NEAR: Neural Electromagnetic Array Response
Authors: Yinyan Bu
Jiajie Yu
Kai Zheng
Xinyu Zhang
Piya Pal
Venue: ICML 2025 poster (2025)
Keywords: Implicit Neural Representation
Untrained Neural Network
Radar Sensing
Super-resolution
Physics-informed Neural Network
Primary area: applications->everything_else

Abstract: We address the challenge of achieving angular super-resolution in multi-antenna radar systems that are widely used for localization, navigation, and automotive perception. A multi-antenna radar achieves very high resolution by computationally creating a large virtual sensing system using very few physical antennas. However, practical constraints imposed by hardware, noise, and a limited number of antennas can impede its performance. Conventional supervised learning models that rely on extensive pre-training with large datasets, often exhibit poor generalization in unseen environments. To overcome these limitations, we propose NEAR, an untrained implicit neural representation (INR) framework that predicts radar responses at unseen locations from sparse measurements, by leveraging latent harmonic structures inherent in radar wave propagation. We establish new theoretical results linking antenna array response to expressive power of INR architectures, and develop a novel physics-informed and latent geometry-aware regularizer. Our approach integrates classical signal representation with modern implicit neural learning, enabling high-resolution radar sensing that is both interpretable and generalizable. Extensive simulations and real-world experiments using radar platforms demonstrate NEAR's effectiveness and its ability to adapt to unseen environments.

