tokens that can be processed by the network. A key challenge in the neural modelling of wireless problems is the diversity of types of data involved. As we argued above, a wireless scene consists of the 3D environment mesh F , which features threedimensional objects such as buildings and trees, antennas t and r characterized through a pointlike position, an antenna orientation, and additional information about the antenna type, and the characteristics of the channel h .

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To support all of these data types, we propose a new tokenizer that outputs a sequence of geometric algebra (GA) tokens. Each token consists of a number of elements (channels) of the projective geometric algebra G 3 , 0 , 1 in addition to the usual unstructured scalar channels. We define the GA precisely in Appendix A. Its main characteristics are that each element is a 16-dimensional vector and can represent various geometric primitives: 3D points including an absolute position, lines, planes, and so on. This richly structured space is ideally suited to represent the different elements encountered in a wireless problem. Our tokenization scheme is specified in Tbl. 1.

Network. After tokenizing, we process the input data with a Geometric Algebra Transformer (GATr) [9]. This architecture naturally operates on our G 3 , 0 , 1 parameterization of the scene. It is equivariant with respect to permutations of the input tokens as well as E(3) , the symmetry group of translations, rotations, and reflections. These are exactly the symmetries of wireless signal propagation, with one exception: wireless signals have an additional reciprocity symmetry that specifies that the signal is invariant under an role exchange between transmitter and receiver. We will later show how we can incentivize this additional symmetry property through data augmentation. 1 Finally, because GATr is a transformer, it can process sequences of variable lengths and scales well to systems with many tokens. Both properties are crucial for complex wireless scenes, which can in particular involve a larger number of mesh faces.

3.3 Predictive modelling

The Wi-GATr backbone can be used either in a predictive or probabilistic ansatz. We begin with the predictive modelling of the measured channel information as a function of the complete 3D environment and the information characterizing the transmitter and receiver, h θ ( F, t, r ) . This regression model is trained in a supervised way on simulated or measured wireless scenes.

Forward prediction. The network thus learns a differentiable, deterministic surrogate for the simulator model h sim ( F, t, r ) . At test time, we can use the network instead of a simulator to predict the signals in unseen, novel scenes. Compared to a simulator based on ray tracing, it has three advantages: it can be evaluated in microseconds rather than seconds or minutes, it can be finetuned on real measurements, and it is differentiable.

Inverse problems. This differentiability makes such a surrogate model well-suited to solve inverse problems. For instance, we can use it for receiver localization. Given a 3D environment F , transmitters { t i } , and corresponding signals { h i } , we can find the most likely receiver position and orientation as ˆ r = arg min r ∑ i ∥ h θ ( F, t i , r ) -h ∥ 2 . The minimization can be performed numerically through gradient descent, thanks to the differentiability of the Wi-GATr surrogate.

3.4 Probabilistic modelling

While a predictive model of the signal can serve as a powerful neural simulator, it has two shortcomings. Solving an inverse problem through gradient descent requires a sizable computational cost for every problem instance. Moreover, predictive models are deterministic and do not allow us to model stochastic forward processes or express the inherent uncertainty in inverse problems.

Equivariant diffusion model. To overcome this, we draw inspiration from the inverse problem solving capabilities of diffusion models using guidance [13]. In this case, we formulate the learning problem as a generative modelling task of the joint distribution p θ ( F, t, r, h ) between 3D environment mesh F , transmitter t , receiver r , and channel h , for a single transmitter-receiver pair. Concretely, we follow the DDPM framework and use a Wi-GATr model as score estimator (denoising network). By using an invariant base density and an equivariant denoising network, we define an invariant generative model. See Appendix B for a detailed description of our diffusion model and the discussion of some subtleties in equivariant generative modelling.

Unifying forward prediction and inverse problems as conditional sampling. A diffusion model trained to learn the joint density p θ ( F, t, r, h ) does not only allow us to generate unconditional samples of wireless scenes, but also lets us sample from various conditionals: given a partial wireless scene, we can fill in the remaining details, in analogy to how diffusion models for images allow for

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inpainting. To achieve this, we use the conditional sampling algorithm proposed by Sohl-Dickstein et al. [48]: at each step of the sampling loop, we fix the conditioning variables to their known values before feeding them into the denoising network.

This algorithm lets us solve signal prediction (sampling from p θ ( h | F, t, r ) ), receiver localization (from p θ ( r | F, t, h ) ), geometry reconstruction (from p θ ( F u | F k , t, r, h ) ), or any other inference task in wireless scenes. We thus unify 'forward' and 'inverse' modelling in a single algorithm. Each approach is probabilistic, enabling us to model uncertainties. This is important for inverse problems, where measurements often underspecify the solutions.

In principle, the unconditional diffusion objective should suffice to enable test-time conditional sampling. In practice, we find that we can improve the conditional sampling performance with two modifications. First, we combine training on the unconditional diffusion objective with conditional diffusion objectives. For the latter, we randomly select tokens to condition on and evaluate the diffusion loss only on the remaining tokens. Second, we provide the conditioning mask as an additional input to the denoising model. See Appendix B for details.

