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Reliable Near-Field Multi-User Positioning Informed by Two-Stage MUSIC
Jiaying Li, Haifeng Wen, Changsheng You, Yuanwei Liu, Hong Xing
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%
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Summary
This paper proposes MUSIC-Net, a deep learning framework for reliable near-field multi-user positioning in mixed LoS/NLoS environments. It integrates two-stage MUSIC to isolate LoS signal subspaces and estimate surrogate distances, bypassing complex NLoS parameter estimation. Additionally, it employs Split Conformal Prediction (SCP) to provide statistically guaranteed confidence sets for user positions, improving reliability and uncertainty quantification compared to existing benchmarks.
Entities (8)
Relation Signals (5)
MUSIC-Net → uses → Two-stage MUSIC
confidence 95% · MUSIC-Net, an end-to-end near-field positioning deep learning (DL) framework informed by two-stage MUltiple SIgnal Classification (MUSIC)
MUSIC-Net → uses → Split Conformal Prediction
confidence 95% · we introduce split conformal prediction (SCP) to move beyond point-estimation-based positioning... The proposed framework directly recovers multi-user positions
Two-stage MUSIC → isolates → LoS-related signal subspace
confidence 90% · embeds the two-stage MUSIC objects into training to isolate the LoS-related signal subspace
Split Conformal Prediction → provides → statistically guaranteed confidence sets
confidence 90% · SCP... construct a collection of confidence sets... jointly cover the true positions of the KK users with probability at least 1−α
MUSIC-Net → achieves → lower mean positioning error
confidence 85% · MUSIC-Net achieves lower mean positioning error (MPER) than existing benchmarks
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Abstract
Abstract:Near-field localization is a promising technique for high-resolution multi-user positioning in future wireless systems, but its performance is often degraded by scattering-induced coherent propagation. Existing near-field localization methods, which require separate parameter estimation and path/source association, suffer from high computation overhead and accumulated errors, and usually do not provide any guarantee on reliability. In this paper, we propose \emph{MUSIC-Net}, an end-to-end near-field positioning deep learning (DL) framework informed by two-stage MUltiple SIgnal Classification (MUSIC) in mixed line-of-sight (LoS) and non-LoS (NLoS) multi-path scenarios, which embeds the two-stage MUSIC objects into training to isolate the LoS-related signal subspace and to identify a surrogate distance. The proposed framework directly recovers multi-user positions without the need for involved NLoS parameter estimation or path/source association. Furthermore, we introduce split conformal prediction (SCP) to move beyond point-estimation-based positioning towards statistically guaranteed (confidence) set estimation for all users. Numerical results show that the proposed MUSIC-Net achieves lower mean positioning error (MPER) than existing benchmarks and yields tighter SCP-calibrated prediction regions, demonstrating both accurate LoS localization and efficient uncertainty quantification (UQ) in coherent multi-path environments.
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- Source: https://arxiv.org/abs/2609.09409v1
- Canonical: https://arxiv.org/abs/2609.09409v1