NEAR: Neural Electromagnetic Array Response
Yinyan Bu'
Jiajie Yu'
Kai Zheng'
Xinyu Zhang'
Piya Pal
Abstract
due to its robustness to adverse weather conditions (e.g. fog, snow, rain). Siniiile-nnut-Muulill--------
We address the challenge of achieving angular super-resolution in multi-antenna radar systems that are widely used for localization, navigation, and automotive perception. A multi-antenna radar achieves very high resolution by computationally creating a large virtual sensing system using very few physical antennas. However, practical con- straints imposed by hardware, noise, and limited number of antennas can impede its performance. Conventional supervised learning models that rely extensive pre-training with large datasets, of- ten exhibit poor generalization in unseen environ- ments. To overcome these limitations, we pro- pose NEAR, an untrained implicit neural repre- sentation (INR) framework that predicts radar re- sponses at unseen locations from sparse measure- ments, by leveraging latent harmonic structures inherent in radar wave propagation. We establish new theoretical results linking antenna array re- sponse to expressive power of INR architectures, and develop a novel geometry-awand regularizer. Our approach inte- grates classical signal representation with modern implicit neural learning, enabling high-resolution radar sensing that is both interpretable and gen- eralizable. Extensive simulations and real-world experiments using radar platforms demonstrate NEAR's 's effectiveness and its ability to adapt to unseen environments.
(MIMO) radar (Li & Stoica, 2008) employs an array of transmit (Tx) antennas which generate signals that are reflected by tar- gets of interest, and received at a receiving (Rx) antenna array. The distance and velocity of these targets are char- acterized by using the radar's Sange-Dopple (RD) map, which computed by applying Discrete-Tinee Fast Fourier Transform (FFT) on digitized receiver signals after Analog- to-Digital Conversion (ADC) (Sun et al., 2020), Direction- of Afrivll (DOA) estimation is then performed exclusively on peaks that pass the constant false alarm rate (CFAR) de- tector (Scharf & Demeure, 1991) to determine the angular orientation of objects. The angular resolution of MIMO radar, which reveals how well it can identify two more closely spaced sources, is fundamentally constrained by the number and configuration of the antenna array. For instance, device equipped with eight uniformly filled antenna ar- rays achieves at most an angular resolution of about 15" (Instruments, 2017). Thus, it is important to develop innova- tive technologies to enhance the angular resolution of radar sensing, without incurring substantial hardware costs.
and latent
For MIMO radar, range and Doppler resolution be im- proved by adjusting signal bandwidth and frame time, which correspond to the frequency range and duration of signal pulses, respectively (Li et al., 2023). However, angular resolution is strictly dependent on the radar hardware speci- fications, and cannot be improved through parameter adjust- ments. Achieving higher angular resolution in both azimuth and elevation requires large aperture in both horizontal and vertical directions, which, for uniformly filled arrays, neces- sitates a significant number of antennas, resulting in high hardware costs. An efficient alternative approach, which is becoming increasingly relevant for next-generation sensing (such as automotive radars) is to use sparse arrays (Pal & Vaidyanathan, 2010; Qin et al., 2015; Sarangi et al., 2023). Sparse arrays deploy reduced number of transmit and re- ceive antennas in order to achieve the same aperture standard uniform array (with quadratically larger number of sensors), which necessitates larger inter-element spacings. They are designed so that their virtual (sum or difference) co-arrays are dense uniform arrays filling the available aper- ture. This property can be utilized in several ways, such as localization of more sources than sensors, and achieving
1. Introduction
In addition Lidar and RGB-cameras, Radar has emerged as a crucial sensing modality for advanced sensing tasks such as driver assistance systems (ADAS) and autonomous vehicles (Bijelic et al., 2020; Caesar et al., 2020), especially
'Department of Electrical and Computer Engineering, Uni- versity of California San Diego (UCSD), La Jolla, United States. Correspondence to: Yinyan Bu cyIbu@ucsd.edd Jiajie Yu fryyss@@ussd.cdus.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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NEAR: Neural Electromagnetic Array Response
very high resolution with sufficient temporal measurements (Cheng al., 2014; Liu & Vaidyanathan, 2017; Wang & Ne- horai, 2017: Qiao & Pal, 2019). Recently, it has been shown that sparse arrays are also near-optimal subspace codes, highlighting their novel connection to channel coding (Mah- davifar et al., 2024). However, naive processing of sparse arra outputs using traditional or ad-hoc methods, can suffer from high sidelobes and degrade DOA estimation accuracy (Sun & Zhang, 2021). Thus, achieving super-resolution angular resolution with low hardware cost and irregular sampling geometries, remains continuing challenge.
millimeter wavelengths necessitate consideration of phase information. Unlike visible light, where the phase is often neglected, the phase in radar signals is crucial for capturing fine-grained details of wave propagation, such as target lo- cations. Ignoring phase would result in a significant loss of critical information. Finally, despite the increasing adoption of INRs in various domains, the theoretical understanding of their properties and their implications in specific appli- cations remain limited. Key aspects, such as the behavior of deep layers in these networks and role of positional en- coding (PE) in representing complex signals, are not yet well-understood.
In this work, we introduce a machine learning framework that tackles the challenge of angular super-resolution at low hardware cost using sparse measurements that employ only a few antennas. Our goal is to predict complex-valued re- sponses at any desired location (that can potentially be used for DOA estimation) within the 2D virtual antenna array domain using only sparse set of responses. One straight- forward approach to accomplish this is to train a machine learning model that maps a spatial location to the corre- sponding antenna response. However, this approach may fail to incorporate the underlying physies of wave signal propagation, and thus still require very dense measurements to achieve reasonable performance. As one of the important breakthroughs in computer vision, NeRF (Mildenhall et al., 2021) has achieved remarkable success in 3D reconstruc- tion and view synthesis tasks by learning a scene's 's radiance field from a set of input images and generating photoreal- istic renderings from novel viewpoints. At its core, NeRF utilizes implicit neural representations (INRs) (Sitzmann et al., 2020; Tancik et al., 2020) to parameterize the radi- ance field as continuous function, modeled by multilayer perceptron (MLP) that maps 3D spatial coordinates to RGB color and volume density. Leveraging volume-rendering techniques to synthesize images, NeRF incorporates the underlying physics of light propagation.
To address these challenges, our work makes the following contributions:
We propose NEAR, the first electromagnetic array re- sponse prediction framework that implicitly integrates signal propagation characteristics into INRs. Our ap- proach enables prediction of array response at unseen receiver locations, facilitating super-resolution angular estimation with low hardware requirements.
We provide tight characterization of the class of func- tions that INR's can represent with certain choices of positional encoding and activation functions. Our results improve upon existing theoretical analysis of INRs.
We evaluate NEAR through both simulation studies and real-world experiments, achieving superior perfor- mance in antenna array response prediction and other downstream tasks such as super-resolution angular esti- mation compared to existing model-based and machine learning methods.
Overall, we believe our findings contribute to advancing research in INRs and their unique applications in radar sens- ing. Our work also marks the first step towards leveraging INRs for predicting unseen antenna responses in radar sens- ing, paving the way for new opportunities to enhance the performance of future sensing and localization systems.
Inspired by recent advances in NeRF and INRs, we pro- pose Neural Electromagnetic Array Response (NEAR), framework utilizing INR, that maps 2D spatial coordinates to complex-valued antenna response at those locations. Sev- eral distinguishing features differentiate our task from tradi- tional NeRF applications, particularly due to fundamental differences between the ways in which visible light and radar signals are processed (Zhao et al., 2023). Firstly, we have access to limited number of complex-valued mea- surements, proportional to the number of deployed anten- nas. Apparently, this conveys significantly less information compared to an image comprising thousands of pixels. As a consequence, training a model using off-the-shelf INR- based algorithms with a limited number of antennas can fail to reliably predict unseen array response at arbitrary spatial locations. Secondly, while optical NeRF frameworks rely solely on light intensity (amplitude), radar signals at
2. Preliminaries
In this section, we provide background knowledge on MIMO radar, virtual array, and implicit neural represen- tation.
MIMO Radar. We consider targets in three-dimensional Euclidean space, represented by spherical coordi- nates as depicted in Figure 1. We consider a L- shaped MIMO radar system (consistent with our hardware) with Nr physical Tx antennas located at ((TT1,0,0) O)..... (TT.N.. O, 0)} and Nr physical Rx antennas located at {0, yn,, O),.". "...,,,, (0,yn,N,, O)). The
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NEAR: Neural Electromagnetic Array Response
2
origin of the coordinate system. The array response at coordinate (f1, (which may be virtual arry element location, or the location of a missing sensor), due to signals impinging from the K targets in absence of noise can be expressed as:
k-th target
TX5
TX
V
0
Σ ane 2 4 (ri siu de cos tra sin du sin 0n).
Mer 12 =
(2)
Tx Rx
Figure 1. Left: Illustration of the target position in Spherical coor- dinate system. Right: One sub-Nyquist sampling pattern with X indicating missing virtual element response.
where 2& is the unknown complex-valued reflection coef- ficient of k-th target. Various algorithms, such as beam- forming (Van Trees, 2002) and subspace-based methods (Schmidt, 1986; Roy & Kailath, 1989), can be applied to k== and {
estimate (0k JK
Tx antennas emit a set of N, orthogonal waveforms, which reflected by K targets and their superposition is received at the Rx antenna array. Each Rx antenna of a MIMO radar is equipped with a bank of N, matched filters, each matched to one of the Nt orthogonal waveforms. This yields a total of N, measurements at the output of N,N, matched filters, which can be used to perform different spatial sensing tasks such as localization, beamforming and so forth (Li & Stoica, 2008).
Implicit Neural Representations (INR). INRs are used to model continuous function 9 : Rddi - Rdout using neural network fe P.... + Rd parameterized by weights O which map input coordinates r E Rd to signal values g(r) E IR dow A significant challenge for INRs is to accurately reconstruct tigh-frequency details, which is needed for radar super-resolution. Classical neural network architectures are known to exhibit strong spectral bias (Ra- haman et al., 2019) towards lower frequencies. Recently, Tancik et al. (2020); Sitzmann et al. (2020) have proposed ar- chitectural solutions to overcome this spectral bias allowing faster convergence and higher accuracy of INRs.
Virtual Array. One of the key features of a MIMO radar is that by using only N,- + N,, physical Tx and Rx antennas, it can create the effect of a much larger antenna array with N,N, virtual sensing elements at the output of the N,,, matched filters. Consider a far-field point target at direction (0, Φ). It can be shown that the noiseless array response at the m-th matched filter output in the n-th receiving antenna can be expressed as
Following the formulation of (Yüce et al., 2022), most INR architectures can be decomposed into a mapping function 7 : R n + R followed by MLP, with weights ((!) R FexF bias ((()= E Re, and activation function p(() R R applied element-wise at each layer & = I,..., L-1.
xe X (Z sin the ros Ox+ + HRR,, sin dhy sin 0z),
(1)
Suppose (()) represents the post activation output at layer l, The INR input-output relationship is given by
where x the unknown amplitude of the signal reflected by the target and A is the wavelength which the narrowband radar operates. Therefore, the array response in (1) is the same as that of (fictitious) two-dimensional receiving array with N, antenna elements located at
z (0)) (()),
0 = (0) (wr(2) (1-1) (w())(t))+bb') l I.
1,
fe(r) = w (4) (1-11) + +b(1)
{ (Zr, m. 0, 0) + (0, Rn,, 0), 1 S m S S
(3)
Tancik et al. (2020) introduced Fourier feature networks (FFNs), which Fourier-based positional encoding 7(r) = sin + Φ). with parameters Ω E IT'xD and @ E TT followed by an MLP with p(0) = ReLU. They demonstrated that by initializing Q ~ ~N(00 o²) with ran- dom Fourier features, and choosing large values of a, one can drive the network response towards realizing higher fre- quencies. SIREN (Sitzmann al., 2020) can also mitigate spectral bias in similar way by choosing different (sinu- soidal) activation function and rescaling certain parameters at initialization.
This two-dimensional antenna array with N, elements is known as the virtual array (Chen & Vaidyanathan, 2008). Figure shows a physical Tx-Rx antenna pair and the asso- ciated two-dimensional virtual array. Notice that depending on the geometry of the Tx-Rx pair, the 2D virtual array need not comprise of elements consecutive locations over uniform grid, and there can be missing elements (or holes) the virtual array, as indicated in Figure 1.
Consider K targets in the far field, with azimuth angle 0 and elevation angle Φk. S k S K. Without loss of gen- erality, let the reference virtual antenna be located at the
In our setting, the 2D virtual array is located in the ryyplane.
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NEAR: Neural Electromagnetic Array Response
3. Related Work
planar wave propagation, we propose novel framework for reconstructing 2D MIMO virtual antenna array response fields using implicit neural representations. In contrast to aforementioned approaches that employ ray tracing for EM waves, our method employs a straightforward yet effective regularization strategy specifically designed to leverage the spectral sparsity of antenna array measurements and the characteristics of planar wave propagation.
Hallucinated Antenna Interpolatin//Ettapplaatton To mitigate the high sidelobes introduced by the sparse arrays and enhance the SNR of antenna array response, Sun & Zhang (2021) propose to recover missing elements or holes in the sparse arrays by completing a low-rank (Block) Han- kel matrix (Chen & Chi, 2013). However, the nuclear norm minimization that they employed often suffers from subop- timal recovery performance (Lu et al., 2015) and exhibits sensitivity to the sampling pattern and noise(Bu et al., 2025; Sarangi et al., 2022). Furthermore, their approach is limited in its applicabiliity to cases involving non-integer-me non-integer-multille sampling. To enhance the azimuth resolution of MIMO radar, Li et al. (2023) proposed malog-to-Digine super- resolution model (ADC-SR) that predicts or hallucinates additional radar signals using signals from only a few re- ceivers, essentially implementing a uniformly filled array extrapolation framework. However, their approach is re- stricted to ID MIMO configurations and relies on a large training dataset, potentially limiting its generalization pabiliity. In contrast, our method implicitly leverages the underlying physics of signal propagation and requires only single-snapshot sparse measurements, eliminating the de- pendence on extensive training data.