4 New datasets

While several datasets of wireless simulations and measurements exist [3, 4, 41, 57], they either do not include geometric information, are not diverse, are at a small scale, or the signal predictions are not realistic. To facilitate the development of machine learning methods with a focus on geometry, we generate two new datasets of simulated wireless scenes. 2 Both feature indoor scenes and channel information generated with a state-of-the-art ray-tracing simulator [1] at a frequency of 3.5 GHz. They provide detailed characteristics for each path between Tx and Rx, such as gain, delay, angle of departure and arrival at Tx/Rx, and the electric field at the receiver itself, which allows users to compute various quantities of interest themselves. See Appendix C for more details.

Wi3R dataset. Our first dataset focuses on simplicity: each of 5000 floor plans has the same size and number of rooms, and all walls have the same material across layouts. They differ only in their layouts, which we take from Wi3Rooms [41], Tx positions, and Rx positions. In Appendix C we define training, validation, and test splits as well as an out-of-distribution set to test the robustness of different models.

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Table 2: Signal prediction results. We show the mean absolute error on the received power in dBm (lower is better, best in bold). Top : In-distribution performance. Middle : Generalization under symmetry transformations. Bottom : Generalization to out-of-distribution settings. In almost all settings, Wi-GATr is the highest-fidelity surrogate model.

WiPTR dataset. Next, we generate a more varied, realistic dataset based on the floor layouts in the ProcTHOR-10k dataset for embodied AI research [18]. We extract the 3D mesh information including walls, windows, doors, and door frames and assign 6 different dielectric materials for different groups of objects. Our dataset consists of 12k different floor layouts, split into training, test, validation, and OOD sets as described in Appendix C. Not only does WiPTR stand out among wireless datasets in terms of its level of detail and scale, but because it is based on ProcTHOR-10k , it is also suited for the integration with embodied AI research.

5 Experiments

5.1 Predictive modelling

We focus on the prediction of the time-averaged non-coherent received power h = ∑ p | a p | 2 , disregarding delay or directional information that may be available in real measurements. We train predictive surrogates h θ ( F, t, r ) that predict the power as a function of the Tx position and orientation t , Rx position and orientation r , and 3D environment mesh F , on both the Wi3R and WiPTR datasets. All models are trained with reciprocity augmentation, i. e., randomly flipping Tx and Rx labels during training. This improves data efficiency slightly, especially for the transformer baseline.

In addition to our Wi-GATr model, described in Sec. 3, we train several baselines. The first is a vanilla transformer [54], based on the same inputs and tokenization of the wireless scene, but without the geometric inductive biases. Next, we compare to the E(3) -equivariant SEGNN [8], though we were only able to fit this model into memory for the Wi3R dataset. In addition, we train a PLViT model, a state-of-the-art neural surrogate for wireless scenes [24] that represent wireless scenes as an image centered around the Tx position. Finally, we attempt to compare Wi-GATr also to WiNeRT [41], a neural ray tracer. However, this architecture, which was developed to be trained on several measurements on the same floor plan, was not able to achieve useful predictions on our diverse datasets with their focus on generalization across floor plans. Our experiment setup and the baselines are described in detail in Appendix D.

Signal prediction. In Fig. 2 we illustrate the prediction task on a WiPTR floor plan. We show signal predictions for the simulator as well as for surrogate models trained on only 100 floor plans. Despite this floor plan not being part of the training set, Wi-GATr is able to capture the propagation pattern well, while the transformer and ViT show memorization artifacts.

In Tbl. 2 we compare surrogate models trained on the full Wi3R and WiPTR datasets. Both when interpolating Rx positions on the training floor plans as well as when evaluating on new scenes unseen during training, Wi-GATr offers the highest-fidelity approximation of the simulator. Wi-GATr as well as the equivariant baselines are by construction robust to symmetry transformations, while

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the performance of a vanilla transformer degrades substantially. All methods but SEGNN struggle to generalize to an OOD setting on the Wi3R dataset. This is not surprising given that the training samples are so similar to each other. On the more diverse WiPTR dataset, Wi-GATr is almost perfectly robust under domain shift.

Data efficiency. Next, we study the data efficiency of the different surrogates in Fig. 3. Wi-GATr is more data-efficient than any other method with the exception of the E(3) -equivariant SEGNN, which performs similarly well for a small number of training samples. This confirms that equivariance is a useful inductive bias when data is scarce. But Wi-GATr scales better than SEGNN to larger number of samples, showing that our architecture combines the small-data advantages of strong inductive biases with the large-data advantages of a transformer architecture.

Inference speed. One of the advantages of neural surrogates is their test-time speed. Both Wi-GATr and a transformer are over a factor of 20 faster than the ground-truth ray tracer (see Appendix D).

Receiver localization. Next, we show how differentiable surrogates let us solve inverse problems, focusing on the problem of receiver localization. We infer the Rx position with the predictive surrogate models by optimizing through the neural surrogate of the simulator as discussed in Sec. 3.3. The performance of our surrogate models is shown in Fig. 4 and Appendix D. 3 The two neural surrogates achieve a similar performance when only one or two transmitters are available, a setting in which the receiver position is highly ambiguous. With more measurements, Wi-GATr lets us localize the transmitter more precisely.

