Channel Estimation Knowledge Map (CEKM)

• CEKM is an environment-aware mapping that links physical locations to predicted channel state information, reducing real-time pilot overhead.
• It leverages offline construction, spatial prediction, and image super-resolution techniques to reconstruct fine-grained channel responses effectively.
• CEKM supports diverse applications including beamforming, interference cancellation, and mobility management through measurement-, model-, and hybrid-based paradigms.

Searching arXiv for recent CEKM/CKM papers to ground the article. Channel Estimation Knowledge Map (CEKM) denotes a site-specific, environment-aware mapping or database that associates physical location—and in some formulations transmitter and receiver positions, time, frequency, environment representation, or transceiver configuration—with channel knowledge useful for estimation, prediction, and control. In the literature, CEKM is sometimes defined as the estimation-oriented component of the broader Channel Knowledge Map (CKM), storing or predicting fine-grained channel impulse response or sufficient statistics, and sometimes as the general map $\mathcal M:\mathcal X\to\mathcal H$ from location to predicted channel state information (CSI). Across these usages, the recurring aim is to reduce or obviate sophisticated real-time CSI acquisition by replacing part of the online estimation burden with offline construction, environment-aware inference, and map querying (Zeng et al., 2020, Liu et al., 30 May 2025).

1. Terminology, scope, and relation to CKM

The literature uses closely related formulations for CEKM and CKM rather than a single universal definition. One early formulation defines a CEKM as “the component of CKM specifically targeted at instantaneous or refined channel estimation,” with a general mapping

$$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$

where $\Theta$ may be instantaneous CSI or a set of path parameters. A later survey defines CEKM more broadly as

$\mathcal M:\mathcal X\to\mathcal H,$

mapping each location to predicted CSI such as impulse responses, power delay profiles (PDPs), or angle-delay tuples. A dual-domain tracking formulation introduces an inverse CEKM,

$$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$

used to infer position and speed from measured channel parameters (Zeng et al., 2020, Liu et al., 30 May 2025, Du et al., 28 Jun 2025).

Formulation	Mapping	Emphasis
CEKM as CKM component (Zeng et al., 2020)	$$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$	Estimation-oriented channel knowledge
CEKM as location-to-CSI map (Liu et al., 30 May 2025)	$\mathcal M:\mathcal X\to\mathcal H$	Prior CSI for estimation and control
CEKM as inverse map (Du et al., 28 Jun 2025)	$$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$	Localization and tracking

The object stored by a CEKM is correspondingly diverse. Reported instances include channel gain maps (CGMs), channel path maps (CPMs), channel impulse response maps, path loss, shadowing-aware gain, angle-of-arrival (AoA), angle-of-departure (AoD), PDPs, joint frequency-space covariance, beam indices, and even location-specific channel probability distributions or score functions (Zeng et al., 2020, Jiang et al., 2024, Ding et al., 2021, Qiu et al., 8 Jul 2025).

This breadth distinguishes CEKM from conventional pilot-based CSI acquisition. In the pilot-centric view, the training overhead scales nearly linearly with channel dimension; one summary expresses this as $T_{\rm pilot}=O(D_c)$ with $D_c=\#$antennas $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$0 #subcarriers. In CEKM-assisted estimation, the residual overhead is instead tied to map-query dimensionality, written as $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$1, and the online task becomes database lookup, interpolation, or lightweight inference rather than full channel reconstruction from scratch (Zeng et al., 2020).

2. Mathematical structure and sample-complexity principles

A central mathematical theme is that CEKM construction is a spatial prediction problem. For large-scale channel gain, one analytical model writes

$$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$2

where $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$3 is log-normal shadowing with spatial autocovariance $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$4 and $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$5 is small-scale fading modeled as spatially uncorrelated at map scales. In this setting, the average mean square error (AMSE) of offline CGM construction depends explicitly on sample density $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$6, and the online AMSE with $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$7 nearest neighbors decreases with diminishing marginal return as $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$8 grows. The same analysis states that grid sampling slightly outperforms random PPP at the same $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$9, while random PPP may be easier in crowdsourcing (Xu et al., 2023).