4. Neural Electromagnetic Array Response
In this section, we present the design of NEAR. Section 4.1 outlines our problem formulation, followed by theoretical results on the expressive power of INRs and their connection to Fourier series in Section 4.2 elucidating why and how array response function in (2) can be effectively learned. Section 4.3 details our novel implicit regularization strategy that integrates signal propagation model while harnessing harmonic structure. Finally, we describe the response pre- diction process of NEAR in Section 4.4.
4.1. Problem Formulation
We consider a environment where all objects are located in the far-field relative to the radar antenna array. In this setup, the propagation of wireless signals can be modeled as planar waves that are emitted from the Tx array, reflected by objects and finally captured by the Rx array. Let S, and S, represent the sparse sets of physical Tx and Rx antennas, respectively. The coordinate set of available virtual antennas is given by {(r,, (()=:Ty)) "y, iyy as explained in Section 2. We define the domain of antenna array response field as D = { y) 0 S x S max(S,,, 0 S y S max(S,)). We represent the continuous complex- valued response field as a function y : D - C, where the input is 2D coordinate r [r1 within the domain D, and the output is complex- valued response yra,r that adheres to the signal model in (2). To approximate this continuous 2D response field, we employ an INR model that maps the input 2D coordinates to a vector in IR2,, where the two components correspond to the real and imaginary parts of the complex- valued response, respectively. Specifically, the model is defined as fe R22 - IR2 and the parameters Q are optimized map each input 2D coordinate its corresponding response. The goal of this puper is to learn the function Se solely from physical antenna measurements (without using any offline training data), by exploiting the harmonic structure of array response in (2). Once the response function is learned, it enables the prediction of array responses at any unseen locations within D, facilitating downstream tasks such as angle estimation and localization.
Expressive power of INRs. INRs have emerged as a ver- satile set of neural architectures for representing and pro- cessing signals on low-dimensional spaces. Understanding the function class that an INR architecture can represent is essential for their application to practical problems. Recog- nizing that polynomials of sinusoids generate linear combi- nations of integer harmonies of said sinusoids, Yüce et al. (2022) analyzed the expressivity of FFN, SIREN and re- lated architectures in (Fathony al., 2020). Subsequently Roddenberry al. (2023) developed a broader theoretical understanding of INR architectures with a wider class of activation functions and provided a superset to which the integer harmonic frequencies characterizing INR functions belong. While their supersel results provide valuable theo- retical insights, our work refines this analysis and derives the exact set (and not superset) of integer harmonics which describe the expressive power of INRs, delivering tight characterization.
Neural Radio-Frequeet Field Reconstruction. Building on the fact that light is a kind of electro magnetic (EM) wave, Zhao et al. (2023) and (Lu et al.) proposed two NeRe-based frameworks, named 2 and NeWRF, re- spectively, for wireless channel modeling based on implicit wireless radiation field reconstruction. Chen et al. (2024) further developed hybrid model that integrates NeRF-like object representation with physics- ahysics-based ray tracing models. These models enable accurate characterization and predic- tion of channel properties. Building on the principles of
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NEAR: Neural Electromagnetic Array Response
4.2. Representational Ability of INRs
This shows that our desired array response indeed belongs to the class of functions representable by INRs. Although the resulting INR architecture appears deceptively simple, it is to be noted that the positional encoding requires the ground- truth DOA and amplitude of each target, which are never available in practice. Hence it is important to investigate the class of functions that INR can approximate using a given mapping 7(r), such as the type of fixed sinusoid positional encodings employed in NeRF (Mildenhall et al., 2021):
While substantial empirical evidence demonstrates the ef- fectiveness of INRs in representing scenes and various vi- sual signals, the theoretical underpinnings of their ability to approximate continuous functions remain underexplored. In this subsection, we establish that many contemporary INRs inherently build upon similar underlying structures and shared fundamental principles, enabling them to repre- sent a certain class of signals.
To rigorously analyze the expressive power of INRs, we follow the formulation outlined in (3). Following (Yüce et al., 2022; Roddenberry et al., 2023; Mehmeti-Gööp Göpel et al., 2020), we restrict our investigation to polynomial activation functions of the form p(x)= = Σg= AgT7, a widely adopted approach in the study of the expressive capacity of INRs.
sin ((rr cos(Srr)
(()
with
(5)
n =
0
2
π
0
2
π
0
0
2 π
n
2 -
π
Theorem 4.1. Let fe RDD + I be an INR given by
(3), where the activation uccivationfuuction for layers l >> is given by (z) -Ego 0gz2.. Let S2t [w1...., WT] E RTXD represent the frequency matrix and ΦT E RT the phase vector used to map the input coordinate r E RD into the feature space via the mapping 7()) sin(Styr +
Consider a non-periodic function g Rdi + Rdow defined over bounded domain 'D (e.g. the height and width of a image, the aperture of 2D-MIMO array). We can define its periodic extension g R - Rddest with period p as follows (Benbarka et al., 2022):
②T). The resulting architecture is capable of representing functions of the form:
g(x + = g(x) Va E D, Vn E 2hn,
(6)
fo(r)= = Σ
casin sin (() do),
(4)
#EST
where o denotes the Hadamard product. By normalizing the input domain to its respective bounds, we assume a period of for each variable, i.e., within the range [-1, 1). The Fourier series expansion for periodic extension ğ R +
where
ST [$1:82..... 831
8t Z, Σ stl s 94-1
R of period 2 is given by (Oppenheim et al., 2010):
Theorem 4.1 gives an exact characterization of the set Sy of all possible integer harmonics of the feature mapping (r).
8 Σ 372,
Amn cos( cos(m((m + Bm, sin ( (mar+ +ny)..
In contrast, Yüce et al. (2022); Roddenberry et al. (2023) only provide a superset to which ST belongs.
It can be shown that if the frequency matrix Ω R 2T of the INR described in Theorem 4.1 is chosen according to (5), then as the number (L) of layers of the MLP/INR increases, fe approximates to certain period-2 functions of the form (7). See Appendix A.7 for more details.
Remark 4.2. Let yn and y1 denote the real and imaginary parts of the response field function (2), respectively. The function yR (r) can be equivalently represented as (4) by applying Theorem 4.1 with the following parameterization:
2T isin du cos 02 sin the sin 0KlISKSK E 4x2
S2 x
4.3. Physics-Informet Implicit Regularization
- = Re(#x), C2k = Im(xs), d2x-1 2'
To model the antenna array response field, we discretize the domain of interest into finite set of points in the 2D plane. Let [o, U1] x [o. Uz] represent the antenna array response field with bounded domain positioned the x - y plane, consider general case of uniform sampling grid of dimensions Mi x M2, with spacing d, - 3 and d2 U, < } Supposing an array snapshot containing M K targets with azimuth angle 0g and elevation angle Φa (A: = 1,... ,K) and leveraging planar wave propagation (2), the (m1, m2) th element of the response with respect to
= ST =ex Φ2k = 0, sy e E RK ST
Under this parameterization,
K
UR( (r) = C sin (12 r) + 622
+ O2k sin ( 8k, r) + 62x).
similar representation also holds for yn-
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NEAR: Neural Electromagnetic Array Response
K targets in the absence of noise can be written as
and its latent representation. To further justify the effec- tiveness, we establish the algebraic properties of the block Hankel matrix corresponding to the ground truth response Y using its harmonic structure.
mm
Iye $ (1m11 -1]d1 sin dh cos (m1g-1)24 du sin 0x)
Theorem 4.5, Consider the ground truth response mairix Y as defined in (8), For K I min( M 1,1 A there exists vectors m E CK and m2 E CK such that the last column of Mr (Y) can be uniquely represented in terms of the first K columns of HM,,My MM-Kk kYY using the corresponding coefficient vectors mî. i.e.
(8)
for 1 I mi S Mj and 1 S m2 S M2. Notably, when d1 =
d2 = 2.. the sampling pattern aligns with the Nyquist sam- pling. Let Y = mmylim S S M2 E CM x M2 be the ground truth response matrix with entries as the an- tenna array response defined in (8).
Definition 4.3. Given Y
(ymimm?) E MMxMa for
| | (M),M = O,
1 1 m2 I M1,1 I m2 I M2. a Block Hankel matrix of Y, Y, N1 V M1, < N₂ S M2 can be constructed as:
where S = [I 0x] E R(K+I)*K, and = [Ok 1] E R((+1)x1 Similarly, an equivalent property holds for Mx(,M given by
NNN.NN (Y)
HN,(y1)
(Y2
HN, (M (M--222
HNs
(Vs)
HNN
(NN1+a)
HN,(VMM
| - HMM.M. = O.
where HNo (ym), S m S M1 is defined as:
Building upon the planar wave signal propagation model, Theorem 4.5 establishes connection between rank property and least squares by leveraging harmonic structure of Block Hankel matrix. However, the global optimizer m and m9 are intrinsically dependent on parameters {(Ok: thx) )== (see Lemma B.3 in Appendix), which are part of the radar sensing task and not known in advance. To address this, as detailed in the next subsection, we integrate the least squares term into the loss function and paramcterize the unknown coefficients, enabling them to be learned adaptively.
ym,,
ym.2
Um, M2 N1+1
3m..2
3m.3
ym. 22+2
HN, (yn) =
:
Nm
Vm,,N2+1
Dm, M2
Remark 4.4. Definition 4.3 defines the block Hankel ma- trix constructed along the row direction. Similarly, a block Hankel matrix can also be constructed along the column direction, denoted as N,Ng (Y) (see definition in Ap- pendix I). Moreover, the rank property remains consistent for block Hankel matrices constructed along both the row and column directions.
4.4. Optimizing NEAR
In practice. the model predicts the real and imaginary parts of the response signal (R J. 3 (ymira H), instead of amplitude and phase ( (A(Be,.z)) ((Dr,r))) .))).. This is because phase is modulo against 2T, which is not differentiable. We perform uniform inference for fe(-) over the bounded domain, using a pre-chosen grid of M, M2 data points. Denote the predieted response the (m1, m2) th element
We emphasize that the number of targets in the same range- Doppler bin that need angle estimation is small since the targets are first separated in range-Doppler domain (Sun et al., 2020). In other words, the targets are sparsely present in the angular domain and, as a result, HN,,N, (Y) (Y) and H. N, (Y) exhibit low rank, with rank equal to K for appropriate choice of N1, Ni, N2 (see Lemma B.1 in Appendix). To characterize such a property, numerous convex/non-conves, rank surrogate functions have been ex- plored in the literature, which include but are not limited to nuclear norm (Candes& & Recht, 2012), schatten-p norm (Mohan& & Fazel, 2012) and truncated nuclear norm (Hu et al., 2012). However, all of these surrogate functions are explicit and requires singular value decomposition (SVD), which can be not only computational expensive but also sub-optimal. In this work, we propose a novel implicit regu- larizer that exploits the structure of the block Hankel matrix
as fm , = fe ((m) - 1) H (m2 1) M and let Y represent the predicted response matrix. All the other notations remain consistent with those introduced in Section 4.3, with an additional to distinguish predicted quantities. Consider two sparse sampling pattern Sg, where the observed noisy response Bra.r, is only available at locations [Tz, Vr E Sn, Vr, E Sy. The overall loss function is defined as
L(O,, (1, m2) La + XLr,
(9)
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NEAR: Neural Electromagnetic Array Response
with
Y Square Error (NRMSE), defined as N n= Ir
Σ
-Y.YY.ll. Y.IlI.
La = Σ Σ
[i) - T2,ll2,
where Y, and Y,, denote the predicted array response and the (noiseless) ground truth full virtual array response at n-th realization, respectively, with || || representing the Frobenius norm. Our method consistently outperforms all baselines across different evaluation settings, demonstrating superior generalization in response recovery tasks. Notably, NEAR achieves even lower enror than the 20 x 20 full vir- tual array benchmark across different SNR levels with a fixed sampling number (Table 1). This can be attributed to the inherent denoising ability of our regularizer that exploits low-dimensional structure of array response and provides cleaner estimate of at a given coordinate, compared to actual noisy measurement at the same location. The poor performance of SIREN and NEAR w/o R across all settings suggests that these models struggle to learn the appropriate continuous response function in the absence of physics- informed regularization. This highlights the importance of incorporating prior knowledge into implicit neural repre- sentations for structured signal recovery. more detailed analysis is provided in the Ablation Study.
JESy iES,
C. -I(HHM,,, (Y)mm) HM,, M2 K (Y)b
+ +I(HMss,M (Y)Sm²- HMM.M, -K (Y)bl,,,
Y = -feeii.))ssssM
(10)
Specifically, La represents data fitting term, which quanti- fies the gap between the predicted and acquired responses at observed locations; f,r corresponds to regularization term, as elaborated in Section 4.3 and Appendix B. Parameters optimized by minimizing the total loss function
θ mf. m = arg min La + AD, mm
(11)
Using the optimal parameters Oº, the predicted array re- sponse can be computed by Dt.5 = (i. Vi.j E D.
5. Experiments
We evaluate the performance of NEAR on both simulated (Section 5.1) and real-world (Section 5.2) tasks. All ex- periments are run on a laptop with CPU AMD Ryzen 9 5900 HS with Radeon Graphics and GPU NVIDIA GeForce RTX 3050 Ti Laptop. See Appendix C for more ex- perimental results. The codes are available at: https: //github. com/ LannJmmmyyuul /NEAR.
Table I. Averaged NRMSE of response at different SNR level. 8x8 sampling is employed for NEAR, EMaC,, NEAR w/O and SIREN
METHOD
dB
20 dB
30 dB
BENCHMARK
0.2608
0.0825
0.0261
NEAR
0.2248
0.0495
0.0189
Baselines and Benchmark. We compare NEAR against four representative baselines: Enhanced Matrix Completion (EMaC) (Chen& & Chi, 2013), SIREN (Sitzmann et al., 2020), 2 (Zhao et al., 2023), and NEAR without Regulariza- tion (NEAR w/o R), more implementation details and analy- sis of these baseline methods can be found in Appendix C.2. For fair comparison, we adopt the hyperparameters recom- mended by the original authors. Additionally, we include 20 x 20 full virtual array response (noisy) as a benchmark reference.
EMAC
0,3537
0.1889
0.0921
NEAR W/O R
1.0663
1.0504
1.0485
SIREN
1.0512
1.0277
1.0244
Table 2. Averaged NRMSE of response for different sampling num- ber 20 dB with targets.
METHOD
6x6
8x8
10x10
NEAR
0.1884
0.0495
0.0362
EMAC
0.5306
0.1889
0.0724
NEAR w/o R
1.0689
1.0504
1.0030
NEAR Architecture. In both simulated and real-world settings, we employ the architecture described in Equa- tion (3), with depth of L = ReLU activation function p(-) = ReLU(-), and positional encoding 7(r) following NeRF's formulation in Equation (5) with T = 10. The hidden layer dimension is set to 256. Additional imple- mentation details and hyperparameter configurations are provided in Appendix C.I.
SIREN
1.0656
1.0277
0.9860
Table 3. Averaged NRMSE of response for differrnt number of
targets at 20 dB. 8 8x8 sampling is employed for all methods.
METHOD
1 TARGET 2 TARGETS
TARGETS
NEAR EMAC
0.0382
0.0461
0.0860
0.1454
0.1941
0.2503
NEAR w/o R
1.0399
1.0501
1.0308
5.1. Simulation Tasks
SIREN
1.0077
1.0262
1.0543
Response Recovery. We evaluate the response recovery performance of NEAR against baseline methods and the full virtual array benchmark, as summarized in Tables 3. The evaluation metric the Normalized Root Mean
Angular Resolution. The resolution probability (defined in Appendix C.1.2) of NEAR compared to baselines and the full virtual array benchmark is illustrated in Figure 2.
Our method consistently achieves the highest resolution
7