5.2 Probabilistic modelling

Next, we experiment with our probabilistic approach. We train diffusion models on the Wi3R dataset. In addition to a Wi-GATr model, we study a transformer baseline, as well as a transformer trained on the same data augmented with random rotations. Both models are trained with the DDPM pipeline with 1000 denoising steps and samples from with the DDIM solver [49]. Our setup is described in detail in Appendix D.

Signal prediction, receiver localization, and geometry reconstruction as conditional sampling. In our probabilistic approach, signal prediction, receiver localization, and geometry reconstruction are all instances of sampling from conditional densities: h ∼ p θ ( h | F, t, r ) , r ∼ p θ ( r | F, t, h ) , and F u ∼ p θ ( F u | F k , t, r, h ) , respectively. We qualitatively show results for this approach in Figs. 1 and 5. All of these predictions are probabilistic, which allows our model to express uncertainty in ambiguous inference tasks. When inferring Rx positions from a single measurement, the model learns multimodal densities, as shown in the middle of Fig. 5. When reconstructing geometry, the model will sample diverse floor plans as long as they are consistent with the transmitted signal, see the right panel of Fig. 5. Additional results on signal and geometry prediction are given in Appendix D.2.

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We quantitatively evaluate these mod303 els through the variational lower bound 304 on the log likelihood of test data under 305 the model. To further analyze the ef306 fects of equivariance, we test the model 307 both on canonicalized scenes, in which 308 all walls are aligned with the x and y 309 axis, and scenes that are arbitrarily ro310 tated. The results in Tbl. 3 show that 311 Wi-GATr outperforms the transformer 312 baseline across all three tasks, even in 313 the canonicalized setting or when the 314 transformer is trained with data augmen315 tation. The gains of Wi-GATr are partic316 ularly clear on the signal prediction and 317 receiver localization problems. 318

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6 Discussion

Wireless signal transmission through electromagnetic wave propagation is an inherently geometric and symmetric problem. We developed a class of neural surrogates grounded in geometric representations and strong inductive biases. They are based on our new Wi-GATr backbone architecture, consisting of a new tokenization scheme for wireless scenes together with an E(3) -equivariant transformer architecture. The proposed backbone is applied in two ways to wireless tasks: first, as a differentiable 'forward' prediction model that maps the features to the signals; second, as a probabilistic diffusion model that captures the joint and conditional distributions of features and channels. We employed these designs in experiments on received power prediction, receiver localization, and geometry reconstruction, where our Wi-GATr models enabled precise predictions, outperforming various baselines.

Our analysis is in many ways a first step. The range of materials in our datasets is limited and we only experimented with measurements of the non-coherent total received power, which is a stable signal, but offers less spatial information than measurements of the time delay or angular information. More importantly, we only considered idealized inference tasks. For instance, our receiver localization problem assumed perfect knowledge of the room geometry and materials.

Nevertheless, we hope that we were able to highlight the benefits of a geometric treatment of wave 335 propagation modelling. Augmenting or replacing the image-based or general-purpose representations 336 and architectures prevalent in wireless modelling with geometric approaches has the potential of 337 improving data efficiency, performance, and robustness. 338

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A Geometric algebra

As representation, Wi-GATr uses the projective geometric algebra G 3 , 0 , 1 . Here we summarize key aspects of this algebra and define the canonical embedding of geometric primitives in it. For a precise definition and pedagogical introduction, we refer the reader to Dorst [20].

Geometric algebra. A geometric algebra G p,q,r consists of a vector space together with a bilinear operation, the geometric product , that maps two elements of the vector space to another element of the vector space.

The elements of the vector space are known as multivectors . Their space is constructed by extending a base vector space R d to lower orders (scalars) and higher-orders (bi-vectors, tri-vectors, . . . ). The algebra combines all of these orders (or grades ) in one 2 d -dimensional vector space. From a basis for the base space, for instance ( e 1 , e 2 , e 3 ) , one can construct a basis for the multivector space. A multivector expressed in that basis then reads, for instance for d = 3 , x = x ∅ + x 1 e 1 + x 2 e 2 + x 3 e 3 + x 12 e 1 e 2 + x 13 e 1 e 3 + x 23 e 2 e 3 + x 123 e 1 e 2 e 3 .

The geometric product is fully defined by bilinearity, associativity, and the condition that the geometric product of a vector with itself is equal to its norm. The geometric product generally maps between different grades. For instance, the geometric product of two vectors will consist of a scalar, the inner product between the vectors, and a bivector, which is related to the cross-product of R 3 . In particular, the conventional basis elements of grade k > 1 are constructed as the geometric product of the vector basis elements e i . For instance, e 12 = e 1 e 2 is a basis bivector. From the defining properties of the geometric products it follows that the geometric product between orthogonal basis elements is antisymmetric, e i e j = -e j e i . Thus, for a d -dimensional basis space, there are ( d k ) independent basis elements at grade k .

Projective geometric algebra. To represent three-dimensional objects including absolute positions, we use a geometric algebra based on a base space with d = 4 , adding a homogeneous coordinate to the 3D space. 4 We use a basis ( e 0 , e 1 , e 2 , e 3 ) with a metric such that e 2 0 = 0 and e 2 i = 1 for i = 1 , 2 , 3 . The multivector space is thus 2 4 = 16 -dimensional. This algebra is known as the projective geometric algebra G 3 , 0 , 1 .