The same paper turns sample density into a design variable. Required offline density $\Theta$0 is obtained by solving an AMSE inequality for a target error $\Theta$1, and the online neighbor count $\Theta$2 is chosen from a corresponding AMSE constraint. Subregion partitioning is then introduced to handle local non-homogeneity: larger regions reduce parameter-estimation variance by increasing sample count, while smaller regions increase homogeneity but raise variance because fewer local samples are available. This analytical layer makes CEKM a design problem in addition to a learning problem (Xu et al., 2023).

A second mathematical theme is the reinterpretation of CKM construction as image super-resolution. One formulation represents the complete high-resolution map as

$\Theta$3

the sparse measurements as

$\Theta$4

and the sampling pattern by a binary matrix $\Theta$5 satisfying

$\Theta$6

The super-resolution factor is

$\Theta$7

so that only $\Theta$8 of the spatial locations are measured. This formulation explicitly supports scalar maps such as path loss and AoA, and it motivates the direct use of image-SR backbones for CEKM reconstruction from sparse field measurements (Wang et al., 2024).

High-dimensional extensions generalize the mapping itself. A hybrid 2D/3D construction defines, for receiver location $\Theta$9,

$\mathcal M:\mathcal X\to\mathcal H,$0

with $\mathcal M:\mathcal X\to\mathcal H,$1 containing sparse channel measurements, the environment representation, and transmitter information. A 6D formulation for varying transmitter and receiver positions writes

$\mathcal M:\mathcal X\to\mathcal H,$2

where $\mathcal M:\mathcal X\to\mathcal H,$3 and the output is the SIMO channel vector or its power gain in dB. These formulations move CEKM from fixed-base-station radio maps to fully transceiver-dependent channel fields (Wang et al., 9 Oct 2025, Zhou et al., 30 Oct 2025).

3. Construction paradigms and representative architectures

A survey classification divides CEKM construction into three families: measurement-based, model-based, and hybrid. Measurement-based methods include K-nearest neighbors (KNN), kriging or Gaussian-process regression, matrix or tensor completion, and deep regression. Model-based methods include statistical path-loss-plus-fading synthesis and deterministic ray tracing. Hybrid methods combine both, either by weighted linear fusion or Bayesian update, with local measurement density or uncertainty determining the relative trust placed in measurement-derived and model-derived maps (Liu et al., 30 May 2025).

A sparse-measurement super-resolution construction based on SRResNet is one of the clearest deep-learning instantiations. The network takes $\mathcal M:\mathcal X\to\mathcal H,$4 as input, uses one initial $\mathcal M:\mathcal X\to\mathcal H,$5 convolution with 64 feature maps, a residual trunk of 5 residual blocks, a global skip, sub-pixel upsampling, and a final $\mathcal M:\mathcal X\to\mathcal H,$6 reconstruction layer. It is trained only with mean-square error because CEKM requires accurate pixel-level intensity rather than perceptual realism. On the RadioMapSeer path-loss test set at $\mathcal M:\mathcal X\to\mathcal H,$7 SR, SRResNet achieves PSNR $\mathcal M:\mathcal X\to\mathcal H,$8, SSIM $\mathcal M:\mathcal X\to\mathcal H,$9, LPIPS $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$0, MSE $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$1, and RMSE $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$2 dB, outperforming nearest neighbour, bicubic interpolation, and SRGAN. The same study reports that only $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$3 of locations need to be measured to achieve approximately $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$4 dB RMSE in path loss, and that AoA reconstruction remains visually strong for $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$5 but degrades faster as $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$6 increases because angular structure is more complex (Wang et al., 2024).