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Reliable Near-Field Multi-User Positioning Informed by Two-Stage MUSICThanks: The work of H. Xing was supported in part by the Guangdong Basic and Applied Basic Research Foundation under Grant 2025A1515010123, and in part by the Guangdong Provincial Key Lab of Integrated Communication, Sensing and Computation for Ubiquitous Internet of Things under Grant 2023B1212010007. Jiaying Li∗, Haifeng Wen∗, Changsheng You § , Yuanwei Liu†, and Hong Xing∗‡ Affiliation: Affiliation: ∗ The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China § Southern University of Science and Technology, Shenzhen, China † The University of Hong Kong, HK SAR, China ‡ The Hong Kong University of Science and Technology, HK SAR, China E-mails: jli989, hwen904@connect.hkust-gz.edu.cn, youcs@sustech.edu.cn, yuanwei@hku.hk, hongxing@ust.hk Affiliation: Abstract Near-field localization is a promising technique for high-resolution multi-user positioning in future wireless systems, but its performance is often degraded by scattering-induced coherent propagation. Existing near-field localization methods, which require separate parameter estimation and path/source association, suffer from high computation overhead and accumulated errors, and usually do not provide any guarantee on reliability. In this paper, we propose MUSIC-Net, an end-to-end near-field positioning deep learning (DL) framework informed by two-stage MUltiple SIgnal Classification (MUSIC) in mixed line-of-sight (LoS) and non-LoS (NLoS) multi-path scenarios, which embeds the two-stage MUSIC objects into training to isolate the LoS-related signal subspace and to identify a surrogate distance. The proposed framework directly recovers multi-user positions without the need for involved NLoS parameter estimation or path/source association. Furthermore, we introduce split conformal prediction (SCP) to move beyond point-estimation-based positioning towards statistically guaranteed (confidence) set estimation for all users. Numerical results show that the proposed MUSIC-Net achieves lower mean positioning error (MPER) than existing benchmarks and yields tighter SCP-calibrated prediction regions, demonstrating both accurate LoS localization and efficient uncertainty quantification (UQ) in coherent multi-path environments. Index Terms: Near-field, localization, non-line-of-sight (NLoS), split conformal prediction (SCP). I Introduction With rapid development of extremely large-scale antenna arrays and higher-band radio-frequency (RF) communications, next-generation wireless systems are envisioned to operate in the radiative near-field regime featuring spherical wavefronts [1], which entails involved multi-path channel estimation. Meanwhile, unlike conventional far-field localization, which mainly resolves angular information under the planar-wave assumption, near-field localization benefits from such high-resolution estimation of both direction of arrival (DoA) and distance of source targets, thus enabling high-precision near-field multi-user positioning [2]. Existing near-field localization methods mainly comprise beamforming [3], compressive sensing [4], subspace-based [5, 6, 7, 8], and learning-based approaches [9, 10, 11, 12, 13]. Among them, subspace-based methods such as 2D MUltiple SIgnal Classification (MUSIC) and its variants are particularly appealing, as they provide both high resolution and physical interpretability [5, 6, 7, 8]. In particular, 2D MUSIC [5] performed near-field localization through a two-dimension angle-range spectral search, whereas its multi-stage variants, including two-step MUSIC [6], RD-MUSIC [7], and two-stage MUSIC [8], reduced the computational overhead by decomposing the original joint angle-range search into multiple stages of one-dimension search. However, two-step MUSIC [6] and RD-MUSIC [7] require restrictive antenna spacing conditions, while two-stage MUSIC [8] avoids imposing this limitation by introducing a surrogate distance for angular search, which decomposed the original two-dimension angle-distance search into two sequential one-dimension searches. On another front, recent deep learning (DL) based methods improve positioning robustness in challenging propagation conditions such as low signal-to-noise ratio (SNR) regimes, model mismatch, and multi-path environments [9, 10, 11]. Among these prior arts, end-to-end (E2E) approaches