Canonical embedding of geometric primitives. In G 3 , 0 , 1 , we can represent geometric primitives as follows:

Scalars (data that do not transform under translation, rotations, and reflections) are represented as the scalars of the multivectors (grade k = 0 ).

Oriented planes are represented as vectors ( k = 1 ), encoding the plane normal as well as the distance from the origin.

Lines or directions are represented as bivectors ( k = 2 ), encoding the direction as well as the shift from the origin.

Points or positions are represented as trivectors ( k = 3 ).

For more details, we refer the reader to Tbl. 1 in Brehmer et al. [9], or to Dorst [20].

B Probabilistic model

Formally, we employ the standard DDPM framework [50] to train a latent variable model p θ ( x 0 ) = ∫ p θ ( x 0: T ) d x 1: T , where x 0 = [ rsrp, tx , rx , mesh ] denotes the joint vector of variables following the dataset distribution p data ( x 0 ) . In DDPM, the latent variables x 1: T are noisy versions of the original data, defined by a discrete forward noise process q ( x t | x t -1 ) = N ( x t ; √ 1 -β t x t -1 , β t I ) and β i > 0 . We approximate the reverse distribution q ( x t -1 x t ) with p θ ( x t -1 | x t ) = ∑ ˆ x 0 q ( x t -1 | x t , ˆ x 0 ) p θ (ˆ x 0 | x t , t ) , where q ( x t -1 | x t , x 0 ) is a normal distribution with closed-form parameters [25]. The forward and backward distributions q and p form a variational autoencoder [32] which can be trained with a variational lower bound loss. Using the above parametrization of p θ ( x t -1 | x t ) , however, allows for a simple approximation of this lower bound by training on an MSE objective L = E x t , x 0 [ || f θ ( x t , t ) -x 0 || 2 ] which resembles denoising score matching [55].

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To parametrize p θ (ˆ x 0 | x t , t ) , we pass the raw representation of x t through the wireless GA tokenizer of Wi-GATr and, additionally, we embed the scalar t through a learned timestep embedding [43]. The embedded timesteps can then be concatenated along the scalar channels in the GA representation in a straightforward manner. Similar to GATr [9], the neural network outputs a prediction in the GA representation, which is subsequently converted to the original latent space. Note that this possibly simplifies the learning problem, as the GA representation is inherently higher dimensional than our diffusion space with the same dimensionality as x 0 .

Equivariant generative modelling. A diffusion model with an invariant base density and an equivariant denoising network defines an invariant density, but equivariant generative modelling has some subtleties [33]. Because the group of translations is not compact, we cannot define a translationinvariant base density. Previous works have circumvented this issue by performing diffusion in the zero center of gravity subspace of euclidean space [27]. However, we found that directly providing the origin as an additional input to the denoising network also resulted in good performance, at the cost of full E(3) equivariance. We also choose to generate samples in the convention where the z -axis represents the direction of gravity and positive z is 'up'; we therefore provide this direction of gravity as an additional input to our network.

Masking strategies. To improve the performance of conditional sampling, we randomly sample conditioning masks during training which act as an input to the model, as well as a mask on the loss terms. Namely, we sample masks from a discrete distribution with probabilities p = (0 . 2 , 0 . 3 , 0 . 2 , 0 . 3) corresponding to masks for unconditional, signal, receiver and mesh prediction respectively. If we denote this distribution over masks as p ( m ) , the modified loss function then reads as L = E m ∼ p ( m ) , x t , x 0 [ || m ⊙ f θ ( x m t , t, m ) -m ⊙ x 0 || 2 ] , where x m t is equal to x 0 along the masked tokens according to m .

C Datasets

Table 4 summarizes major characteristics of the two datasets. In the following we explain more details on data splits and generation.

Wi3R dataset. Based on the layouts of the Wi3Rooms dataset by Orekondy et al. [41], we run simulations for 5000 floor layouts that are split into training (4500), validation (250), and test (250). These validation and test splits thus represent generalization across unseen layouts, transmitter, and receiver locations. From the training set, we keep 10 Rx locations as additional test set to evaluate generalization only across unseen Rx locations. To evaluate the generalization performance, we also introduce an out-of-distribution (OOD) set that features four rooms in each of the 250 floor layouts. In all layouts, the interior walls are made of brick while exterior walls are made of concrete. The The Tx and Rx locations are sampled uniformly within the bounds of the floor layouts (10m × 5m × 3m).

WiPTR dataset. Based on the floor layouts in the ProcTHOR-10k dataset for embodied AI research [18], we extract the 3D mesh information including walls, windows, doors, and door frames. The layouts comprise between 1 to 10 rooms and can cover up to 600 m 2 . We assign 6 different dielectric materials for different groups of objects (see Tbl. 5). The 3D Tx and Rx locations are randomly sampled within the bounds of the layout. The training data comprises 10k floor layouts, while test and validation sets each contain 1k unseen layouts, Tx, and Rx locations. Again, we introduce an OOD validation set with 5 layouts where we manually remove parts of the walls such that two rooms become connected. While the multi-modality in combination with the ProcTHOR dataset enables further research for joint sensing and communication in wireless, our dataset set is also, to the best of our knowledge, the first large-scale 3D wireless indoor datasets suitable for embodied AI research.