---

NEAR: Neural Electromagnetic Array Response
probability among baselines and closely follows the full benchmark. While EMaC achieves comparable resolution probability for larger angle separations, its performance degrades significantly as the angle separation decreases. This is because convex relaxation techniques, such as the Nuclear Norm used in EMaC, impose separation conditions that inherently limit resolution (even in noise-free scenarios) (Dai& & Milenkovie, 2009). In contrast, NEAR demonstrates robust resolution across different separations.
bandwidth is set to be 67-69 GHz. The antenna spacing is roughly half of the wavelength. After collecting the 20 x 20 full array response matrix, we select a subset of data and treat it as a sparse set of measurements. Our proce- dure for active sensing using NEAR is depicted in Figure 4. Additional details on radar data processing (such as analog- to-digital conversion) across range and Doppler cells are included in the Appendix C.3.
Angular resolution. To measure the angular resolution, we put two corner reflectors at the boresight of the radar and gradually reduced the spacing between them. We employ the same signal processing pipeline (e.g., beamforming) and record the angular separation when the two targets merge in the radar angular spectrum. Table 4 shows the measured angular resolution with different setups. As the distance increases, the SNR reduces, and the reflected signal becomes weaker. NEAR achieves similar performance as the full array across all the range settings, confirming its robustness at lower SNR conditions in real-world environments.
Brodado Prtuaitt 6-
Plae SPREN
DON
Anglo Suparation (dog)
Hange Seppler
Figure 2. Angular resolution performance vs. angle separa- tions.
Figure 3. DOA estimation racy vs. different number of tar- gets.
DOA Estimation. The DOA estimation error of NEAR, compared to baselines and the full virtual array bench- mark, is presented in Figure 3. Both NEAR and EMaC achieve similar performance to the benchmark when esti- mating the DOA single target. However, for multiple targets, NEAR significantly outperforms EMaC, demonstrat- ing superior robustness in resolving closely spaced sources. Notably, EMaC's performance deteriorates as the number of targets increases, whereas NEAR maintains lower esti- mation error, showing its capacity to generalize effectively to more complex scenarios.
Table 4. Smallest angular separation that can be resolved across various distance (SNR).
METHOD
2M
3M
4.5M
BENCHMARK
5.7248
6.6769
6.9941 "
NEAR
5.7248
6.6769
6.9941
EMAC
8.5783
9.5273
10.1592
NERF
8.5783
8.5783
8.8948
Target localization. We put several corner reflectors (1 4) in random positions in the field view of the radar and perform radar localization. The location of the reflectors spans I to 4 m in range, -45% to 45 in the azimuth angle, and -20º to 20 in the elevation angle. A total of 70 loca- tion samples are collected and their localization errors are calculated. Table shows that NEAR outperforms the full array baseline, NeRF² and EMaC in terms of mean absolute error. NEAR exhibits denoising effect that improves the localization accuracy compared with the full array baseline. This denoising was achieved by using only an upper bound (and not the exact value) on the number of targets to de- sign the regularizer. The results further confirm NEAR's s capability to work in a complicated real-world environment with multiple reflectors. See more experimental results in Appendix C.4.
Ablation Study. Tables 3 present a comprehensive ab- lation study assessing the impact of the prssics-informed regularizer NEAR. Without this regularizer, implicit neu- ral representations (INRs) merely perform data fitting the observed array responses but fail to capture the inher- ent low-rank structure in the Hankel matrix of the noise- less full virtual array response. This limitation severely affects the model's ability to generalize beyond observed data. The findings confirm that leveraging thysics-informed constraints allows NEAR achieve superior signal recon- struction and DOA estimation accuracy, particularly in chal- lenging multi-target scenarios.
5.2. Real-world Experiments
We further conduct experiments using a commercial MIMO radar platform (IMAGEVK-74) as shown in Figure 5. IMAGEVK-74 employs 20 Tx antennas on a vertical line and 20 Rx antennas on a horizontal line, resulting in a vir- tual array of 20 x 20. IMAGEV- 7-74 transmits a Stepped- Frequency Continuous Wave (SFCW) waveform and the
Computation Time. Table 6 reports the averaged running times: NEAR (our approach) finishes in roughly 9 minutes, whereas EMaC and NeRF require about 20 and 21 minutes, respectively. These results highlight the potential for real- time implementation of our approach with future advances in algorithms and computing hardware.
8