aim for directly learning the mapping from measurements to positions using, e.g., convolutional neural networks (CNNs) [9, 10] and Transformers [11], but often rely on highly parameterized DL models with limited generalization capability. Meanwhile, the model-based DL methods [12, 13] design classical signal processing algorithms informed neural-network architectures, thereby combining data-driven flexibility with physical interpretability. Prototypical examples include NF-SubspaceNet [12], which adopted a surrogate covariance matrix for subsequent 2D MUSIC processing, and DCD-MUSIC [13], which learned separate surrogate covariance matrices for angle and distance estimation, respectively. However, all the above methods may suffer from scattering-induced multi-paths, of which the line-of-sight (LoS) path and the non-line-of-sight (NLoS) paths associated with the same user can be highly coherent, thus distorting the structure of the LoS-only induced subspace and making conventional localization methods demanding if not impossible [14]. Existing methods for coherent-source localization mainly start with estimating all path-related parameters, and then performing further association schemes to group those multi-paths associated with the same source in a multi-user setting [15]. However, subsequent LoS-path identification for each group entails additional computation overhead and possibly accumulated positioning error. These limitations motivate a near-field multi-user positioning scheme that directly targets at LoS parameters estimation bypassing the coherent NLoS estimation and subsequent path/source association steps. Furthermore, the reliability of the positioning scheme also plays a crucial role especially in safety-critical localization, such as autonomous driving and tactile networks. Split conformal prediction (SCP) [16] is known to provide a model-agnostic, distribution-free and training-free uncertainty quantification (UQ) framework that moves beyond point-parameter estimation by constructing post-hoc confidence sets with statistically guaranteed true-position coverage. For instance, SCP has recently found its applications in, e.g., channel prediction [17], wireless resource allocation [18], and beam selection [19]. Based on the motivations described above, the contributions of this paper are summarized as follows: • We propose MUSIC-Net, a near-field multi-user positioning framework in coherent multi-path conditions, which introduces two learning modules to estimate the LoS-related subspace and the surrogate distance, respectively, while preserving the interpretability of the two-stage MUSIC, enables users’ positioning without explicit NLoS parameter estimation or path/source association. • We integrate SCP into the proposed multi-user positioning framework to construct a robust position set against uncertainties underlying MUSIC-Net inference with statistical coverage guarantees for all users. • Numerical results show that MUSIC-Net achieves the lowest mean positioning error (MPER) across different SNR regimes and yields the tightest prediction regions among the benchmarks, demonstrating reliable and efficient UQ in coherent multi-path environments. I System Model and Reliable Problem Formulation I-A System Model As shown in Fig. 1, we consider a multi-user narrowband near-field channel for uplink localization, where an access point (AP) equipped with a uniform linear array (ULA) with M~=2M+1 M=2M+1 antennas receives signals from K single-antenna user equipments (UEs), denoted by k∈1,…,K≜[K]k∈\1,...,K\ [K]. The users are assumed to lie in the near-field region, i.e., the distance r from users to the center of the ULA satisfying r<2D2/λr<2D^2/λ, where D=(M~−1)dD=( M-1)d is the array aperture; d is the antenna spacing; and λ is the carrier wavelength. Fig. 1: An illustration of the system for near-field multi-user positioning with multiple coherent NLoS paths. Under