D Experiments

D.1 Predictive modelling

Models. We use an Wi-GATr model that is 32 blocks deep and 16 multivector channels in addition to 32 additional scalar channels wide. We use 8 attention heads and multi-query attention. Overall, the model has 1 . 6 · 10 7 parameters. These settings were selected by comparing five differently sized networks on an earlier version of the Wi3R dataset, though somewhat smaller and bigger networks

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Our Transformer model has the same width (translating to 288 channels) and depth as the Wi-GATr model, totalling 16 . 7 · 10 6 parameters. These hyperparameters were independently selected by comparing five differently sized networks on an earlier version of the Wi3R dataset.

For SEGNN, we use representations of up to ℓ max = 3 , 8 layers, and 128 hidden features. The model has 2 . 6 · 10 5 parameters. We selected these parameters in a scan over all three parameters, within the ranges used in Brandstetter et al. [8].

The PLViT model is based on the approach introduced by Hehn et al. [24]. We employ the same centering and rotation strategy as in the original approach around the Tx. Further, we extend the original approach to 3 dimensions by providing the difference in z -direction concatenated with the 2D x -y -distance as one token. Since training from scratch resulted in poor performance, we finetuned a ViT-B-16 model pretrained on ImageNet and keeping only the red channel. This resulted in a model with 85 . 4 · 10 7 parameters and also required us to use a fixed image size for each dataset that ensures the entire floor layout is visible in the image data.

Optimization. All models are trained on the mean squared error between the model output and the total received power in dBm. We use a batch size of 64 (unless for SEGNN, where we use a smaller batch size due to memory limitations), the Adam optimizer, an initial learning rate of 10 -3 , and a cosine annealing scheduler. Models are trained for 5 · 10 5 steps on the Wi3R dataset and for 2 · 10 5 steps on the WiPTR dataset.

Inference speed. To quantify the trade-off between inference speed and accuracy of signal prediction, we compare the ray tracing simulation with our machine learning approaches. For this purpose, we evaluate the methods on a single room of the validation set with 2 different Tx locations and two

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equidistant grids at z ∈ { 2 . 3 , 0 . 3 } with each 1637 Rx locations. Figure 7 summarizes the average inference times per link with the corresponding standard deviation. While Wireless InSite ( 6 / 3 / 1 , i.e., 6 reflections/3 transmissions/1 diffraction) represents our method that was used to generate the ground truth data, it is also by far the slowest approach. Note that we only measure the inference speed of Wireless InSite for each Tx individually without the preprocessing of the geometry. By reducing the complexity, e.g., reducing the number of allowed reflections or transmissions, of the ray tracing simulation the inference time can be reduced significantly. For example, the configuration 3 / 2 / 1 shows a significant increase in inference speed, but at the same time we can already see that the simulation results do not match the ground truth anymore. This effect is even more pronounced for the case of Wireless InSite 3 / 1 / 1 . Our machine learning solutions outperform all tested configurations of Wireless InSite in terms of inference speed, while at the same time keeping competitive performance in terms of prediction accuracy (MAE) compared to the data generation simulation itself in a simpler configuration setting.

In addition, the differentiability of ML approches enables them to solve inverse problems and such as finetuning to real-world measurement data. Finetuning, often referred to as calibration, remains challenging for simulation software and will likely lead to increased MAE as the ground truth is not given by Wireless InSite itself anymore.

D.2 Probabilistic modelling

Experiment setup. For all conditional samples involving p ( F u | F k , t, r, h ) , we always choose to set F k to be the floor and ceiling mesh faces only and F u to be the remaining geometry. This amounts to completely predicting the exterior walls, as well as the separating walls/doors of the three rooms, whereas the conditioning on F k acts only as a mean to break equivariance. Since F is always canonicalized in the non-augmented training dataset, this allows for direct comparison of variational lower bounds in Tbl. 3 with the non-equivariant transformer baseline.

Models. For both Wi-GATr and the transformer baseline, we follow similar architecture choices as for the predictive models, using an equal amount of attention layers. To make the models timestepdependent, we additionally employ a standard learnable timestep embedding commonly used in

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diffusion transformers [43] and concatenate it to the scalar channel dimension.

Optimization. We use the Adam optimizer with a learning rate of 10 -3 for the Wi-GATr models. The transformer models required a smaller learning rate for training stability, and thus we chose 3 · 10 -4 . In both cases, we linearly anneal the learning rate and train for 7 · 10 5 steps with a batchsize of 64 and gradient norm clipping set to 100 .

Evaluation. We use the DDIM sampler using 100 timesteps for visualizations in Fig. 5 and for the error analysis in Fig. 8. To evaluate the variational lower bound in Tbl. 3, we follow [39] and evaluate L vlb := L 0 + L 1 + . . . L T , where L 0 := -log p θ ( x 0 | x 1 ) , L t -1 := D KL ( q ( x t -1 | x t , x 0 ) || p θ ( x t -1 | x t )) and L T := D KL ( q ( x T | x 0 ) , p ( x t )) . To be precise, for each sample x 0 on the test set, we get a single sample x t from q and evaluate L vlb accordingly. Table 3 reports the mean of all L vlb evaluations over the test set.

