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Covariance-Aware MM-PGD for Mixed Near-/Far-Field Activity Detection

Xinjue Wang, Zhi-Yong Wang, Sergiy A. Vorobyov, Esa Ollila, Gayan Amarasuriya Aruma Baduge, Mojtaba Vaezi

arXiv:2609.10207v1eess.SP

TL;DR

Mixed NF/FF activity detection must account for structured near-field covariance that breaks the standard far-field snapshot approximation. The paper formulates a relaxed likelihood detector with joint MM-PGD updates; at high NF fractions it outperforms the strongest coordinate-wise baseline, while matching it in the all-FF case.

  • Problem

    Near-field users induce structured, device-specific covariance that breaks the standard snapshot approximation used by far-field detectors.

    Source: §I · p. 1

  • Method

    The paper formulates mixed NF/FF detection as relaxed maximum-likelihood estimation under a unified Rician model and uses MM-PGD to update the full activity vector jointly.

    Source: A. Related Work · p. 1, B. Contributions · p. 2

  • Results

    At ηNF = 1.00, MM-PGD achieves PMD = 0.010 while CWO-MMLE saturates near PMD ≈0.20, a 20× advantage; at ηNF = 0, both achieve PMD = 0.014.

    Source: B. SNR Sweep · p. 5, D. NF-Ratio Sweep · p. 5

  • Takeaways & Limitations

    The reported benefit comes from exploiting structured NF covariance rather than a generic optimization effect.

    Source: §VI · p. 5

    Developing scalable implementations for larger arrays and device pools, and reducing exact-likelihood cost in strongly coupled hybrid settings, remains future work.

    Source: §VI · p. 5

Abstract

from arXiv · show

Grant-free activity detection with mixed near-field (NF) and far-field (FF) devices is an important problem that can be addressed via covariance-based detectors. The difficulty is that NF users induce device-specific structured spatial covariances, whereas FF users are well approximated by isotropic covariances. Under a unified Rician model, we first formulate activity detection as a relaxed maximum-likelihood problem in the full LM-dimensional vectorized observation space. We then develop a covariance-aware majorization-minimization projected gradient descent (MM-PGD) detector. It updates the full activity vector jointly and avoids the per-coordinate high-rank subproblems that arise in the NF regime. Numerical results over SNR, antenna-count, and NF-ratio sweeps show that MM-PGD achieves up to 20x lower miss-detection probability than the strongest coordinate-wise baseline. The advantage is most pronounced at high NF ratios, while in the all-FF case MM-PGD performs on par with the strongest coordinate-wise baseline.

I. INTRODUCTION

Mixed near- and far-field activity detection requires accounting for structured, device-specific near-field covariance rather than relying on the usual far-field snapshot model. The paper formulates a unified covariance-aware detector and evaluates it across operating sweeps.

I. INTRODUCTION

  • Grant-free random access requires the base station to infer which small subset of registered devices is active from short uplink pilots.The setting is motivated by massive machine-type communications.Source: §I · p. 1
  • Mixed NF/FF populations break the standard snapshot approximation because NF users have structured spatial covariances, motivating detectors that use joint space–time covariance.The long-term channel means and covariances are assumed available at the base station.Source: §I · p. 1

A. Related Work

  • Prior covariance-based methods largely target FF fading, while NF-aware coordinate-wise solvers face heavier high-rank subproblems.The paper contrasts its global MM updates with isotropic-covariance Rician estimation and coordinate-wise structured NF covariance.Source: A. Related Work · p. 1

B. Contributions

  • The study emphasizes detector formulation, global MM-based updates, and detection performance rather than large-scale implementation details.It focuses on a compact single-cell detector under representative operating points.Source: B. Contributions · p. 2
  • The paper formulates mixed NF/FF detection as relaxed maximum-likelihood estimation under a unified Rician model and jointly updates the full activity vector with MM-PGD.The joint update avoids per-coordinate high-rank subproblems in the NF regime.Source: B. Contributions · p. 2
  • At high NF fractions, MM-PGD shows its largest gains over four representative baselines across SNR, antenna-count, and NF-ratio sweeps.The NF ratio is the fraction of near-field devices in the registered pool.Source: B. Contributions · p. 2

II. SYSTEM MODEL

The system model represents a single-cell uplink with registered devices that may be near-field or far-field. Their long-term channel statistics differ by regime and are assumed known to the detector.