the narrowband near-field assumptions, the steering vector (θ,r)∈ℂM~×1 a(θ,r) M× 1 for a target user at DoA θ and distance r is given by [1] (θ,r)=[e−j2πλ‖−M‖2,…,e−j2πλ‖−M‖2]T, a(θ,r)= [e^-j 2πλ\| p- s_-M\|_2,…,e^-j 2πλ\| p- s_M\|_2 ]^T, (1) where =r[cosθ,sinθ]T p=r[ θ,\, θ]^T and n=[0,nd]T s_n=[0,nd]^T denote the positions of the user and the n-th antenna element, respectively. We assume that, for the signal transmitted by the k-th user, there exists one LoS propagation path and Lk≥0L_k≥ 0 NLoS ones generated by scatterers. Let sk(t)s_k(t) denote the signal transmitted by user k∈[K]k∈[K] at time slot t∈1,…,T≜[T]t∈\1,…,T\ [T], where T is the total number of time slots with T>KT>K. We also assume that sk(t)s_k(t) is independent, denoted by sk(t)∼(0,Pk)s_k(t) (0,P_k). Then, the received signal (t)∈ℂM~×1 $y$(t) M× 1 can be expressed as (t)=LoS(t)+NLoS(t)+(t) y(t)= y_LoS(t)+ y_NLoS(t)+ n(t) (2) where LoS(t)=∑k=1K(θk0,rk0)sk(t), y_LoS(t)= _k=1^K a( _k0,r_k0)s_k(t), (3) and NLoS(t)=∑k=1K∑l=1Lk(θkl,rkl)cklsk(t), y_NLoS(t)= _k=1^K _l=1^L_k a( _kl,r_kl)c_kls_k(t), (4) respectively, where θk0 _k0 and rk0r_k0 denote the DoA and distance associated with the LoS path of the k-th user; θkl _kl, rklr_kl, and cklc_kl denote the DoA, distance, and complex reflection coefficient associated with the l-th NLoS path of the k-th user, where l∈1,..,Lk≜[Lk]l∈\1,..,L_k\ [L_k] and k∈[K]k∈[K]; and (t)∼(,σ2) n(t) ( 0,σ^2 I) is the additive white Gaussian noise (AWGN). Accordingly, the covariance matrix of (t) y(t) is given by =[(t)H(t)]=LoS+NLoS+cross+σ2, R=E[ y(t) y^H(t)]= R_LoS+ R_NLoS+ R_cross+σ^2 I, (5) where LoS=[LoS(t)LoSH(t)] R_LoS=E[ y_LoS(t) y_LoS^H(t)], NLoS=[NLoS(t)NLoSH(t)] R_NLoS=E[ y_NLoS(t) y_NLoS^H(t)], and cross=[LoS(t)NLoSH(t)]+[NLoS(t)LoSH(t)] R_cross=E[ y_LoS(t) y_NLoS^H(t)]+E[ y_NLoS(t) y_LoS^H(t)] denote the covariance matrices of the LoS and NLoS components and their cross-correlation term, respectively. In practice, one can approximate the true covariance matrix R by the sample covariance matrix (SCM) as ^=1TH R= 1T Y Y^H (6) where =[(1),…,(T)]∈ℂM~×T Y=[ y(1),…, y(T)] M× T is the observations matrix over T consecutive time slots. I-B Reliable Multi-User Positioning Based on the observation model in (2), we aim for estimating all K users’ location-related parameters associated with the LoS paths, i.e., LoS=(θk0,rk0)k=1K _LoS=( _k0,r_k0)_k=1^K. We assume that the number of users K is known a priori, which can be estimated by conventional methods such as Akaike information criterion (AIC) and minimum description length (MDL) [12]. The parameters associated with the NLoS paths, i.e., NLoS=((k)k=1K,σ2) _NLoS=(( _k)_k=1^K,σ^2) where k=(θkl,rkl,ckl)l=1Lk _k=( _kl,r_kl,c_kl)_l=1^L_k, are treated as nuisance parameters. To estimate the location-related parameters LoS $ $_LoS, the standard approach is to apply subspace-based methods such as the 2D MUSIC algorithm [5] and its variants [8, 7] directly to the SCM R, which generally work well in LoS only or LoS dominant scenarios. However, in the considered multi-path scenario, the term NLoS+cross R_NLoS+ R_cross in (5) deteriorates the subspace structure induced solely by the LoS components. Besides, the estimated locations ^k p_k of all users k∈[K]k∈[K], where ^k=r^k0[cos(θ^k0),sin(θ^k0)]T p_k= r_k0[ ( θ_k0), ( θ_k0)]^T, also deviate from their true positions k,k∈[K] p_k,k∈[K], due to random factors including, e.g., remained NLoS traces, AWGN, and the positioning algorithms due to the limited number of observations. In this paper, we develop in scenarios with multiple coherent NLoS paths a multi-user positioning scheme that is robust against uncertainty induced by channel modeling or localization methods. Specifically, we are interested in constructing a collection of confidence sets (^k)k=1K\C( p_k)\_k=1^K such that they jointly cover the true positions of the K users with probability at least 1−α1-α, i.e., ℙ(∃π∈ΠK:k∈(^π(k)),∀k∈[K])≥1−α,P\! (∃\,π∈ _K: p_k ( p_π(k)),~∀ k∈[K] )≥ 1-α, (7) where ΠK _K denotes the set of all permutations of [K][K], and the probability is taken with respect to the joint distribution of the test data and calibration data, which will be elaborated in later sections. I Two-stage MUSIC Informed MUSIC-Net To address the near-field multi-user positioning with multiple coherent NLoS paths, we propose MUSIC-Net, a deep learning framework informed by the two-stage MUSIC algorithm [8]. I-A Motivation In standard two-stage MUSIC algorithm [8], since the LoS paths of the users dominate uplink transmissions, the covariance matrix of the received signal R, estimated by the SCM $R$, can be well approximated by =∑k=1KPk(θk0,rk0)H(θk0,rk0)+σ2 R= _k=1^KP_k a( _k0,r_k0) a^H( _k0,r_k0)+σ^2 I in (5), and therefore the eigenspace of R is decomposed, by eigenvalue decomposition (EVD), into a signal subspace s $U$_s and a noise subspace n $U$_n as assumed in conventional far-field MUSIC algorithm. In a near-field setting, the MUSIC based multi-user localization problem thus translates into a two-dimension search for the top-K largest peaks based on the spectrum function given by P(θ,r)=1‖H(θ,r)n‖2. P(θ,r)= 1\| a^H (θ,r ) U_n\|_2. (8) Instead of an exhaustive two-dimension search over [−π2,π2]×[rmin,rmax][- π2, π2]×[r_ ,r_ ], two-stage MUSIC decouples the above search into two one-dimension spectral searches, and performs them in the following two stages. 1) DoA search: Based on a surrogate distance r0r_0, which is obtained by either maximizing the minimum correlation (MMC) rule that maximizes the worst-case steering vector correlation or exceeding the minimum correlation threshold (EMCT) rule that enforces this correlation to exceed a prescribed threshold [8], estimate the DoAs of the users by searching for the K peaks with r replaced by r0r_0 in (8); and 2) Distance search: Based on the estimated DoAs θ^k0 θ_k0, k∈[K]k∈[K], estimate the distance corresponding to each user k∈[K]k∈[K] by searching for one peak with θ replaced by θ^k0 θ_k0 in (8). However, we cannot trivially tailor the two-stage MUSIC algorithm to multi-user localization with NLoS, as the structure of the R becomes fundamentally different from LoS R_LoS due to involved unknown NLoS parameters (cf. (5)), thus disabling direct estimation of the noise subspace n $U$_n, which motivates a novel data-driven design as introduced in the next subsection. I-B MUSIC-Net Training We propose a novel multi-user localization scheme enabled by MUSIC-Net, which substitutes two dedicated trainable modules for the estimated noise subspace ^n $U$_n and the surrogate distance r^0 r_0 in the standard two-stage MUSIC, respectively. As illustrated in Fig. 2, prior to performing the two-stage MUSIC procedure, we adopt MUSIC-Net training that consists of 1) a LoS feature extractor module and 2) a surrogate distance estimation module to obtain an LoS-plus-noise subspace estimation ^n $U$_\!n and a surrogate distance estimation r^0 r_0, respectively, from the input SCM $R$. Fig. 2: The proposed two-stage MUSIC informed MUSIC-Net. Specifically, the LoS feature extractor, denoted by (⋅,1) f(·; W_\!1) with trainable parameters 1 W_\!1, maps the input R to a complex-valued feature matrix 1∈ℂM~×M~ X_\! W_\!1 M× M, i.e., (^,1)=1 f( R; W_\!1)= X_\! W_\!1, which predicts denoised signals induced by a virtual subspace dominated by the LoS paths. Then, we perform the EVD on 11H X_\! W_\!1 X_\! W_\!1^H: 11H=[^s,^n]^[^s,^n]H, X_\! W_\!1 X_\! W_\!1^H= [ U_\!s,~ U_\!n ] [ U_\!s,~ U_\!n ]^H, (9) where ^s∈ℂM~×K U_\!s M× K contains the eigenvectors associated with the K largest eigenvalues spanning over the estimated LoS-induced subspace; ^n∈ℂM~×(M~−K) U_\!n M×( M-K) contains the remaining eigenvectors; and is a diagonal matrix containing all eigenvalues in descending order. Next, we adopt the subspace distance [20] between the target LoS-induced signal subspace s $U$_\!s, which is obtained by EVD on LoSLoSH A_LoS A_LoS^H with LoS A_LoS composed of steering vectors associated with only LoS paths of all K users, and the learned LoS-induced signal subspace ^s $U$_\!s as the supervised (empirical) loss given by ℒ1(1)=1||∑k=1Karccos(σk(sH^s)), aligned L_1 ( W_\!1 )= 1|D| _D _k=1^K \! ( _k\! ( U_s^H U_\!s ) ), aligned (10) where D is the training dataset composed of |||D| labelled data (^,LoS)( R, _LoS), and σk() _k( A) denotes the k-th largest singular value of the matrix A. It is worth noting that there are other losses like SCM-fitting [20] and affine invariant distance [21], which are, however, sensitive to SNR of the received signal, causing possibly bad generalization. The surrogate distance estimator, denoted by g(⋅,2)g(·; W_\!2) with trainable parameters 2 W_\!2, maps the SCM R to a surrogate distance r^0 r_0, i.e., r^0=g(^,2). r_0=g( R; W_\!2). To ensure that the surrogate LoS-induced subspace ~s $U$_\!s associated with a single distance r^0 r_0, which is obtained by EVD on r0r0H $A$_r_0 $A$_r_0^H with r0=[(θ10,r^0),…,(θK0,r^0)] A_r_0= [ a( _10, r_0),…, a( _K0, r_0) ], aligns well with the estimated subspace ^s U_\!s associated with the users’ K distances, the empirical training loss is therefore chosen as the subspace distance between ~s U_\!s and ^s U_\!s, which is given by ℒ2(1,2)=1||∑k=1Karccos(σk(~sH^s)). aligned L_2 ( W_\!1, W_\!2 )= 1|D| _D _k=1^K ( _k\! ( U_\!s^H U_\!s ) ). aligned (11) The model parameters 1 W_\!1 and 2 W_\!2 of the two modules are trained jointly, by minimizing the following combined multiple objectives: min1,2ℒ1(1)+λℒ2(1,2), _ W_\!1, W_\!2\;L_1( W_\!1)+λ\,L_2( W_\!1, W_\!2), (12) where λ>0λ>0 is a hyperparameter that balances the loss induced by noisy NLoS paths and by surrogate distance estimation. In the inference stage, the MUSIC-Net takes an SCM sample $R$ as an input and outputs the estimated ^n $U$_\!n and r^0 r_0, which are then fed into the two-stage MUSIC to obtain the estimates of DoA θ^k0k=1K\ θ_k0\_k=1^K and distance r^k0k=1K\ r_k0\_k=1^K, resulting in estimated positions ^k0=r^k0[cos(θ^k0),sin(θ^k0)]T p_k0= r_k0[ ( θ_k0),\, ( θ_k0)]^T for all users in [K][K]. However, like most DL-based localization methods, the proposed localization scheme produces only point estimates of users’ positions without any robustness guarantee against uncertain behavior of MUSIC-Net due to limitations of training. In the next section, we augment point estimates with confidence sets. IV SCP based Positioning Calibration IV-A Preliminary Among the various conformal prediction methods, SCP [16] features simple implementation and computational efficiency, thus being suitable for post-hoc calibration [16]. Suppose that we have a calibration dataset cal =(X(i),Y(i))i=1nD_cal =\(X^(i),Y^(i))\_i=1^n, where X(i)∈X^(i) and Y(i)∈Y^(i) , drawn independently and identically distributed (i.i.d.) from a distribution, denote the feature and the corresponding label, respectively. Next, given a prescribed nonconformity score function s:×→ℝs:X×Y associated with a predictive model, define the calibration score for each data point (X(i),Y(i))(X^(i),Y^(i)) by S(i)=s(X(i),Y(i))S^(i)=s (X^(i),Y^(i) ), which measures how poorly the Y(i)Y^(i) conforms to the X(i)X^(i). Then, for a target mis-coverage level α∈(0,1)α∈(0,1) and a test feature X(n+1)X^(n+1), SCP constructs the following prediction (confidence) set: (X(n+1))=y∈:s(X(n+1),y)≤q^1−α,C(X^(n+1))=\y :s(X^(n+1),y)≤ q_1-α\, (13) which contains labels whose scores do not exceed the empirical quantile q^1−α q_1-α given by q^1−α=Q((1−α)(1+1/n),S(i)i=1n), q_1-α=Q\! ((1-α) (1+1 / . -1.2ptn );\S^(i)\_i=1^n ), (14) where Q(γ,S(i)i=1n)Q(γ;\S^(i)\_i=1^n) is the ⌈nγ⌉ nγ -th smallest value in the set S(i)i=1n\S^(i)\_i=1^n. Under the assumption of exchangeable calD_cal and (X(n+1),Y(n+1))(X^(n+1),Y^(n+1)), the marginal coverage of the SCP set (13) is proven to satisfy ℙ(Y(n+1)∈(X(n+1)))≥1−αP\! (Y^(n+1) (X^(n+1)) )≥ 