---

NEAR: Neural Electromagnetic Array Response
Input
Ouiuu
Raw radar frames
Range spectrum
Physical Arrey
Full Virtual Armey
Angular spectrum
3D Point Cloud
Response (Low lesolution)
NEAR
Responce (High Resolution)
Range Processing
Select range and arrange frames
Predict Response
Calculate coordinates
MUSIC
Figure 4. Radar active sensing workflow.
Table Computation time comparison.
METHOD
NEAR
NERF
EMAC
AVERAGED TIME COST (s)
550.83
1278.31
1226.15
computational overhead while maintaining accuracy.
Multi-Modal Sensor Fusion. Integrating NEAR with Li- DAR, camera, and RF-based sensing to enhance robustness in complex environmental conditions.
Acknowledgements
Figure 5. Left: 2D MIMO radar platform.
Right: Real-world
We thank the reviewers for their insightful comments. Ad- ditionally, we would like to thank Xingyu Chen for helpful discussions and instructions on implementation and exper- imentation. We also thank Parthasarathi Khirwadkar and Mohamed Hamdy for their valuable comments and feedback.
experimental setup.
Table 5. Localization accuracy for different number of targets (AC)
the environment.
This work is generously supported by the UC San Diego Center for Wireless Communications, by ONR N00014-19- 1-2227, DOE DE-SC0022165, and by NSF under grants CNS-2128588, CNS-2312715, CNSS-240324, and NSF
METHOD
K 1
I. =
IC = 3
IC = 4
BENCHMARK
0.0827
0.0903
0.0965
0.0964
NEAR
0.0744
0.0770
0.0762
0.0718
EMAC
0.1062
0.1158
0.1170
0.1157
N
0.4902
0.5096
0.4346
0.3898
2124929.
Impact Statement
6. Conclusions and Future Work
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences our work, none which we feel must be specifically highlighted here.
We proposed NEAR, the first framework that leverages Im- plicit Neural Representations to model and predict antenna array responses with sparse measurements without training data. By integrating harmonic signal structure and planar wave propagation models, NEAR effectively enables en- hanced angular resolution and robustness radar sensing applications. We believe NEAR represents the first step towards bridging the gap between deep learning-based neu- ral fields and classical eiectromagnetic sensing and signal processing, unlocking radar, wireless and autonomous sensing applications. Future work will focus on addressing the following challenges and improvements:
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NEAR: Neural Electromagnetic Array Response
A. Proof of Theorem 4.1
A.I. Notations and Definitions
Let A, B C I be two sets in n-dimensional Euclidean space. The Minkowski sum or difference A and is denoted by
+ B, B respectively, and defined as:
A+ B = (a
be be
Additionally, we define D(A, B) as the union of the Minkowski sum and difference of A and B, given by:
B).
We define (((9) and B(a) as follows:
T Z
4(10)
((1)
(9)
8T
(4)
$t E
(4)
(4)
S
1=1
8/1)
(9) 81 ...
8T
(0)
86
(0) E
Z
86
(0)
9
Recall that we are interested in analyzing the expressive power of INR architectures, which consist of a mapping function y IRD. -> R (positional encoding) followed by multilayer perceptron (MLP). The MLP is parameterized by weights (#) Fex 1, biases (( R and activation functions p(t) R IR applied elementwise at each layer
l = 1, 1,....I. L - 1. Specifically, denoting the post-activation output of each layer as 2(0), most INR architectures compute:
(0) = 7(r),
20 = /(0) ()0()))))(())))))))))))(())))1 t= I....,1--1 1,
fe(r) ((-)z(b-1) bbL))
where r E R is the input coordinate. As it is plausible to normalize the inputs (r) to their bounds, we assume that each variable's 's period is (normalized to [0, 1)) or (normalized to [-1, 1)).
A.2. Lemma A.1 and Proof
Lemma A.1. Les Ω = [wi....,, wr] E RTxD be frequency matrix, and let S₁ C R denote two sets of weights corresponding to these frequencies. Additionally, let (&e E Th 81 E {1} and (Φx, E R 82 E S2} represent two collections of scalar phases, and (3e, E IR 81 E S1} and (3ez E R 82 E S2) two corresponding sets of scalar coefficients. For any v E R.D, the following holds:
Σ Bo, cos (( r) + (e.))
Σ S3
Be, cos ((22 82, + 0ee))
Σ
By cos ((ST 8', r) + 52))
81 E
o (EE((S.
(12)
where
( ( -S))
(13)
and (6. E RR s' E S2) and (3g R E D(S,, S2)) denote the resulting scalar phases and coefficients, respectively.
12

---

NEAR: Neural Electromagnetic Array Response
Proof.
Σ
Sai cos ((2) r).r) + (on))
Σ A2ESs
Baz cos (1st 82, + 2as)
81 81
Be, Bor cos ((2) 81, r) + do.) cos ((2 82, + -oa.)
S1 $2E
Σ
Be, Bo2 2
(cos ((52 (s1 + 82), + do, + )e,) + cos - 82), + do. -0az))
$1ESi $2ESg
Σ
B& cos
( sT + +
8.)
The last equality combines terms with the same frequency, where By and Φ represent the resultant magnitude and phase, respectively, obtained through phasor addition after grouping. This technique will be used repeatedly in the following subsections to simplify analogous expressions.
A.3. Lemma A.2 and Proof
Lemma A.2. Let = [W]...... WT] E R be a frequency matrix, and let ET {e, R t Z,1 denote the set of canonical basis vectors RT. Define (() as the augmented set of Er, given by S(1) {ee, -e velee e E ET).
Additionally, let (dga) E R | s(1) E ESU)) be a collection of scalar phases, and (Bam) E R ((()) E ( the corresponding set of scalar coefficients. For any r E R and E N, the the following equality holds:
Bacu cos (IT (Q)))))))))))) + datx)) )aat))
Σ (
Σ
Bals cos ((1)s(11 .r) + +
(14)
alv) EStw)
where
s(1) = D(S s()), B(a) c cu
(15)
for some ( E R s(1) E S and (3,00 E R s(9) E st1
Proof. We will use induction to prove our statement. The statement trivially holds for since B(1) = 8(1) c (()) Assume it also holds for >> 1, we first show that S(++1) C Cu(++), Using the induction hypothesis and Lemma A.I, we have:
9+1
Σ (L))
Baa cos (1))
(',,r) .)) + )501))
Σ WesO
Baa cos (oT&(1)) ())) + )o()))
Σ ((1))())))
Bai cos ((2T r) + 22(1))
Σ ((v)
Bow cos
(S2T (()) r) + (occ))
Σ
Ban cos (S2 (()) r) + )o(1))
Σ
Bootn cos ( s(o+1), r) + ('aiitt))
el g(1))
Σ
B Ba cos (12 ((n ((0+1), r) + )
13

---

NEAR: Neural Electromagnetic Array Response
Moreover,
((9+1) = + s(1)
si1) E s(9), s(1) {sa)
8 (0+1) (695 + s(1)
s(1) E 2(1)) 2 E {
T is E & Σ
T
g( = 11
t te
(q)
S 9, E, ES(1)
t=1
c (9+1) [
81
(9+1)
8T
(0+1)
2,
Σ
10+1)
S q + 1 110t2),
t=1
where the last line follows from triangle inequality. Next, we show that given the assumption, we have Bl++1) S s(a+1);
gl++1) = -1(86) ,00)
(1++1) = + s22) s/0) E sl9), s(1) E {
2 8 gl(t)) s(() + s())
8l0) E Bla),
s(1) E {st)]
8(0+1)
11 ... ,8
(4) T
(()) E Z 1 et
T D
166
(9)
= er E S(I)
t=1
T
2 s(0+1) ((0
87
8t
(9+1) E 2,
(g+l)
= q + 1
= B(++1)
1=1
Therefore we have Bl++1) C s(a+1) C ((0+1)) Thus by induction (14) holds 4g E N.
A.4. Lemma A.3 and Proof
Lemma A.3. Let J = [wi....., wr] E RRTxDD be a frequency matrix, and let Ex = (e, E R² E 2,1 T) denote the set of canonical basis vectors of TR.T. Define S(1) as the augmented set of Er,, given by ((1) = (et, -e I ee E F Additionally, let (dam.) E R b E {()) be a collection of scalar phases, and (Bam, E R ( E st) the corresponding set of scalar coefficients. For any r E R., Q EET N and Ogg E ERA (q 1, Q), the following equality holds:
Bach cos (ST 800) + )acx)) boo1) ))²
9
Σ ag
Σ (0ls
Σ
Ba cos (S2
(I
5(0), .r) + @are))
(16)
9=0
((1)
where
((Q)_ = U ((1)
Q
8(Q)
( (0)
T
(
E
Σ
T
(
S Q = (9))
(17)
q=1
for some ( E R ({Q) E ((Q) and (3ac0 E (Q)) { E 8
Proof. According to Lemma A.2, we have:
Q
Σ 0ig
Σ
Ban cos (12
s''), (S2Ts(1), + )ac))
Beco cos ( (()) .T) )0(m))
Q2
9=0
(()))(1)))
U=
a
Baie cos ( (0() ,r) + )g()))
where 5(Q) U s(i), Since Bla) S(a) C ((1) for q N, we then have:
Q
Q c c Uua
U
e B(a)
q=1
g=l
14

---

NEAR: Neural Electromagnetic Array Response
According to the definition of B( and (11)), have:
Q
Q 8=1
U
5(9)
(0)
(1)
(v)
(9)
$1
8t
=
9
g=1
(
(Q)
ST
(Q)
12
(
Z
(Q) (1,2, .01.
1=1
Q
Q
Jul
8/9)
(
(9) E
Σ
(9)
ST
51
S q
q=1
q=1
1=1
8(1)
81
(0)
[9]
8t
(9) Z Σ
8t
(9)
Q
-1
Moreover, ((Q) B(0)
(00)
[
81
(0)
ST
00)
St
(
Z. Σし
1/
(0)
00
since it easy verify that 8(2)
( Therefore, we have
4 = U B( (B( (( ( = 5(9) = 19)
gm1
A.5. Lemma A.4 and Proof
Lemma A.4. Let (wi..... WT] E RTDD be a frequency matrix, and let ET (er E RD tE T) denote the set of canonical basis vectors of R.T Define (()) as the augmented set of Er, given by 5(1) = fer. e le e E Er}.
Additionally, let (dgas E R s(1) E { be a collection of scalar phuses, and (Bgm E (()) E ((i)) the corresponding set of scalar coefficients. For any r E TR. and q.P E N, the following equality holds:
Boin) cos (12 s(1), .r) + "
9
Beleps cos (S2 (()))) (am)rr) + ) ))
Σ aiv) (i)
Σ
(18)
for some (3gram) glap) E u(m) and (6gcom s(ap) E ulm).
Proof. Again will use induction prove the statement. The statement trivially holds for q Assume it also holds for q > 1, then
9+1
Σ aip) Eld(n)
Bein cos (I2
s(1), (
r) + decn))
Σ aip) El/(n)
Bewn cos (sn s()) (())s r) + (ben))
Σ ((l)
Born cos (STT r) + )o(m))
Σ
Batees cos (12 (S2 s(op) r) + (slun)
Σ Edelo)
Bawn cos (122 s(D) r) )aww))
Σ
By cos (12T s' + to.)
8'ED(U(O)), ((()))
15

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NEAR: Neural Electromagnetic Array Response
where the second equality holds by assumption, and the last equality holds by Lemma A.I. Moreover, have
D(((9)) , = 88 = 8 (9p)
+ g(P)
gl9p) E E ,(1)
E Eu(P)
8' b/
(1p) T
+ 8t
[
(1p) ,t
(p)
T EZA A
Σ
(ap)
S qp
T
S 8t
(p)
p
ISIST
ISIST
t=1
t=1
8' BilSSST
8/ EZA ZA
Σ
8 S (q 1)p
((()+1))),
where the third line follows from triangle inequality. Next, we will show that U((++1)p) S D(((Ar)) ((1())) For u = [uI,... Wr] E Ulo++1)) we would like to construct two vectors such that v = (0 E U(w)) w [w..... wrl T E 2(()) and u =-=- w.
Let = Ei=lyy / and min(ū, qp). It is easy to see ū S (g + 1)p by definition. Suppose t' s T is the largest integer that satisfies Et= uel S ü. If = T, it follows that u, w 0, and hence the statement holds. If < T, we can argue that |ur+1| > Et= [el - u', otherwise E [ue] S contradicts the assumption that t' s i' the largest integer that satisfies Σi= uuel= S u. Thus, we can construct such v and w as:
1t 1 (isepw + (1= sgn(u=+))()))) V VENN,,ST
W/ = 1 (1=++1) ((nuee+))(uu++1) u') + 1150-1144 Vt E N+,N,,
where (.) denotes the indicator function and sgn(-) represents the sign function. is easy follow that ē Σt |val and
u 0 + ET= /wp/- Then will verify that ((9)) and (((r), By construction, Ei=lesl = Ei=iluel + v' 8 S min(ū, (ū. qp) S qp, and E/ |w,| = ú = ύ - min(ū, qp) = max(0, û - qp) S p since û S (q 1)p. which completes the construction rule. Therefore, we have ((o 1)9) C D(L(()) (,))), and hence U((o+ 1)p) D((((a)) (46).
A.6. Main Proof of Theorem 4.1
Proof. To begin with the proof of Theorem 4.1, we will prove the following two statements first.
Define g(1) (0) + b(1) as the pre-activation output of the first layer, where 20)) >()) = sin + ΦT). Let
(Dam IR (1)) E } be collection of scalar phases, and (Ba0) E R s(2) 8 the corresponding set of scalar coefficients. Let =(1) and of2) denote the 5th entries of {(1) and ((1), respectively, and let w represent the 7th row of
( We first would like to show that given (dem) E R (1)) E s(0) and {8km) E R s()) E s()) (S(1) defined in Lemma A.2):
Σ Baa cos (12Fs()) .)) + (aw)))+ + ==== = w(()) (inr + ΦT) + b (2) ((1) sa))
for some Wi E x and (1) E R. Note that adding constant does not affect frequency and interchanging sines with cosines only affects the phase terms. We can express the summation as follows:
Σ (())
Bas cos (127 .r) $0(11)
3 cos((wrr r) + +c) + 3 cos((-wi,r)) Wi, r) db-))
31 sin (W))T) Φ: =) + 3_ cas ((w,, 2-1)
Σ R2 sin (wwe, r) 6%)
16