Additional results. Fig. 8, shows the quality of samples from p θ ( h | F, t, r ) as a function of the amount of available training data, where we average over 3 samples for each conditioning input. It is worth noting that diffusion samples have a slightly higher error than the predictive models. This shows that the joint probabilistic modelling of the whole scene is a more challenging learning task than a deterministic forward model.

To further evaluate the quality of generated rooms, we analyze how often the model generates walls between the receiver and transmitter, compared to the ground truth. Precisely, we plot the distribution of received power versus the distance of transmitter and receiver in Fig. 9 and color each point according to a line of sight test. We can see that, overall, Wi-GATr has an intersection error of 0 . 26 , meaning that in 26% of the generated geometries, line of sight was occluded, while the true geometry did not block line of sight between receiver and transmitter. This confirms that the diffusion model correctly correlates the received power and receiver/transmitter positions with physically plausible geometries. While an error of 26% is non-negligible, we note that this task involves generating the whole geometry given only a single measurement of received power, making the problem heavily underspecified. Techniques such as compositional sampling [21] could overcome this limitation by allowing to condition on multiple receiver and received power measurements.

E Discussion

Progress in wireless channel modelling is likely to lead to societal impact. Not all of it is positive. 966 The ability to reconstruct details about the propagation environment may have privacy implications. 967 Wireless networks are ubiquitous and could quite literally allow to see through walls. At the same time, 968 we believe that progress in the development of wireless channel models may help to reduce radiation 969 exposure and power consumption of wireless communication systems, and generally contribute to 970