1-α [16]. IV-B Uncertainty Quantification for MUSIC-Net We provide in this subsection a UQ method for users’ positions leveraging SCP. First, define a held-out calibration dataset composed of n samples, denoted by cal=^(i),(θk0(i),rk0(i))k=1K(i)i=1nD_cal=\ R^(i),\( _k0^(i),r_k0^(i))\_k=1^K^(i)\_i=1^n, where (θk0(i),rk0(i))( _k0^(i),r_k0^(i)) is the ground-truth DoA and distance associated with the LoS path of user k∈[K(i)]k∈[K^(i)] in the i-th sample. Next, we define the nonconformity score function s(⋅,⋅)s(·,·) as s(^,(θk0,rk0)k=1K)=maxk∈[K]‖π⋆(k)−^k‖2, s ( $R$,\( _k0,r_k0)\_k=1^K )= _k∈[K]\| $p$_π (k)- $p$_k\|_2, (15) where ^k $p$_k is the estimated position obtained by the proposed MUSIC-Net based positioning in Section I, and π⋆∈ΠKπ ∈ _K denotes the optimal permutation obtained by matching the estimated and ground-truth positions of the K users leveraging the permutation-based matching method, e.g., the Hungarian algorithm [22]. Then, for a newly observed K-user SCM $R$, we construct the confidence region for each estimated user position based on the empirical quantile q^1−α q_1-α obtained via (14) as (^k)=∈ℝ2∣‖−^k‖2≤q^1−α,k∈[K],C( $p$_k)= \ p ^2 \| p- p_k \|_2≤ q_1-α \, k∈[K], (16) which corresponds to a circular region centered on ^k p_k with radius q^1−α q_1-α. Finally, we derive the following joint coverage guarantee for the predicted regions (^k)k=1K\C( $p$_k)\_k=1^K. Proposition IV.1 ( Joint coverage guarantee) Assume that the calibration dataset calD_cal and an upcoming test sample (^,(θk0,rk0)k=1K)( R,\( _k0,r_k0)\_k=1^K) are exchangeable. Then, the predicted regions (^k)k=1K\C( p_k)\_k=1^K constructed in (16) satisfy (7). Proof: By Theorem 1 [16], maxk∈[K]‖π⋆(k)−^k‖2≤q^1−α _k∈[K]\| p_π (k)- p_k\|_2≤ q_1-α holds with probability at least 1−α1-α. Hence, (16) gives k∈(^k) p_k ( p_k) for all k∈[K]k∈[K]. ∎ V Experiment results The experiment considers near-field multi-user localization via uplink transmissions with a ULA of M~=25 M=25 antennas operating at carrier frequency f=1f=1 GHz with half-wavelength antenna spacing. The number K of users is randomly drawn from K∈1,2,3,4K∈\1,2,3,4\. For each user k∈[K]k∈[K], its DoA θk0 _k0 is drawn uniformly as θk0∼(−60∘,60∘) _k0 (-60 ,60 ), with a minimum angular separation constraint mini≠j|θi0−θj0|≥10∘ _i≠ j| _i0- _j0|≥ 10 , and its distance (in meters) rk0r_k0 is generated uniformly over a fan-shaped sector, denoted by rk02∼(52,552)r_k0^2 (5^2,55^2). For the NLoS paths l∈[Lk]l∈[L_k], k∈[K]k∈[K], the number LkL_k of NLoS paths associated with user k is taken from 0,1,2\0,1,2\; the DoA θkl _kl is generated by θkl∼(θk0−5∘,θk0+5∘) _kl ( _k0-5 , _k0+5 ) with a minimum angular separation no larger than 0.5∘0.5 ; the corresponding distance rklr_kl is generated by rkl2∼((rk0−10)2,(rk0+10)2)r_kl^2 ((r_k0-10)^2,(r_k0+10)^2); and the complex reflection coefficient is generated as ckl=ρklejϕklc_kl= _kle^j _kl with the phase following ϕkl∼(0,2π) _kl (0,2π) and the amplitude following ρkl∼(0,1) _kl (0,1), respectively. Based on the generated LoS and NLoS parameters, i.e., LoS _LoS and NLoS _NLoS, and given a prescribed transmit SNR=10log10(∑k=1KPk/(Kσ2))SNR=10 _10( _k=1^KP_k/(Kσ^2)) with equal source powers (P1=P2=⋯=PKP_1=P_2=·s=P_K), the SCM R is synthesized according to (6) with T=25T=25 snapshots. As a result, we generate a total of 1.25×1051.25× 10^5 data points (^,LoS)( R, _LoS) with 10510^5 for training and 1.25×1041.25× 10^4 each for calibration and test in SCP. We employ ResNet-based networks [23] as the neural architectures for the LoS feature extractor and surrogate distance estimator. Specifically, a two-channel tensor is formed by the real and the imaginary parts of the SCM $R$ as the input, and then is processed by a 3×33× 3 convolutional layer followed by N residual stages to extract high-level features. Each residual stage consists of a shortcut branch with a 1×11× 1 convolutional layer and a