---

NEAR: Neural Electromagnetic Array Response
The final line follows from the Auxiliary Angle Formula, with:
Bi
R₂ = v Ai + Bi, of = arctan
A
A. = 3r cos Φ: + 2 + __, sin( B. 3r sin(
2 + 3_r cos(
where we assume A, 0 for all 1,2,...... T. R& represents the magnitude, and o; is the adjusted phase angle. For the case where A, S O, we leave the derivation as an exercise for the reader. Let w [ Φा = bils and b(") = 5. Then, we can conclude that the statement holds for = 1...., Fi. Second, we would like to show that given w E RIXT and b(")) I.
w (2nr + TT) + +.00) = =(1)
Σ
Baw cos ((DFs"),)) + )ow)) +
for some (8am) }. { with cardinality 27 and Ç. We can re-express the summation as follows:
w! sin(S2r + ΦT) 1,th w:" cos Φ 2 + b,
] r) + 0e - a
+ +b
(1)
2
Σ (w: 5) cos(( cos 1) + vi 2
Σ Ecos(( Wi, r) -Øx 5)
(oos((-w1,0))
+ 1!!)
for E R. Let Ç (1 [3,m0 0 s(1) E (th) (-01 2' ØT + 2. 2. ... ΦT # be two ordered sets. Then, we can conclude that the
R (( E s = {s. ... E. W - ... .... (w - {} and {orm E R
statement holds for
Fi.
Next, we will prove Theorem 4.1 by induction using the previous statement.
Base case Let denote the pre-activation vector at layer & as 2()), i.e., x0 = (x)))2())) Consider the pre-activation of a node at the first layer of the neural network for any mapping of the form in (3). Then
(1) (r))
Σ Wst cos ((w), r) Φt),
with some Wi E E depending on the first layer weights connected to that node and d, E IR. Also note that interchanging sines with cosines only affects the phase terms. After applying the activation function, and using the previous statement and the result of Lemma A.3, the output of each node at the first layer is given by
Q
Q
9
2 - (2) (2)")
Σ
Σ Wht cos ((Wr, r) + vt))
Og
Oiq
g=0
BaCe)
cos
(1NTs r) + Ta()))
B(Q))
where ((Q) = U 8(4) (1()) is defined in Lemma A.3. Therefore, the statement trivially holds, i.e.,
(( = (100
=
Q
(Q)
T EZZ
Σ
8
Induction step Assume the output of the nodes at layer satisfy the following expression:
Σ
3 cos (12 .r) + 22))))
17

---

NEAR: Neural Electromagnetic Array Response
where
((Q")
{
10'0
T ZA
oP
t=1
Then, the pre-activation of any node at the (E + 1)th layer can be expressed as:
(f+1)
Σ
Baoh, cos (
r) ,r) )gaa))
g090))
since the sum of sines/cosines with the same frequency only result in sine/cosine with the same frequency but with modified phase and amplitude. Hence, after applying the activation function, the output of the 2th node the (E 1)th layer can be written as:
3 Boun, cos (127 ,r) + 2 @asrs) ))
(++1) ((tt1) 2
(((+1)
= Σ
ag
9=0
By using Lemma A.4, we have:
3 a009 cos (1275109) (
Σ
Baves, cos (0RT , r))
o (gwor)s
where 5(099) =
{ [S000s
5
E Z
Σ ===
(00%)
)
Now, let us use the above result to complete the proof of the inductive step. particular, we can now express #; as: (f+1)
((+1)
Σ
3 Soo0e) cos (1T33199, r) (9s10e7))
2
q=0
Σ
By sin (12 ()) +
8' '
where
S' U = q=1
= -1
(110011
A
18
(Q
S Q
1=1
This sequence of inclusions concludes the proof of induction. Thus, considering >(r) sin(S2rr ΦT), the INR architecture of the form (3) can only represent functions of the form:
fe(r) Σ
cv sin (112 Φz).
BEST
where ST
( 82, ST] 8t Z, ΣT Ein1s S Q4-11.
{1s1, 82,
T
A.7. Proof of the connection between the expressive power of INRs and certain period-2 functions.
Proof. As we previously mentioned, interchanging sines with cosines only affects the phase term, we can rewrite the positional encoding in (5)
(an))
S2r 12Tx1)
(())
sin(Sir + +),
[S2r)
SSrr
18

---

NEAR: Neural Electromagnetic Array Response
where Si
|2 Ω
E RRTxx2 and Φ 02TxI T _ " R We use the same architecture of the form (3).
Directly using the result of Theorem 4.1, the expressive power of this architecture is of the form:
fe(r) Σ co' sin ( 8 r) + 6s))
e'EST
where
S7 = 16 114.000
14........,. -4. 1T]] by Z,
4T Σ S
Using the Trigonometric Sum and Difference Formulas, we can rewrite the above as:
fe(r)= Σ
cer sin (i s',r) + 0e))
e'ES,
Σ e'ES,
Cer cos (sn (10' s' .2) co sin (0or) cos
Σ x'E
de sin ( Ss cos
The inner product (Ω2 s', r), where r [r., y)_ can be expressed linear combination the corresponding components, involving coordinate x and y scaled by the respective elements of 52 s'. Then, we have
se(r)
Σ
DLs sin (T(i + jv)) + Fig cos +
(19)
where Di.j, Fi,, are some constants with respect to i.j, and = 0(2 This can be easily verified using the idea of binary representation, since the frequency matrix Ω only contains coordinate-win wise frequencies 2'm,, O,..., Hence, the layer of MLPs/INRs goes infinity, i.e. - oc, we have fr( approaching (7).
B. Proof of Theorem 4.5
B.1. Notations and Definitions
Consider an array snapshot containing K targets with azimuth angle 0₂ and elevation angle 6% (k = 1, ... ,K) Let [0, U1] x [o, U2] represent the antenna array response field with bounded domain positioned in the x y plane, consider a general case of an uniform sampling grid of dimensions M1 x M2, with spacing dN = M According to (2), the (m1, m2) element of the response with respect to K targets in absence of noise can be written as
U < 2 and d2 M, M < 2
U12 à
Σ rg e 3 ((mi 1)d, du 02 (m2- de sin 0c)
Yms, m2
for S my & M2 and 1 < m2 S M2. Notably, when d₁ = d2 = aें the sampling pattern aligns with the Nyquist sampling. Let Y mmm,malism M.. E M x Ma be the response matrix with entries as the antenna array response defined (8).
Given Y [ym M2 for mi S M1, S m2 M2. Block Hankel matrix of Y can be constructed
Hn, (y)) HN2
Hn, (Y2) HN, (ya)
HN, (YM,- +11 HN, LYM,-N14
(Y)
(yn,()N.)
(x.
(YM,)
19

---

NEAR: Neural Electromagnetic Array Response
where H. (ym) is defined as:
ym,,
ym,
12+1
ym,2
ym,3
M2
2222
HN2 (ym)=
Mm.N2
Mm.NN++1
M2
block Hankel matrix can also be constructed along the column direction, which defined as:
HNN(((())
H (x(0) HN, (N(O))
H (y(Mss-Ni+t)
H&r, (W(ee)
H. N₂ Ly(Nss-Ne+20
H AN (Y)=
: H (y(()))
His (1((82))
H His (y((i+ti))
where Hxx, (y())) is defined as:
V1,m
12.m
10 M 12
No (y(m)))
12,m
y3,m
VM -
Namm
y N 1,m
yM
Note that NN, (YT)= HN,,N, Moreover, we have rank (Y) rank (8 (Y)) based on the condi- tions in Lemma B. For the sake of clarity, we define
[ 0k
[0x] E (K++))x
S
{K+T)) b=
The matrices S and are column selection matrices, selecting the first K columns and the last column, respectively,
B.2. Lemma B.1, Lemma B.2, Lemma B.3 and their Proofs
Lemma B.1. For the Block Hankel matrix in Definition 4.3, if S N1 S M1 - K and K S N2 S M2 K + then we have rank(H (Y)) K, and the first K columns of HNI.Nz (Y) serve as a basis of R(HN.. (Y)). Similarly, and S No I M1 - K + 1. we have N,, = K, and the first K columns of H N, (, (Y) serve as a basis of ( No (Y)).
if S N₁ < M2 - K
Proof. Proof followed by (Hua, 1992).
Lemma B.2. For Hankel matrix HNg (ym) (1 S m S Mi). S N S M2-K 11, then rank(HH, (ymm)) K, and the first K columns of H., my)) serve as a basis of RHHN, (ym)). Similarly, for Hankel matrix H Ni (y(m)) (1 S m S M2). if K +1, rank(H, No (y(m)) = K, and the first K columns of H No (y(m)) serve as basis of R No (x(())) MM-KKH
K N2 < M1
Proof. We will prove the statement for HN, (ym), while the proof for H (y(m)) follows the same way therefore omitted here. According to (8), we have
I e 2r (m-1)d; sin di cos 0,
ed Pr di dy sin
ed dy sin de 0g
e Br di de sin Ox
Ize 30t (m-1)dd sin do cos 0,
Ym
e 3 (M2 - 1)d; sin di 0i 0,
ed (M (Mg-1)ds sin dh sin 0g
e 2 (Ma - 1)d, sin dx sin Ox
Le 3 (m-1)dds sin ox cos dx
AM, (0, Φ $)sm
20

---

NEAR: Neural Electromagnetic Array Response
can be shown that HNz (mm) admits Vandermonde decomposition
Hvz(ym) = A. (0, ()diag(smm)) (so (A (AM.-N+1(00.0)
where A Na (0, and AMM-NN++ (0, Φ are both Vandermonde matrix. is easy to verify that if K S N2 < M2 K + rank(H, (y.N,(y)) (ym)) = K. And the first K columns of HN, (ym) have the form AN. (0, ()diag(sm) be verified to be rank-K due to the Vandermonde structure. Thus, the first columns form a linearly independent set,
Φ)) which can
therefore serve as basis for R(H (HH, (ym)).
Lemma B.3. Consider Hankel matrix HM2 x (Ym) (1 S m S Mi) generated using (8), there exists a unique mi E C such that
M 11 xx (ym))mm HM, x (ym)blla = 0.
(20)
m=1
Similarly, consider Hankel matrix Hw, x (y((m)) (1 m S M2) generated using (8), there exists a unique m2 E CK such that
Mg HMM - ((y(m))))m HM, - (n(m))bll2 (
O.
(21)
Proof. We will prove the statement for HM2-K K (ym) S m S M1). while the proof for H HM, - ((y(m)) follows the same way therefore omitted here. According Lemma B.2, we have rank (H x (y,)) = K where its first K columns serve as basis of R(HM,-K mm))), This means there exists a [ai. ax]" x such that HMr-K (ym))at H.M-- K (y (ym) bb. Now let analyze whether depend on Using Vandermonde decomposition and explicit form of least squares solution, we have
HyI,-K(Fm)5 AM,-x d)dag(s (sm) (Ax 6))²
e 2n Kd, sin sin 0g
HM AM (0, (idag(sm.)
e 2 Kd, sin ox sin 0x
a (H 11 (HHM, (ymm)))
(MMs- (ym)
Kd, sin di sin 6g
= (0,
2m Kd, sin d sin
And we can see that a E KK does not depend on m but only depend on and d. This means for Hankel matrix HM (ym) S m < Mj), there exists a unique m1 CK such that
Mi
1 K (ym Sm; 7MMM )bl2 = 0.
B.3. Main Proof of Theorem 4.5
Proof. We will prove the statement for H.,, .Ng (Y), while the proof for H. (Y) follows the same way therefore omitted
here. Let N1 = Mj K + 1 and N₂ M2 K, consider the matrix HAs -K (Y). According to Lemma B. we have rank(H 1, xc (Y)) = K, and its first K columns
(y,,K(y1)s HM,-K(y2)S
HM, HM--KHIM,-K(Y)5-
HM, K MVM,,KK11)
21