II. SYSTEM MODEL

  • The modeled uplink has an M-antenna base station, N single-antenna devices, K ≪ N active devices, and unit-norm pilots of length L.The received-signal model includes additive noise with i.i.d. complex Gaussian entries.Source: §II · p. 2
  • Near-field channels use a spherical-wave multipath model with a line-of-sight component and scatterer-dependent steering vectors.The model describes each scatterer by its gain, distance, and angle.Source: §II · p. 2
  • The NF covariance is a sum of L_n rank-one terms and has rank r_n ≤ L_n < M, whereas FF devices use an isotropic covariance specialization.Large-scale fading is represented by g_n.Source: §II · p. 2
  • The base station is assumed to know each device’s NF/FF status and to have its long-term mean and covariance descriptors learned a priori.NF covariance learning is more demanding than fitting the FF scalar fading level; the detector does not jointly learn the environment or re-estimate covariances in each access slot.Source: §II · p. 2

A. Vectorized Observation and Joint Statistics

The model uses the vectorized observation because near-field covariance couples antenna and pilot dimensions, unlike the far-field snapshot simplification. Per-device statistics can be precomputed for online detection.

A. Vectorized Observation and Joint Statistics

  • The all-FF isotropic model permits the antenna observations to act as repeated snapshots, but this simplification breaks in mixed NF/FF settings.The detector therefore works directly with the vectorized observation.Source: A. Vectorized Observation and Joint… · p. 2
  • FF devices contribute Kronecker-structured covariance terms, while NF covariance couples antenna and pilot dimensions rather than decomposing into independent snapshot-domain terms.The aggregate covariance is structurally heterogeneous in a mixed population.Source: A. Vectorized Observation and Joint… · p. 3
  • Per-device mean vectors and covariance matrices are fixed by the long-term descriptors and pilots, so they can be precomputed for online detection.The online detector then uses the current observation with those precomputed quantities.Source: A. Vectorized Observation and Joint… · p. 3

B. Relaxed Maximum-Likelihood Detection

The detector begins with the likelihood of a binary activity pattern, then relaxes the activity variables to obtain a tractable bounded optimization problem.

B. Relaxed Maximum-Likelihood Detection

  • The likelihood is defined for a binary activity pattern using its associated mean and covariance.The negative log-likelihood is obtained by dropping a constant term.Source: B. Relaxed Maximum-Likelihood Detection · p. 3
  • Because exact ML searches over the combinatorial binary set, the formulation relaxes activity variables to values in [0, 1] and reuses the mean and covariance forms.This produces a relaxed maximum-likelihood problem.Source: B. Relaxed Maximum-Likelihood Detection · p. 3
  • The relaxed objective combines the log determinant of the covariance with a covariance-weighted residual between observation and mean.The optimization constrains each relaxed activity variable to [0, 1].Source: B. Relaxed Maximum-Likelihood Detection · p. 3

IV. MM-PGD DETECTOR

Coordinate-wise covariance updates are efficient under FF isotropic fading but become costly in the NF regime, motivating joint updates of the full activity vector. MM-PGD factorizes the covariance and uses projected updates before selecting the K largest activity entries.

A. Why Full-Vector Updates

  • The cited passage supports that under far-field isotropic fading Cn has Kronecker structure, expressed as Cn = gn(IM ⊗snsH; it does not establish coordinate-wise inverse updates or their role in solvers.Source: A. Why Full-Vector Updates · p. 3
  • NF device covariances have rank greater than one, so coordinate-wise updates lose the FF rank-one simplification and must operate on the full LM × LM covariance.The resulting cost becomes increasingly unattractive as the NF fraction grows.Source: A. Why Full-Vector Updates · p. 3
  • MM-PGD factorizes the aggregate covariance, computes a gradient step, chooses the step size by backtracking, and projects the activity vector onto [0, 1]^N.The algorithm then returns the indices of the K largest activity values.Source: A. Why Full-Vector Updates · p. 3

B. Gradient of the NLL

The NLL gradient combines expected covariance contribution, observed contribution from whitened data, and a LoS-mean term; its components are coupled through the whitened residual.