residual branch with L residual blocks, where each residual block contains two “ReLU–3×33× 3 convolution” layers. The extracted high-level feature is further passed through a ReLU activation, average pooling, and a fully connected layer. The networks are trained using the Adam optimizer [24] with a mini-batch size of 128128. The test accuracy is evaluated by the MPER, computed by averaging (15) over all users and all samples in the test dataset. We consider the following benchmark positioning schemes: 2D MUSIC, which directly estimates the K users’ positions via two-dimension angle-distance spectral search; Two-phase MUSIC [25], which applies spatial smoothing to decorrelate coherent DoAs but does not decorrelate coherent ranges; NF-SubspaceNet [12], which employs an autoencoder to learn a surrogate covariance matrix for subsequent 2D MUSIC-based localization; and DCD-MUSIC [13], which employs two autoencoders to learn surrogate covariance matrices for cascaded DoA and range estimation, and further trains the cascaded architecture using positioning errors. Note that Two-phase MUSIC, NF-SubspaceNet, and DCD-MUSIC are aimed for estimating all parameters including both LoS _LoS and NLoS _NLoS. To enable a fair comparison with the LoS-positioning objective considered in this paper, we introduce the following two evaluation approaches for these schemes: Indirect LoS, by which all path-related parameters are first estimated and then grouped user-wise using the path association strategy in [15, Algorithm 1]; then, for each associated path group, the delay-and-sum algorithm [26] is applied to select the LoS angle-distance pair estimates of the corresponding user; and Direct LoS, by which the outputs of the original schemes are replaced by E2E predictions of the multi-user positions. First, we evaluate the point estimation accuracy using the MPER between the estimated position p and the ground truth p, defined as ‖−^‖2E\| p- p\|_2, where the expectation is taken over all samples in the test dataset. Fig. 3 shows that the proposed MUSIC-Net outperforms all other methods across the entire SNR range. Moreover, the direct-LoS baselines generally outperform their indirect-LoS counterparts, indicating that directly targeting the LoS-positioning objective is more effective than first estimating all paths and then extracting the LoS component, due to the accumulated errors introduced in path estimation and post-processing. Fig. 3: The MPER versus the SNR. (a) 1−α=0.71-α=0.7 (b) 1−α=0.81-α=0.8 (c) 1−α=0.91-α=0.9 Fig. 4: Box plots for the radius q^1−α q_1-α with different target coverage levels (1−α1-α), SNR=5SNR=5 dB, and ⋆ marks the empirical mean over 100100 Monte Carlo trials. Fig. 5: Visualization of the regions (^k)k=1K\C( p_k)\_k=1^K constructed by (16) with SNR=5SNR=5 dB, α=0.1α=0.1, K=3K=3, and the region size obtained by different schemes. Next, Fig. 5 illustrates the confidence regions (^k)k=1K\C( p_k)\_k=1^K constructed by (16) for all users in a typical sample. It shows that the proposed MUSIC-Net produces a much smaller predicted region covering the true location p compared to other benchmark schemes. Finally, we evaluate the efficiency of the confidence regions (^k)k=1K\C( p_k)\_k=1^K under different target coverage levels 1−α1-α by comparing the value of radius q^1−α q_1-α across different benchmarks in Fig. 4. The results show that the proposed MUSIC-Net yields the smallest radius q^1−α q_1-α among all benchmarks, indicating that it achieves the tightest predicted regions while satisfying the prescribed coverage level described in (7). Moreover, increasing the target coverage level 1−α1-α comes at the cost of enlarging the prediction set. VI Conclusion This paper proposed MUSIC-Net, an E2E DL framework informed by two-stage MUSIC for reliable near-field multi-user positioning in mixed LoS and NLoS multipath environments. 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