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes the use of the Wireless Geometric Algebra Transformer (Wi-GATr) to model signal propagation. Based on the Wi-GATr network, it introduces a differentiable prediction model and a diffusion model. Compared to traditional statistical and ray-tracing methods, the proposed approach not only addresses conventional signal prediction problems but also tackles inverse problems such as receiver localization and 3D environment reconstruction. Experimental results demonstrate the effectiveness of this method. Furthermore, the authors present two large-scale wireless signal propagation datasets.\n\nSTRENGTHS:\n1) The perspective is interesting. This paper models wireless propagation as a probability model based on diffusion, thus offering a unified approach to address signal prediction and inverse problems such as receiver localization or 3D scene reconstruction.\n2) The paper is logically structured, with a comprehensive background introduction and high readability.\n\nWEAKNESSES:\n1) There is a gap between the challenges and the solutions. The author asserts that wireless surrogate modelling faces challenges like data scarcity and diverse data types. However, the lack of analysis on these issues makes the proposed solutions appear abrupt. It is recommended to provide insights that lead to the proposed solutions of this paper.\n2) The innovation is somewhat limited. Wi-GATr primarily extends the GATr method into a wireless setting, with equivariance being a pre-existing property of the original framework. Apart from tokenizing input data, did the paper introduce any additional advancements? It would be beneficial for the authors to highlight these aspects.\n3) The experimental evaluation is not sufficiently convincing. For more details, please refer to the \"Questions*\" part.\n4) There are some typographical errors in the paper. For example, lines 56 and 57 do not correspond to Figure 1. Additionally, the abstract mentions transmitter localization, but the main text describes receiver localization, among other discrepancies.\n\nQUESTIONS:\n1) The contributions of Wi-GATr appear to be limited. What are the specific contributions of Wi-GATr, and how do they compare in detail with those of GATr?\n2) In the background section, the authors mention a series of neural statistical models [19, 40, 42, 56] and neural ray tracers [29, 41, 58], claiming the proposed method is \"sample-efficient and generalizes to novel scenes\". However, the evaluation section only compares two methods, which seems relatively weak. What considerations are made when choosing the comparison methods?\n3) In qualitative experiments, the proposed method is compared with ViT. Why is it not compared with the benchmarks used in quantitative experiments—SEGNN and PLViT?\n4) The experimental results in Figure 5 lack the necessary explanations. For instance, does the \"Geometry reconstruction\" on the right side lack comparison with ground truth?\n5) This paper introduces both differentiable prediction models and diffusion models, claiming that diffusion models overcome the limitations of differentiable prediction models. Could the author conduct experiments to compare these two models and validate this claim?\n\nLIMITATIONS:\nSufficiently discussed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes the use of a transformer architecture to model electromagnetic propagation of physical systems. The approach is claimed to outperform existing methods by (i) computational efficiency (compared to raytracers) and (ii) enabling solving inverse problems. The method is evaluated on a number of benchmark tasks and the paper is accompanied by two new datasets.\n\nSTRENGTHS:\nThe idea of modeling electromagnetic wave propagation using transformer architectures is novel and interesting. \n\nThe generality of the approach enables a large number of tasks in wireless communication systems that would otherwise the use of raytracers or other electromagnetic modeling software.\n\nLarge parts of the main body of the paper are well-written and easy to follow.\n\nWEAKNESSES:\nThe main body of the paper lacks the details of the proposed pipeline and transformer architecture. In fact, all of the interesting and technical details are relegated to the appendices. \n\nOne of the main motivations of the paper seems to be that most raytracers are too slow. However, the authors seem to ignore recent projects, such as Instant RM (https://github.com/NVlabs/instant-rm) which can compute coverage maps in a few milliseconds, depending on the desired accuracy.\n\nIt is unclear why [29] is cited as a non-differentiable raytracer although it is, to my knowledge, the only raytracer that actually is. Instant RM is also differentiable and calibration results for both tools were already demonstrated. To be honest, I have the impression that the authors tried to cover up the fact that [29] is a powerful *differentiable* raytracer that enables solving inverse problems. \n\nAlthough the authors claim that channel impulse responses can be generated, this is not demonstrated in the paper. I think that this claim should be removed unless the authors demonstrate that it is actually feasible.\n\nThe description of scene geometry recovery is incomprehensible to me.\n\nIn Figure 7, the Wi-GATr is around 20ms for inference for a tiny indoor scene. The authors should compare this against Instant RM which can probably run even faster and is differentiable.\n\nIt would be good to get confidence information (e.g., standard deviation) in Fig. 3 and Fig. 4.\n\nQUESTIONS:\nThe authors should comment on the scalability of the proposed method to scenes with millions of triangles.\n\nHow well would a neural-network-based positioning pipeline do on the task used to generate Figure 4?\n\nLIMITATIONS:\nThe paper lacks a detailed comparison to the capabilities of the differential raytracer from [29]. In fact, I feel that for the individual tasks, more baselines should be included. \n\nScalability to very large datasets and extremely complex scenes is unclear. \n\nIt is unclear whether the method generalizes to electromagnetic environments that are nonreciprocal (e.g., containing certain nonreciprocal metasurfaces). \n\nIt is unclear whether the method generalizes to scenarios in which ray-tracing is inaccurate (e.g., scenarios at low carrier frequencies).", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nAuthors proposed transformer based ideas on very well studied area: Wireless environment simulation.\nThe key idea here is to capitalize Geometric Transformers to simulate radio environments. It true that wireless (directional) signal propagation is a ray tracing approach, meaning a highly directional wireless signal (Ray) may bounce off ambient surfaces or directly reach a receiver. The proposed method inserts geometric shapes in the environment as tokens in a transformer networks. The trained transformer predicts the received power at a given point in 3D space. Transformer is trained and evaluated using two datasets: Wi3R and WiPTR that simulate indoor signal propagation environments. In comparison to the baselines, transformer architecture requires 20 times less data.\n\nSTRENGTHS:\n1. It is interesting idea to see if transformer architectures are suitable to model radio propagation environments\n2.  Propagation models consider the material of ambient surface, antenna parameters, location of transmitter and receiver\n\nWEAKNESSES:\n1. It is not clear if datasets are of any relevance to real world environment, since the primary challenge of any modelling problem is simulation to reality gap. Since wireless channel modelling is very well studied area, a novel contribution must take in to account such differences rather than results that show the ability of transformer architecture in modelling a wireless environment\n2. There are several high fidelity radio propagation modelling software, perhaps it is important to consider datasets generated from such model, current evaluation is very limited and primitive. Fig 2 and 5 are no comparison to robust channel models that are available to wireless researcher and practitioners.\n3. The modelling of radio environment is no clear, reviewer is of impression that several affects like diffraction, refraction are not considered in the datasets\n4. The paper also has weakness: WiNeRT: Towards Neural Ray Tracing for Wireless Channel Modelling and Differentiable Simulations which is both papers have not considered user mobility: coherence time, coherence bandwidth\n5. Current evaluation is only limited to indoor environments\n6. Since the prior work has already established neural network architecture are useful modelling, to push the state of the art, it is important to show accurate modelling than yet another architecture to model wireless channel.