B. Gradient of the NLL

  • The residual is defined as rγ ≜ y − µγ, and the whitened residual is v ≜ Σγ^-1 rγ.Source: B. Gradient of the NLL · p. 3
  • Each gradient component combines expected covariance contribution, observed contribution via whitened data, and a term accounting for the LoS mean.All components depend on v = Σγ^-1 rγ, coupling activity variables through Σγ.Source: B. Gradient of the NLL · p. 3

C. Majorization–Minimization Update

The detector uses a majorization–minimization surrogate to produce a projected gradient update with monotone objective descent. The direct implementation evaluates the covariance terms by Cholesky factorization, with a lower-cost exact reduction in the all-FF case.

C. Majorization–Minimization Update

  • MM-PGD uses a quadratic upper bound and minimizes its surrogate over the box constraint to obtain a closed-form projected gradient step.The MM principle yields a monotonically descending update.Source: C. Majorization–Minimization Update · p. 3
  • Backtracking doubles the step parameter until sufficient decrease holds, guaranteeing monotone descent; objective, gradient, and trace terms use direct Cholesky factorization.The first iteration starts with Lt = 1, and later iterations warm-start from the previously accepted value.Source: C. Majorization–Minimization Update · p. 3
  • The supplied figure captions specify PMD versus SNR, antenna count, and NF ratio, but do not state plotted outcomes.Source: Figure 1 · p. 4, Figure 2 · p. 4, Figure 3 · p. 4
  • The mixed and all-NF cases use exact factorization at cost O((LM)^3), while the all-FF case reduces to an L × L factorization with dominant cost O(L^3).For (L, M) = (20, 48), the general factorization is of a 960 × 960 matrix.Source: C. Majorization–Minimization Update · p. 4

V. NUMERICAL RESULTS

The numerical study uses a mixed NF/FF single-cell setting and evaluates detection with miss-detection probability. Its comparisons use four baselines and an oracle top-K operating point.

A. Setup

  • The simulations use a mixed NF/FF single-cell setting at 3 GHz with a 500 m cell radius.Source: A. Setup · p. 4
  • Each NF device has four scatterers, and the baseline setup uses an NF fraction of 0.5 and a −5 dB LoS-to-scattering power ratio.The NF ratio is the fraction of NF devices in the registered device pool.Source: A. Setup · p. 4
  • The study averages 500 trials, compares MM-PGD with CWO, CWO-MMLE, CL-MP, and SBL, and uses PMD with oracle top-K support recovery.CWO-MMLE is identified as the strongest competing covariance-based baseline.Source: A. Setup · p. 5

B. SNR Sweep

MM-PGD performs on par with CWO-MMLE in the all-FF limit and gains as the NF ratio grows. The sweeps report lower miss-detection probability across SNR and antenna counts, with the largest reported gap at a high NF ratio.

B. SNR Sweep

  • At ηNF = 1.00, CWO-MMLE reaches PMD ≈0.20 while MM-PGD improves to 0.010, a 20× advantage.MM-PGD maintains the lowest PMD across the full SNR range and all three NF compositions.Source: B. SNR Sweep · p. 5

C. Antenna-Count Sweep

  • MM-PGD reduces PMD from 0.168 at M = 16 to 0.007 at M = 64, a 24× reduction, while CWO-MMLE remains higher.The sweep fixes ηNF = 0.5; CWO-MMLE decreases from 0.272 to 0.063 as M grows.Source: C. Antenna-Count Sweep · p. 5

D. NF-Ratio Sweep

  • As ηNF rises to 1, CWO-MMLE degrades to PMD = 0.213 while MM-PGD remains nearly flat at 0.024.At ηNF = 0, both methods achieve PMD = 0.014.Source: D. NF-Ratio Sweep · p. 5

VI. CONCLUSION

  • MM-PGD performs on par with CWO-MMLE in the all-FF limit and achieves clear gains as the NF ratio grows.The conclusion attributes the gains to exploiting structured NF covariance rather than a generic optimization effect.Source: §VI · p. 5
  • More scalable implementations for larger arrays and device pools, including lower-cost exact likelihood evaluation in strongly coupled hybrid NF/FF settings, remain future work.Source: §VI · p. 5

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