---

NEAR: Neural Electromagnetic Array Response
linear independent. If we keep appending columns the bottom of HM, -M (Y),, the columns of the resulting
matrix
HM5- HMz--((y))
HM, (0.,-K(MM---
HM)))
HM,, K -
HM.-K (yM,)vMM)))))) )
is still linear independent. Using Lemma B.2 and Lemma B.3, if we add the K + 1-th column at the right of the above matrix, the resulting matrix M, (Y) ) is still rank-K K with its first K columns serve as a basis of R(HM, M (Y)).
Thus, there exists unique global optimizer mi such that
S)m HM,,M, K (Y)6l12 0.
C. Further Experimental Results and Details
C.I. Experimental Setup
C.I.I. SIMULATION DATA GENERATION
For the signal model in (2), we assume the reflection coefficients follow a circularly symmetric complex Gaussian distribution, given by 2 c.) with σr for k 1..... K. The additive noise is modeled ni, CN o where on determined by the specified SNR levels. The SNR is defined as:
Pz Pa
SNRAB 10 logio
10 logio
d
For tasks with different sampling configurations, we use the following selected indices sub-sampling:
6 S. =, (0, 2, 11, 19).
8 S = S, =
14, 19).
10 x 10: S. = = (0,1,,,,,33 10, 13, 19).
Remark. When the index sets are mapped to the world coordinate system, each discrete index (i,j) corresponds to the physical position
ri = (i 2 2
where À is the wavelength.
For angle resolution experiments, we define the azimuth and elevation angles of two targets as [10, 20) and [10 + A, 20 + A] degrees, where represents the angular separation, which varies from to 10 degrees. For the other tasks, the azimuth and elevation angles of each target are randomly sampled from a uniform distribution over [-60, 60) degrees. Each experiment conducted with N 50 Monte-Carlo trails.
C. 1.2. EVALUATION METRICS
The Normalized Root Mean Square Error (NRMSE) is defined as:
Y Y
NRMSE
where Y and Y denote the predicted array response and the ground truth full virtual array, respectively, and ||- represents the Frobenius norm.
22

---

NEAR: Neural Electromagnetic Array Response
The Resolution Probability is defined as:
N
RP
(B.) with
N
==1
A
A
A
A
E, =
gm a,1
E
0a.1
1,
+
0%,2 E
0a,2
0,, +
2
2
2
2
E
0m e, E
Oe,1
A
O0,1
+
A
0e E
00.2
00,2 +
A
2
2
2
2
where pr: 6n, pm i = 1,2 denote the prediction azimuth and elevation angles for each target in n-th Monte-Carlo
a.1. c,i
trail, 0. = 10 0e.s)/² = 1,2 represent the ground truth azimuth and elevation angles, 1()) is the indicator function, and
separation A - 0..2 1,, = 6..2 - fe,1
C.1.3. OPTIMIZATION AND HYPERPARAMETERS
We optimize the loss function defined in (11) through a two-stage training process. In the initial warm-up stage, we set À = 0 and optimize using the Adam optimizer with (0.9, 0.999) and a weight decay of 10-4. Letting O, = arg mine La, we use the obtained parameters as the initialization for the next stage. In the adaptation/training stage, we optimize O m,, and my using Adam with the same configuration as in the warm-up stage. In both the simulation and real-world experiments, we normalized the input coordinates to the range (-1,
We provide detail hyperparameter settings for both simulation and real-world experiments. The model architecture remains consistent with that described in the Experiments section. For simulation tasks, we use learning rate of 10 -4 and train for 5, 000 epochs in the warm-up stage. the adaptation stage, we set À = 0.5, Ire- = 10-3 Irm..m = 3 x 10 10-3, and train for 25, 000 epochs, with K max set to the exact number of targets for each scenario.
For world experiments, we adopt a learning rate of 10-1 and train for 10, 000 epochs in the warm-up stage. In the adaptation stage, we set À 1, Ire 10-3, mmmmm, 3 x 10 and train for 50, 000 epochs. Here, set K mamax 4 as an upper bound on the number of targets in each range bin, as typically, the number of targets within a single range bin is very small (Sun et al., 2020).
C.2. Implementation Details and Analysis of Baseline Methods
EMaC. We adopt equation (9) from the original paper (Chen & Chi, 2013) as the optimization problem, which can be solved using CVX (CVX Research, 2012; Grant & Boyd, 2008) toolbox.
SIREN.
We adopt the same architecture and recommended hyperparameters from the original paper (Sitzmann et al., 2020).
For fair comparison, we match NEAR's network size, using a depth of L and a hidden layer width of 256.
NeRF2 We adopt the same architecture and recommended hyperparameters from the original paper (Zhao et al., 2023). To match our experimental setup, the location of TX is fixed (co-located with RX), and the unknown antenna response is inferred based on their spatial coordinates. The loss function is calculated as the mean-squared error (MSE) between predicted arra responses and observed array response, rather than using RSSI values.
Remark. The inferior performance of N is attributed to some important distinctions between radiance-field reconstruc- tion and our method, which are listed below:
Our setting uses far fewer measurements (see below) in the form of antenna array response, compared to N This renders measurement heavy methods like NeRF² somewhat inferior in our settings. Hence we need to heavily utilize the underlying wave propagation model and the harmonic structure of measurements received at antenna arrays, in order to successfully regularize the problem with so few measurements. This is major contribution of our work which sets us apart from direct use of N
In fact, NEAR targets different objective than 2 Our approach emphasizes more on the (super-resolution) localization of the targets, while N cares more about the physical property of all objects in a 3D scene order to model signal propagation. This also serves a crucial reason why we opt to directly predict the response from the antenna coordinates rather than modeling all the voxels' properties as a continuous volumetric function.
23

---

NEAR: Neural Electromagnetic Array Response
As explained earlier, NeRF requires large set of measurements for training. According to (Zhao et al., 2023), it uses around 6000 x 21 measurements and 80% for training/testing splitting, while we only use a sparse set of 8 x 8 measurements for training. Under the same setting of training, NEAR uses less than half of the training time of NeRF due to our proposed regularization rather than the ray tracing strategy, which is well known for its heavy computational cost.
C.3. Additional Details on Radar Data Processing
To sense the environment, the system emits sequence of waveforms, commonly referred to chirp signals, through the Tx within a short time interval. These signals propagate, interact with objects in the environment, and are subsequently reflected back to be captured by the Rx. The received signals are then processed to generate an intermediate frequency (IF) signal by mixing the transmitted and received signals from each Tx-Rx antenna pair. This mixed signal is then sampled by an ADC to generate discrete samples for each chirp. By aggregating ADC samples across all chirps and Tx-Rx antenna pairs, the sensing system constructs a three-dimensional (3D) complex data cube for each frame. This data cube is organized into three dimensions: fast time, slow time, and channel, which correspond range, range rate, and angle, respectively (Kramer et al., 2022).
To process the acquired ADC samples, fourier techniques are applied along the fast time and slow time dimensions to extract detailed information. The first range processing is performed across the fast time axis to isolate objects at different distances into distinct frequency responses within range bins defined by hardware specifications. Subsequently, Doppler processing along the slow time axis decodes phase puriances-Doplle bins-to to derive relative radial velocities, producing a range- Sagge-Dpplerr (RD) map (Ding et al., 2024). An additional CFAR target detector is usually employed to detect peaks that stand out prominently from their surroundings in the range-Doppler velocity heat-map by comparing local signal power to an adaptive threshold. DOA processing is then performed only for the peaks detected by the CFAR detector.
C.4. More Experimental Results
Some more experimental results of target localization are shown below.
25
2
2-
2991
Y392
7390
Figure 6. Point cloud visualizations for target localization with K 1 (scenario 1). Left: Full array, Middle: NEAR, Right: EMaC.
400
2000
25
X01
Y011
V.355
X000
Y390
Figure 7. Point cloud visualizations for target localization with K = 1 (scenario 2). Left: Full array. Middle: NEAR, Right: EMaC.
24

---

NEAR: Neural Electromagnetic Array Response
1001
-35
+00
055
+00
-35
X09)
y011
V011
Figure& 8. Point cloud visualizations for target localization with K = 2 (scenario 1). Left: Full array, Middle: NEAR, Right: EMaC.
R
X00
3.5
+00
155
435
+00
-35
5.001
1/01
Figure 9. Point cloud visualizations for target localization with K = 2 (scenario 2). Left: Full array, Middle: NEAR, Right: EMaC.
15-
25-
2000
230
+.00
155
15
Figure 10. Point cloud visualizations for target localization with K- = 3 (scenario 1). Left: Full array, Middle: NEAR, Right: EMaC.
4
0
18
15
0
K091
-55
1.5
K200
-1.
K000
Y00
you
Figure/ 11. Point cloud visualizations for target localization with = (scenario 2). Left: Full array, Middle: NEAR, Right: EMaC.
25

---

NEAR: Neural Electromagnetic Array Response
4000
-15
-25
-05
-12
-65
200)
X000
Figure 12, Point cloud visualizations for target localization with K = 4 (scenario 1). Left: Full array, Middle: NEAR, Right: EMaC.
-15
-05
-05
-55
405
Figure 13. Point cloud visualizations for target localization with K = 4 (scennrio 2). Left: Full array, Middle: NEAR, Right: EMaC.
26