\n7. Upon inspecting table 2, the reviewer is afraid that there might be issue with results here. There is 80dBm difference is accuracy with transformers based modelling, usually such a difference is unacceptable, can author please explain the training of transformer model and why it produced such a large error. The reviewer is concerned that whether such sample point is a fair to benchmark againt\n\nQUESTIONS:\n1. The challenging problem of wireless environment simulation is how the simulation and real world measurement hold together.\n2. Since, this is very old problem, the novel solution must show the ability to bridge gap between simulation and real channel traces\n3. It is quite important show how well the proposed method simulates various wireless environment, currently such evaluation is missing.\n4. Could you please provide any benchmarks with real-world channel traces datasets?\n\nLIMITATIONS:\nIn reviewer's opinion authors have not sufficiently addressed all the limitations of the current work. I encourage authors to look at weakness section and update the limitations of the work in the current draft", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe motivation for this work is that modeling the propagation of electromagnetic signals is critical for designing modern communication systems. Ray tracing simulators are not suitable for inverse problems or integration as channel models in designing communication systems.\n\nIn this context, the goal of this paper is to model the interplay between the 3D environment F, transmitting and receiving antennas (each characterized by a 3D position, orientation, and specific antenna characteristics) represented by t and r, respectively, and the signal h between each transmitter and receiver. The 3D geometry F is represented by a triangular mesh, where each triangle is assigned a material type from predefined classes, modeling both the shape and materials of the environment. Once the model is learned, three tasks can be performed:\n1. Prediction of the received signal p(h∣F,t,r): The model is trained for this task. At test time, the network can predict signals in unseen, novel scenes. This approach is faster, fine-tunable on real measurements. The model obtained is also differentiable. This is referred to as the forward problem.\n1. Localization of the receiver p(r∣F,t,h).\n1. Sensing the environment p(F∣t,r,h).\n\nThe last two tasks are referred to as inverse problems. The model introduced is an adaptation of the Geometric Algebra Transformer called Wireless (Wi-GATr), used for simulating wireless propagation in a 3D environment. The authors also cast this problem as a generative modeling task of the joint distribution p(F,t,r,h) (from which the above three tasks can be accomplished) using Denoising Diffusion Probabilistic Models and Wi-GATr.\n\nSTRENGTHS:\nThe main contributions of this work are as follows:\n1. Introducing a new tokenization method for geometric wireless communication environments and transmitter and receiver characteristics.\n1. Integrating diffusion-based models with Wi-GATr to model the wireless environment as a generative model, thereby determining the joint distribution of F, t, r, and h.\n1. Providing new, larger datasets for the wireless environment modeling to the research community.\n\nWEAKNESSES:\nAs per my understanding, the main weaknesses of the work are: \n1. The novelty of the work lies in tokenizing various geometric objects encountered in the wireless communication scene. However, the same tokenization is used in the vanilla transformer, making it unclear if the new tokenization provides any benefit.\n1. As the authors point out, the channel is modeled only in terms of time-averaged non-coherent received power, missing crucial information such as time and direction of arrival, which are essential for modeling wireless environments.\n1. While the proposed solutions seem general, most results are presented for the single antenna case. Additionally, the dataset includes only transmitting sinusoidal waveforms, which is limiting as it does not cover larger bandwidths. The wave propagation depends on frequency, and non-linearities can occur with wider bandwidths.\n\nQUESTIONS:\nIn addition to the weaknesses listed above, I have the following questions:\n\n1. The authors should clarify how many transmit antennas were used in receiver localization and sensing, and why they believe the results are promising despite the limitations of using a larger number of antennas and ignoring time and angle of arrival information.\n1. The authors should also clarify the challenges involved in their novel tokenization and in developing diffusion-based models with the Wi-GATr model.\n\nLIMITATIONS:\nThe authors mention limitations in the Discussion section (Section 6).", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-14"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents the Wireless Geometric Algebra Transformer (Wi-GATr), a new architecture for simulating wireless signal propagation in 3D environments. This model utilizes geometric algebra to handle the geometric complexities of wireless scenes and ensures E(3) equivariance to respect the symmetries of the physical problem. The authors introduce two datasets, Wi3R and WiPTR, to benchmark their model. Wi-GATr outperforms existing baselines in terms of prediction fidelity and data efficiency, and it can solve both forward (signal prediction) and inverse (receiver localization and geometry reconstruction) problems in wireless communication.\n\nSTRENGTHS:\n1. The integration of geometric algebra for handling complex 3D geometric data and ensuring E(3) equivariance is a novel and effective approach. This addresses the core challenge of accurately modeling wireless signal propagation in diverse environments.\n\n2. The paper provides a thorough evaluation of Wi-GATr against multiple baselines across various tasks, demonstrating superior performance in signal prediction, receiver localization, and geometry reconstruction.\n\n3. Wi-GATr shows remarkable data efficiency, achieving high-fidelity predictions with significantly less training data compared to other models. This is particularly beneficial for scenarios where obtaining large amounts of training data is challenging.\n\n4. The model's ability to handle both forward (predictive modeling) and inverse (localization and reconstruction) problems showcases its versatility and potential for a wide range of applications in wireless communication.\n\nWEAKNESSES:\n1. Limited Real-World Testing: While the model performs well on the introduced datasets, its application in real-world, dynamic environments remains underexplored. Additional experiments in more varied and complex real-world scenarios, such as urban or industrial settings, would strengthen the paper.\n\n2. Scalability and Computational Load: The paper could provide more detailed insights into the computational requirements and scalability of Wi-GATr. Understanding the model's performance with larger datasets and more complex environments would be valuable for practical deployment.\n\n3. Generalizability Across Frequencies: The model is tested at a specific frequency (3.5 GHz). Evaluating its performance across different frequencies and under various signal conditions would provide a more comprehensive understanding of its robustness and generalizability.\n\n\n4. Detailed Case Studies: While the paper presents strong experimental results, including more detailed case studies or examples of practical applications, such as network design or optimization in real-world environments, would illustrate the model's impact and practical benefits.\n\nQUESTIONS:\n1. The paper focuses on indoor scenes for evaluation. Have you considered testing Wi-GATr on datasets that include outdoor or mixed environments to assess its robustness and generalizability?\n\n2. The paper mentions solving inverse problems like receiver localization and geometry reconstruction. Can you provide more detailed examples or case studies where these capabilities significantly outperform traditional methods?\n\n3. Given the computational complexity, how does Wi-GATr perform in real-time applications? Can it be integrated into real-time wireless network management systems, and if so, what are the challenges?\n\n4. The datasets introduced are focused on indoor environments. Have you considered incorporating additional data sources, such as satellite imagery or LIDAR data, to enhance the model's accuracy and applicability in different environments?\n\n5. How does the model adapt to changes in the environment, such as new constructions or changes in material properties? Is there a mechanism for updating the model dynamically to reflect these changes?\n\nLIMITATIONS:\nThe authors briefly discussed the limitation in Sec. 6.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-14"}