Interference-aware CEKM construction introduces a different two-stage pattern. IMNet first estimates interfering signal strength (ISS) at sampled locations by subtracting a desired-signal-strength map derived from environment topology, then reconstructs the full ISS map using a negative-correction module and a 17-layer U-Net. The loss is MSE in a normalized log domain. In the reported A2G setting, IMNet achieves ISS-map NMSE of roughly $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$7 dB at $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$8 sampling and about $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v),$$9 dB at $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$0, outperforming U-Net and interpolation baselines; the resulting SINR map is about $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$1 dB worse than ISS reconstruction but still about $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$2 dB better than baselines, and interferer localization error drops from more than 10 grid units for interpolation methods to about 2–3 grid units at high sampling (Zhao et al., 2024).

Hybrid constructions increasingly embed explicit environmental structure. One point-cloud-based framework fuses 2D semantic labels and 3D point-cloud attributes, performs confocal ellipsoid filtering to select relevant scatterers per time-of-arrival bin, extracts PointNet++-style hierarchical features, and regresses RSS or PDP via MSE. On the reported campus dataset, the proposed vector-valued CKM achieves mean error $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$3 dB versus $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$4 dB for point cloud ray tracing and $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$5 dB for Wireless InSite, while scalar RSS RMSE is about $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$6 dB versus $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$7 dB for Kriging, $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$8 dB for RadioUNet, and $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto\Theta$$9 dB for ray tracing (Wang et al., 9 Oct 2025).

A 6D extension, bidirectional wireless Gaussian splatting (BiWGS), represents the environment by learnable 3D Gaussian ellipsoids that model scatterer clusters and obstruction loss, with bidirectional spherical-harmonic scattering coefficients and wireless rendering along Tx-side and Rx-side paths. In 6D power-gain prediction, BiWGS improves over a classical MLP in all reported rooms: MAE drops from $\mathcal M:\mathcal X\to\mathcal H$0 to $\mathcal M:\mathcal X\to\mathcal H$1 dB in a conference room, from $\mathcal M:\mathcal X\to\mathcal H$2 to $\mathcal M:\mathcal X\to\mathcal H$3 dB in a bedroom, and from $\mathcal M:\mathcal X\to\mathcal H$4 to $\mathcal M:\mathcal X\to\mathcal H$5 dB in an office, while spatial-spectrum SSIM remains close to WRF-GS in 3D fixed-Rx settings (Zhou et al., 30 Oct 2025).

4. CEKM-assisted estimation and inference algorithms

In estimation-oriented formulations, the CEKM is not merely a storage object; it supplies structured priors to Bayesian or optimization-based estimators. One interference-cancellation framework discretizes the base-station coverage area into grids and stores, for each cell, dominant multi-path delays, azimuth and elevation angles, and path powers. CKM construction is cast as Bayesian inference under strong inter-cell interference, with a block-sparsity prior for interferers and a hybrid message-passing algorithm that combines expectation propagation and variational message passing. The resulting CKM then yields a joint frequency-space covariance

$\mathcal M:\mathcal X\to\mathcal H$6

for a CKM-assisted MMSE-IRC estimator, and Woodbury-based factorization reduces the per-slot complexity from $\mathcal M:\mathcal X\to\mathcal H$7 to

$\mathcal M:\mathcal X\to\mathcal H$8

In the reported UMa-NLOS simulations, CKM accuracy improves from $\mathcal M:\mathcal X\to\mathcal H$9 dB for an OMP-based naive map to $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$0 dB at $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$1 dB, and the CKM-assisted MMSE-IRC estimator reaches NMSE approximately $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$2 dB at $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$3 dB, whereas Turbo-CS and variational-Bayes baselines require at least $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$4 dB and $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$5 dB, respectively, to reach the same NMSE (Jiang et al., 2024).

A different line of work replaces explicit covariance modeling with learned channel distributions. The Channel Score Function Map (CSFM) stores, for each location $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$6, the conditional channel density $$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$7, its score function

$$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$8

or an equivalent denoiser linked to the score through Tweedie’s formula. Regularized MAP estimation is then written as

$$f_{\rm CEKM}:(\{\tau_i\},\{\mu_i\},\{\theta_i\})\mapsto(q,v)$$9

and solved by a plug-and-play scheme that alternates a closed-form data-fidelity step with denoiser-based prior refinement. In the reported XL-MIMO setting, CSFM-PnP improves NMSE by about 5–10 dB over LS/ML and 3–7 dB over LMMSE under limited pilots, low SNR, and even no-pilot conditions; at $T_{\rm pilot}=O(D_c)$0 dB with $T_{\rm pilot}=O(D_c)$1 pilots, NMSE is approximately $T_{\rm pilot}=O(D_c)$2 dB for CSFM-PnP versus about $T_{\rm pilot}=O(D_c)$3 dB for LS/ML and about $T_{\rm pilot}=O(D_c)$4 dB for LMMSE (Qiu et al., 8 Jul 2025).