Reviews:
[{"review_id": "SPWr9g5ynD", "reviewer": "Reviewer_5fdJ", "summary": "Problem Statement:\nThe paper tackles the challenge of achieving angular super-resolution in multi-antenna radar systems using only sparse measurements. In radar systems, hardware constraints (i.e. having only a few physical antennas) and noise limit the achievable angular resolution. Traditional supervised methods often require large, high-quality training datasets and may not generalize well to new environments.\n\nProposed Solution (NEAR):\nThe authors propose NEAR—a framework based on untrained implicit neural representations (INRs) that predicts complex-valued antenna responses at unseen locations from sparse measurements. The method is physics-informed, as it leverages the latent harmonic structure inherent in electromagnetic wave propagation, and it incorporates a novel latent geometry–aware regularizer.\n\nMain Claims:\n\n- High-Resolution Prediction: NEAR can predict full virtual array responses (both amplitude and phase) with high accuracy despite limited measurements.\n- Generalization & Interpretability: By integrating physical laws (such as planar wave propagation and harmonic structure), the model generalizes well to unseen environments and remains physically interpretable.\n- Superior Angular Super-Resolution: NEAR improves upon conventional methods in resolving closely spaced targets (i.e., achieving super-resolution) while keeping hardware costs low.\n- Theoretical Foundations: The paper provides new theoretical insights into the expressive power of INR architectures when equipped with appropriate positional encodings and shows how these relate to the Fourier harmonic representation of array responses.", "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 sincerely thank the reviewer 5fdJ for the time and effort in reviewing our paper. We greatly appreciate the positive feedback. We hope the following responses can resolve your questions and concerns.\n\n**1. I'm not sure why Theorem 4.5 is needed --- if the Hankel matrices are low rank and the first $K$ columns are independent, then doesn't the statement of the theorem automatically follow from the definition of rank? (this is a minor concern, I don't have major issues with it being included)**\n\nThere are two values which Theorem 4.5 add. Firstly, the low-rank property of $\\mathcal{H}\\_{N\\_1,N\\_2}(\\boldsymbol{Y})$ depends on the choice of $N\\_1$ and $N\\_2$ according to the theoretical results from [1]. However, we cannot directly use the sufficient condition from [1] to claim that $\\mathcal{H}\\_{M\\_1,M\\_2-K}(\\boldsymbol{Y})$ is rank-$K$ since in this case $N\\_1=M\\_1\\notin[K,M\\_1-K+1]$. Secondly, Theorem 4.5 reveals an important relation between submatrices of this low-rank matrix. It shows that all small Hankel matrices $\\mathcal{H}\\_{M\\_2-K}(\\mathbf{y}\\_{m})\\text{ for }1\\leq m\\leq M\\_1$ are not only rank-$K$ with their first $K$ columns serving as a basis, but also share the *same linear coefficients $\\boldsymbol{m}\\_1$*. More details can be found in Appendix B. This is a significant result, since it helps us to re-use the parameters $\\boldsymbol{m}\\_1$ for representing subsequent new entries at unseen locations. Such theoretical results are non-trivial and no existing results can be found in the literature according to our best knowledge.\n\n**2. The significance of the Theorems is doubtful --- they seem unsurprising.**\n\nWe mainly have two theoretical results, one is about expressive power of INRs, the other is about our physics-informed regularizer. We obtain a tighter characterization on the expressability of INR by refining existing analysis and derive the exact set of integer harmonics which describe the expressive power of INRs. One the other hand, our regularizer utilizes physical model (planar wave propagation) and mathematical properties of harmonics structured matrices, which establish a connection between low rank and linear predictability. This results in a regularizer which is both computationally efficient and significantly improves overall performance, as shown by our experimental studies.\n\n**3. One question is: In the regularized loss function, I don't see the point of having $\\boldsymbol{m}\\_1,\\boldsymbol{m}\\_2$ as variables. For each $\\boldsymbol{\\theta}$, you can compute the prediction $\\hat{\\boldsymbol{Y}}$ and use the pseudoinverse projection to compute $\\boldsymbol{m}\\_1,\\boldsymbol{m}\\_2$ in closed form. Why would you run an optimizer on as well?**\n\nThank you for the good question. Please note that $\\boldsymbol{\\theta}$ and $\\boldsymbol{\\phi}$ are angles of arrival from multiple point scatterers, and do not represent incident angles. The unique global optimal $\\boldsymbol{m}\\_1^o$ and $\\boldsymbol{m}\\_2^o$ depend on target angles $\\boldsymbol{\\theta}$ and $\\boldsymbol{\\phi}$, which are part of the sensing task and not known beforehand. In fact, the goal is to first complete a virtual array (by predicting array response at unseen locations) and then use the physical and predicted measurements to obtain a more accurate estimate of the angles $\\boldsymbol{\\theta}$ and $\\boldsymbol{\\phi}$ (and not the other way). If we were to first estimate the angles  $\\boldsymbol{\\theta}$ and $\\boldsymbol{\\phi}$ from a *limited number of sparse* physical antennas (without completing the virtual array), the estimation error would be much larger. Therefore, in order to predict the virtual array response, we use the combined power of INR and the latent variables $\\boldsymbol{m}\\_1$ and $\\boldsymbol{m}\\_2$, which exploits low-rank relationship between entries of the Block Hankel matrix and guides the INR to approach the ground truth array response. However, if $\\boldsymbol{m}\\_1$ and $\\boldsymbol{m}\\_2$ are computed using the pseudoinverse projection after predicting $\\hat{\\boldsymbol{Y}}$ purely using INR, then the whole algorithm runs the risk of overfitting the  observed data, especially given the scarce number of spatial measurements.\n\n**References:**\n\n[1] Hua, Y. (1992). Estimating two-dimensional frequencies by matrix enhancement and matrix pencil. IEEE Transactions on Signal Processing, 40(9), 2267-2280.", "claims_and_evidence": "I found the claims to be well supported.", "methods_and_evaluation_criteria": "The methods are described well and the authors do experimental studies on simulated as well as real world datasets with impressive results.\n\n- Implicit Neural Representation (INR):\n\nNEAR employs an untrained INR that maps 2D spatial coordinates to complex-valued radar responses.\nThe architecture follows a typical INR design: a multilayer perceptron (MLP) with positional encoding (similar to NeRF) is used to represent the continuous response field over a virtual antenna array.\n\n- Physics-Informed Regularization:\n\nA key innovation is the introduction of an implicit regularizer that exploits the harmonic and low-rank structure of the radar array response.\nThe authors show that when the response field is expressed in a domain where the underlying physics (planar wave propagation) holds, its corresponding block Hankel matrix exhibits low rank.\nThe regularizer is integrated into the loss function to enforce consistency with these known physical properties.\n\n- Loss Function and Training:\n\nThe overall loss is composed of a data fitting term (quantifying the difference between the predicted response and the available sparse, noisy measurements) and a regularization term (enforcing the harmonic/low-rank structure).\nImportantly, NEAR is trained without extensive offline training data—it uses only the sparse measurements obtained during normal operation.\n\n- Theoretical Analysis:\n\nThe paper provides rigorous theoretical results that characterize the class of functions representable by the INR architecture, linking it to Fourier series.\nThis analysis not only justifies the choice of positional encoding and network architecture but also explains how the harmonic structure of radar signals can be effectively captured.", "theoretical_claims": "I did not check the correctness of the proof. The theorem statements are sensible, although I'm not sure why Theorem 4.5 is needed -- if the Hankel matrices are low rank and the first K columns are independent, then doesn't the statement of the theorem automatically follow from the definition of rank? (this is a minor concern, I don't have major issues with it being included)", "experimental_designs_or_analyses": "Simulation Studies:\n\n- Setup: The simulations use virtual antenna array responses under various SNR conditions and different sparse sampling patterns (e.g., 6×6, 8×8, 10×10 grids).\n- Baselines: NEAR is compared against Enhanced Matrix Completion (EMaC), a SIREN-based approach, and a variant of NEAR without the physics-informed regularizer.\n- Results:\nNEAR consistently achieves lower NRMSE across various SNR levels and sampling patterns.\nIn angular resolution tests, NEAR maintains high resolution probability even at small angle separations.\nFor multi-target DOA estimation, NEAR shows significantly lower estimation errors than baselines, demonstrating its robustness in complex scenarios.\n\nReal-World Experiments:\n\n- Setup: A commercial MIMO radar platform (IMAGEVK-74) with a 20×20 virtual array is used. Sparse subsets of the full array response are treated as input.\n- Evaluation:\n- Angular Resolution: Experiments with two corner reflectors measure the minimum resolvable angular separation at various distances. NEAR achieves performance close to the full array benchmark.\n- Target Localization: The system is tested with multiple reflectors. NEAR outperforms both the full array baseline and EMaC in terms of localization error, attributed to its denoising capability.\n\n- Ablation Study:\n\nRemoving the physics-informed regularizer leads to significantly worse performance, underscoring the importance of incorporating physical constraints into the model.", "relation_to_broader_scientific_literature": "The paper seems like a good contribution. The proposed method highlights the challenges associated with employing INR models to RADAR, and designs regularizers that respect the harmonics present in RADAR. The loss function designed is straightforward, but I think the authors do a good job evaluating their method and comparing it to baselines on simulated and real-world data. The significance of the Theorems is doubtful -- they seem unsurprising.", "essential_references_not_discussed": "I think the related work section is detailed and comprehensive.", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "I don't have major concerns.\n\nOne question is: In the regularized loss function, I don't see the point of having $m_1, m_2$ as variables. For each $\\theta$, you can compute the prediction $\\widehat{Y}$ and use the pseudoinverse projection to compute $m_1, m_2$ in closed form. Why would you run an optimizer on $m_1, m_2$ as well?", "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": "kxmi66wTKO", "reviewer": "Reviewer_MVwc", "summary": "The authors utilize a new INR-based framework to achieve angular super resolution in multi-antennae radar systems. The authors further propose a physics-informed regularizer and provide theoretical insights into what functions can be represented by INRs under certain, in previous literature established, constraints. The authors provide extensive synthetic and real-world results as well as an ablation study, showing the superiority of the proposed approach over other solutions as well as the positive performance impact of individual components.", "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 sincerely thank the reviewer MVwc for the time and effort in reviewing our paper. We appreciate your positive comments on our work and have fixed the typo you pointed out. If there are any additional areas where you believe we could further improve our manuscript, we would greatly appreciate your insights. Please let us know if we can address any specific concern that could help justify a higher score. Thank you again for your consideration and support.", "claims_and_evidence": "The performance claims made in this work are sufficiently supported by the experiments conducted. I have not extensively validated the proofs for claims about INRs representational power.", "methods_and_evaluation_criteria": "I am not familiar with the application domain, as such I cannot speak on whether the evaluation criteria are appropriate. The ablation study is sufficient evidence to at least support baseline claims about the framework's efficacy and necessity of individual components (i.e. the additional regularizer).", "theoretical_claims": "I did not check the correctness of proofs provided in the paper.", "experimental_designs_or_analyses": "The ablation study is appropriately designed.", "relation_to_broader_scientific_literature": "This work extends the literature in two directions, to the best of my knowledge. It introduces two new methods, NEAR broadly and the physics-informed regularizer specifically. It also extends the literature on INRs representational capabilities.", "essential_references_not_discussed": "I am not familiar enough with the literature in this domain.", "other_strengths_and_weaknesses": "The experiments in this paper seem well-selected and demonstrate the strong performance of the proposed method. On first glance the proofs also seem detailed and, given their veracity, make interesting contributions to the literature that elevates the overall significance of the paper substantially.", "other_comments_or_suggestions": "page 3, line 111, right: Typo in \"reconstruc\" - missing \"t\"", "questions_for_authors": "No questions.", "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": "oeewYB79us", "reviewer": "Reviewer_xAKz", "summary": "This paper addresses the challenge of achieving high-resolution angular estimation in multi-antenna radar systems using sparse measurements. The authors propose NEAR (Neural Electromagnetic Array Response), an innovative framework that leverages implicit neural representations (INRs) to predict complete antenna array responses from limited physical antenna data, effectively creating a large virtual sensing system with few physical antennas.\n\nThe core technical contribution lies in seamlessly integrating INRs with a physics-informed regularization strategy, specifically designing a novel Block Hankel matrix-based constraint that captures the inherent harmonic structures of radar wave propagation. By developing a theoretically grounded approach that maps spatial coordinates to complex-valued antenna responses, the authors enable a continuous representation of the antenna array response field, which allows for super-resolution angular estimation while maintaining computational efficiency and generalizability.\n\nExperimental validation across both simulated and real-world scenarios demonstrates NEAR's superior performance, consistently outperforming baseline methods in response recovery, angular resolution, and direction-of-arrival estimation. The results showcase the method's robustness across different sampling configurations, noise levels, and number of targets, with significant improvements observed particularly in complex multi-target environments. Comprehensive evaluations using commercial MIMO radar platforms further validate the framework's practical applicability and potential for enhancing radar sensing technologies.", "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 sincerely thank the reviewer xAKz for the constructive comments and suggestions. We provide additional experimental comparisons below to address your concerns:\n\n**1. Experimental comparison between our approach and data-driven method (NeRF$^2$).**\n\nWe add a state-of-the-art data-driven baseline called NeRF$^2$[1], which represents scenes as neural radiance fields by optimizing an underlying continuous volumetric scene function using a set of input electromagnetic signal measurements. We compare NeRF$^2$ and our method (NEAR) in terms of angular resolution, target localization accuracy (same as we did in Section 5.2) and average running time using the real-world collected data. We adopt hyperparameters recommended in [1], and run all experiments on a laptop with CPU AMD Ryzen 9 5900 HS with Radeon Graphics and GPU NVIDIA GeForce RTX 3050 Ti Laptop.\n| Method  | Angular resolution for 2m/3m/4.5m | Localization error for 1/2/3/4 target(s) | Average running time |\n|:-------:|:--------------------------------:|:--------------------------------------:|:--------------------:|\n| NEAR    | 5.7248$^\\circ$/6.6769$^\\circ$/6.9941$^\\circ$ | 0.0744m/0.0770m/0.0762m/0.0718m | 550.83s |\n| NeRF$^2$ | 8.5783$^\\circ$/8.5783$^\\circ$/8.8948$^\\circ$ | 0.4902m/0.5096m/0.4346m/0.3898m | 1278.31s |\n\nFrom the table, we can see that NEAR is able to resolve smaller angle separation at different distances (SNR) and achieve a much smaller target localization error, compared to NeRF$^2$, highlighting the effectiveness and efficiency of our method. This improvement is primarily attributed to judicious use of signal processing ideas in designing the regularizer, which fully exploits the underlying harmonic structure in planar wave propagation from far-field targets.\n\n**2. Experimental comparison of computational efficiency between our approach and EMaC.**\n\nFor the average running time, EMaC takes **1226.15s** while our approach only takes **550.83s**, indicating the potential of real-time implantation with future algorithmic and computing hardware improvements.\n\n**References:**\n\n[1] Zhao, X., An, Z., Pan, Q., \\& Yang, L. (2023, October). Nerf2: Neural radio-frequency radiance fields. In Proceedings of the 29th Annual International Conference on Mobile Computing and Networking (pp. 1-15).", "claims_and_evidence": "**Claim: Extensive simulations and real-world experiments using radar platforms demonstrate NEAR’s effectiveness**\n\nAssessment:  Lack of comparison with data-driven algorithms and lack of computational efficiency comparison experiments make it impossible to determine the method's effectiveness.", "methods_and_evaluation_criteria": "Yes, using simulations and real-word experiments to evaluate method's performance is reasonable.", "theoretical_claims": "The constraints used in the paper are common constraints in its application domain, and can effectively constrain INR.", "experimental_designs_or_analyses": "**Issue 1**: Lack of comparison with data-driven algorithms.\n\n**Issue 2**: Lack of computational efficiency comparison experiments, making it impossible to determine the method's usability.", "relation_to_broader_scientific_literature": "The paper applies cutting-edge AI technology INR to traditional problems, and combines domain-specific prior knowledge as constraints, successfully solving domain problems. This problem-solving approach may potentially be extended to many other fields.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. Please provide detailed experimental comparisons or a comprehensive analysis of your method's advantages over data-driven algorithms in terms of accuracy and computational efficiency.\n\n2. Please provide experimental comparisons or a detailed analysis of the computational efficiency differences between your method and EMaC.", "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": "OLIusXfEGZ", "reviewer": "Reviewer_moMV", "summary": "Multi-antenna radar systems face challenges in achieving high angular resolution due to hardware constraints, noise, and limited physical antennas. Traditional supervised learning methods for super-resolution struggle with generalization in unseen environments and require extensive training data. \n\nThe authors propose Neural Electromagnetic Array Response (NEAR), an untrained implicit neural representation (INR) framework that predicts complex radar responses at arbitrary 2D spatial coordinates using sparse antenna measurements. NEAR predicts radar responses at unobserved locations by exploiting latent harmonic structures in radar wave propagation. It integrates physics-based signal processing with neural networks, avoiding reliance on large datasets.\n\nThey claim that this is the first time to establish a link between antenna array response and the expressive power of an INR architecture. Additionally, they introduces a novel regularizer that incorporates radar physics and latent geometry to bridge the gap between traditional INR and NEAR. They include a 20 ×20 full virtual array response (noisy) as a benchmark reference. Evaluated through simulations and real-world (a commercial MIMO radar platform IMAGEVK-74) radar experiments, NEAR demonstrates superior performance in unseen environments compared to conventional methods.\n\nOverall, this research bridges signal processing and INR, offering a data-efficient, physics-aware solution for radar super-resolution without compromising interpretability.", "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 are grateful for your recognition of our work’s strengths, notably our theoretical analysis that precisely characterizes the expressive power of INR, as well as our development of an efficient and effective regularization strategy. In response to your concern regarding the rationale behind directly predicting the response value from the coordinates—as opposed to modeling it as a radiance field-like function—we have conducted a comprehensive comparison between our approach (NEAR) and NeRF$^2$[1], an innovative extension of NeRF[2] into the electromagnetism domain using ray tracing. NeRF$^2$ constructs a continuous volumetric scene function that interprets the propagation of RF signals and can tell what signal is received at any position after training with a set of input signal measurements.\n\nWe compare NeRF$^2$ and our method (NEAR) in terms of angular resolution, target localization accuracy (same as we did in Section 5.2) and average running time using the real-world collected data. We adopt hyperparameters recommended in [1], and run all experiments on a laptop with CPU AMD Ryzen 9 5900 HS with Radeon Graphics and GPU NVIDIA GeForce RTX 3050 Ti Laptop.\n| Method  | Angular resolution for 2m/3m/4.5m | Localization error for 1/2/3/4 target(s) | Average running time |\n|:-------:|:--------------------------------:|:--------------------------------------:|:--------------------:|\n| NEAR    | 5.7248$^\\circ$/6.6769$^\\circ$/6.9941$^\\circ$ | 0.0744m/0.0770m/0.0762m/0.0718m | 550.83s |\n| NeRF$^2$ | 8.5783$^\\circ$/8.5783$^\\circ$/8.8948$^\\circ$ | 0.4902m/0.5096m/0.4346m/0.3898m | 1278.31s |\n\nFrom the table, we can see that NEAR is able to resolve smaller angle separation at different distances (and signal to noise ratio, SNR) and achieve a much smaller target localization error, compared to NeRF$^2$. This is attributed to some important distinctions between radiance-field reconstruction and our method, which are listed below:\n\n- Our setting uses far fewer measurements (see below) in the form of a antenna array response, compared to NeRF$^2$. This renders measurement-heavy methods like NeRF$^2$ somewhat inferior in our settings. Hence we need to heavily utilize the underlying wave propagation model and the harmonic structure of measurements received at antenna arrays, in order to successfully regularize the problem with so few measurements. This is a major contribution of our work which sets us apart from direct use of NeRF$^2$.\n\n- In fact, NEAR targets a different objective than NeRF$^2$. Our approach emphasizes more on the (super-resolution) localization of the targets, while NeRF$^2$ cares more about the physical property of all objects in a 3D scene in order to model signal propagation. This also serves a crucial reason why we opt to directly predict the response from the antenna coordinates rather than modeling all the voxels' properties as a continuous volumetric function.\n\n- As explained earlier, NeRF$^2$ requires a large set of measurements for training. According to [1], it uses around $6000 \\times 21$ measurements and 80\\%/20\\% for training/testing splitting, while we only use a sparse set of $8\\times8$ measurements for training. Under the same setting of training, NEAR uses less than half of the training time of NeRF$^2$ due to our proposed regularization rather than the ray tracing strategy, which is well known for its heavy computational cost.\n\nIn summary, NEAR outperforms NeRF$^2$ by directly predicting antenna responses while leveraging the harmonic signal structure. Our approach significantly reduces the required training measurements, leading to improved angular resolution, enhanced localization accuracy, and reduced runtime.\n\n**Reference:**\n\n[1] Zhao, X., An, Z., Pan, Q., \\& Yang, L. (2023, October). Nerf2: Neural radio-frequency radiance fields. In Proceedings of the 29th Annual International Conference on Mobile Computing and Networking (pp. 1-15).\n\n[2] Mildenhall, B., Srinivasan, P. P., Tancik, M., Barron, J. T., Ramamoorthi, R., \\& Ng, R. (2021). Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1), 99-106.", "claims_and_evidence": "The authors proves that predicting complex-valued responses at any arbitrary location within the 2D virtual antenna array domain is indeed a problem that falls within the class of functions representable by INR.\nFurthermore, they show that this mapping can be effectively approximated using multiple layers of Multi-Layer Perceptrons.\n\nTo further justify the effectiveness of the physics-informed regularizer, the authors establish \nthe algebraic properties of the block Hankel matrix corresponding to the ground truth response, \nleveraging its harmonic structure. \n\nThe experiments demonstrate that the proposed method outperforms the baseline in terms of generalization, \nexhibits robustness to noise, achieves superior performance in multi-objective Direction-of-Arrival (DOA) estimation, \nand significantly reduces hardware requirements.", "methods_and_evaluation_criteria": "The proposed method does not rely on large-scale data, exhibits low computational consumption, and demonstrates greater generalizability. \n\nThey include a 20 $\\times$ 20 full virtual array response (noisy) as a benchmark reference. \nA series of validations are conducted in both simulated and real-world environments, encompassing scenarios with varying noise intensities, multiple targets, and different sampling densities.\n\nIn conclusion, the authors demonstrate the advantages of the proposed method in addressing the radar response super-resolution problem through a series of well-conducted experiments.", "theoretical_claims": "Theorem 4.1 gives an exact characterization of the set $S_T$ of all possible integer harmonics of the feature mapping $γ(r)$. \nThe mapping from 2D coordinates to the complex values of the radar response is shown to belong to the class of INRs, as supported by Remark 4.2 and Theorem 4.1.", "experimental_designs_or_analyses": "The validity of the experimental designs can be referenced in the section \"Methods and Evaluation Criteria.\"\n\nIt is suggested to include a comparison with a NeRF-like method, \nwhere the rendering operation of NeRF can be modified to model radar waves. \nBy adapting the NeRF approach for radar response, \nit could serve as a valuable baseline for evaluating the proposed method.", "relation_to_broader_scientific_literature": "The approach of combining implicit fields with physical constraints is a key feature in many recent works on 3D reconstruction. \nFor example, the incorporation of continuous medium dynamics to describe the evolution of the Gaussian distribution illustrates how physical principles can be leveraged to enhance the accuracy and realism of reconstructed models\\[1\\].\n\n\\[1\\] Tianyi Xie, Zeshun Zong, Yuxin Qiu, Xuan Li, Yutao Feng, Yin Yang, and Chenfanfu Jiang. Physgaussian: Physics-integrated 3d gaussians for generative dynamics. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2024.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "*Strengths*:\n\nTheir work refines the INR superset analysis \\[1\\] by providing the exact set, \nrather than a superset, of integer harmonic frequencies that characterize INR functions. \nThis derivation delivers a precise and tight characterization of the expressive power of INRs.\n\nThey assert that their method employs a straightforward yet effective regularization strategy, in contrast to the more complex ray tracing approach\\[2\\].\n\n\n\\[1\\] Roddenberry, T. M., Saragadam, V., de Hoop, M. V.,\nand Baraniuk, R. G. Implicit neural representations\nand the algebra of complex wavelets. arXiv preprint\narXiv:2310.00545, 2023.\n\n\\[2\\] Chen, X., Feng, Z., Sun, K., Qian, K., and Zhang, X. Rf-\nCanvas: Modeling RF channel by fusing visual priors\nand few-shot RF measurements. In Proceedings of the\n22nd ACM Conference on Embedded Networked Sensor\nSystems, 2024.\n\n*Weaknesses*: \n\nIn this task, both the radar and environmental settings (similar to light and camera parameters in NeRF) are kept constant, \nwith the goal of predicting higher-resolution radar response maps. \nHowever, the authors do not provide a rationale for why they choose to directly predict the value from the coordinates, \nrather than modeling it as a radiance field-like function. \nLack of experiments comparing the two.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Why opt to directly predict the value from the coordinates, instead of modeling it as a radiance field-like function? \nPlease refer to \"Experimental Designs Or Analyses\".\nThe superiority of the proposed method would be more effectively highlighted through an experimental comparison with radiance field-like method.", "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: The paper received consistently positive feedback following the authors’ rebuttal. Reviewers highlighted the paper's theoretical foundation (5fdJ, moMV, MVwc), the quality of the work (MVwc, 5fdJ) with promising results. While there were initial concerns about the lack of comparisons regarding data-driven algorithms and computational efficiency, these were effectively addressed in the rebuttal, leading to stronger overall reviews.

After carefully reviewing the paper, the reviews, and the rebuttal, The AC concurs with the reviewers' positive consensus and recommends acceptance of this paper.

For the camera-ready version, the authors should integrate all discussions from the rebuttal into the main paper and supplementary materials. Specifically, the authors should implement the following changes:
1. Experimental comparison between our approach and data-driven method (NeRF2)
2. Experimental comparison of computational efficiency between our approach and EMaC.
Decision: Accept (poster)