Clustered channels motivate a third estimation-oriented variant. ClusterCKM maps transmitter and receiver positions to parameter ranges

$T_{\rm pilot}=O(D_c)$5

for each physical scatterer cluster, using historical tensors, equivalent-scatterer localization, and DPSS-based subspace restriction. Channel estimation is then performed cluster by cluster with local range refinement and deflation. In the reported MIMO-OFDM simulations, ClusterCKM maintains RMSE around $T_{\rm pilot}=O(D_c)$6 dB up to pilot period $T_{\rm pilot}=O(D_c)$7, while LS and OMP fail once $T_{\rm pilot}=O(D_c)$8, corresponding to a $T_{\rm pilot}=O(D_c)$9 reduction in pilot overhead for the target error level. In dense-clustered scenarios at $D_c=\#$0 dB and $D_c=\#$1, ClusterCKM also outperforms LS, OMP, and CoarseCKM (Zhang et al., 26 Feb 2026).

5. Applications in beam management, mobility, and autonomous systems

One of the most direct applications is beam selection. In IRS-aided communication, the Beam Index Map (BIM) is a CKM instance mapping a user location $D_c=\#$2 to the optimal active/passive beam pair $D_c=\#$3. Training-free beam selection retrieves the most likely beam pair from nearby stored samples, while light-training beam selection restricts beam sweeping to a small candidate set derived from the map. In the reported ray-tracing-based system with an $D_c=\#$4 BS UPA and a $D_c=\#$5-element IRS, BIM training-free reaches about $D_c=\#$6 bps/Hz and BIM light-training about $D_c=\#$7 bps/Hz, close to the perfect-CSI rate of about $D_c=\#$8 bps/Hz, while reducing pilot overhead by $D_c=\#$9 relative to full cascaded CSI training with $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$00 pilots (Ding et al., 2021).

UAV communication uses CEKM differently: not to select beams, but to compensate for positioning uncertainty. A noise-aware CKM maps UAV position, noisy ground-user position, environment features, and a noise flag to predicted path loss, and the predictions are integrated into a PPO-based trajectory and power-allocation controller. In the reported simulations averaged over 500 Monte Carlo runs, PPO+CKM converges after 40,000 episodes to flight time about $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$01 s with reward about $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$02. Under CEP values of $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$03, $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$04, and $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$05 m, the PEC-PPO flight times are $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$06, $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$07, and $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$08 s, respectively, versus $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$09, $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$10, and $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$11 s for LoS-PPO. With $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$12 of tall buildings removed and online CKM update, flight time decreases from $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$13 s to $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$14 s, and CI-based power control saves up to $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$15 energy (Zhang et al., 2024).

High-mobility networks use CEKM as a bridge between coordinate and beam domains. In one dual-domain architecture, the CKM maps $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$16 to path gains, delays, Doppler shifts, and AoAs, while the inverse CEKM infers $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$17 from measured channel parameters. An EKF operates in the coordinate domain, a Markov beam tracker operates in the beam domain, CKM-derived AoA priors shape beam-transition models, and predictive beamforming minimizes the worst-case AoA CRB subject to transmit-power constraints. In the reported 30 GHz vehicular scenario, the CKM-assisted scheme keeps coordinate tracking RMSE below about $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$18 m even during 1 s of LoS outage, versus more than $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$19 m failure for a LoS-only EKF baseline; multipath AoA errors remain below about $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$20 with CKM priors versus more than $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$21 without, and predictive beamforming with optimized power allocation reduces worst-case AoA CRB by about $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$22 over equal-power steering (Du et al., 28 Jun 2025).

These examples show that CEKM applications are not limited to scalar radio maps. The same framework appears in beamforming, proactive handover, network planning, resource allocation, integrated sensing and communication, trajectory optimization, localization, and predictive tracking, with the exact map output chosen to match the downstream decision variable (Liu et al., 30 May 2025).

6. Dynamic maintenance, misconceptions, and current research directions

A recurring practical problem is map freshness. One update framework formalizes this with a Map Efficacy Function (MEF) that combines gradual exponential aging within a segment, abrupt multiplicative entry loss at environmental boundaries, and zero efficacy during update downtime. Update scheduling is posed as maximizing

$$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$23

and solved either exactly by Delta-P, a two-parameter Dinkelbach method with Pareto-frontier dynamic programming, or approximately by Delta-L, a Taylor-linearized Dinkelbach method with near-linear complexity. The reported simulations state that Delta-P achieves the Pareto frontier exactly, while Delta-L lies within 1–2% of the frontier at 1/100 th the runtime. For unpredictable environments, the threshold condition

$$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$24

characterizes when immediate update is optimal (Wang et al., 17 Dec 2025).

Dynamic MIMO-OFDM CEKM construction addresses a different aspect of non-stationarity: coexistence of quasi-static and dynamic scatterers, antenna rotation, and synchronization error. The proposed two-stage approximate Bayesian inference framework first estimates quasi-static delays and angles from historical grid-level data, then reuses them as informative priors to infer dynamic parameters from limited real-time pilots. In the reported 28 GHz LoS setting, Stage I achieves NMSE about $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$25 dB with $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$26 paths versus $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$27 dB for OMP-initialized estimation, and Stage II converges in 5 iterations while improving MSE by more than $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$28 dB over schemes without dynamic-path estimation or synchronization-error estimation. Reducing pilots from $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$29 to $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$30 costs only $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$31 dB (Jiang et al., 29 Dec 2025).

Several misconceptions can be addressed directly from the literature. First, CEKM is not restricted to path-loss prediction; reported outputs range from scalar RSS and path loss to full tensor-valued CSI, PDPs, beam indices, score functions, and clustered parameter ranges (Liu et al., 30 May 2025, Wang et al., 9 Oct 2025, Ding et al., 2021, Qiu et al., 8 Jul 2025). Second, CEKM is not inherently pilot-free. Some systems explicitly target “training-free” or zero-pilot operation once the map is built, whereas others use CEKM priors inside Bayesian, PnP, or covariance-assisted estimators with residual pilots (Wang et al., 9 Oct 2025, Zeng et al., 2020, Jiang et al., 2024). Third, CEKM construction is not purely data-driven. Current methods span interpolation, ray tracing, hybrid geometric filtering, Bayesian inference, Gaussian splatting, and physics-inspired diffusion (Liu et al., 30 May 2025, Zhou et al., 30 Oct 2025, Zhu et al., 2 Dec 2025).

Recent research pushes CEKM toward physically constrained generative modeling. A physics-inspired diffusion framework for CKM construction in large-scale fading augments diffusion loss with edge, regional-propagation, and multi-scale feature constraints, and reports on RadioMapSeer static CGM test data an NMSE of $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$32, RMSE of $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$33, and PSNR of $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$34 dB, outperforming RadioUNet, RME-GAN, and RadioDiff; in dynamic scenes with vehicles, the reported NMSE is $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$35 and inference time is $$\mathcal K:(r_{\rm TX},r_{\rm RX},f,t)\mapsto \Theta,$$36 s per map (Zhu et al., 2 Dec 2025). This suggests that current CEKM research is moving simultaneously toward stronger physics priors, higher-dimensional transceiver-dependent mappings, dynamic updating, non-uniform and noisy sampling, multi-channel maps, and cross-domain transfer, while continuing to treat scalability, robustness, and uncertainty quantification as open design constraints rather than solved problems (Wang et al., 2024, Liu et al., 30 May 2025).