Channel Space Gridization (CSG)

• Channel Space Gridization (CSG) is a method that partitions wireless environments into grids with similar channel characteristics, enabling scalable network optimization.
• It unifies channel estimation and gridization by using beam-level RSRP measurements to infer latent Channel Angle Power Spectra (CAPS) and construct interpretable, sparse grids.
• CSG frameworks, including methods like the CSG Autoencoder and adaptive spatial representations, offer measurable gains in prediction accuracy and reduced inference complexity.

Channel Space Gridization (CSG) denotes a wireless-channel-centric partitioning of space into grids whose members share similar channel characteristics, with the explicit aim of supporting scalable network optimization, channel knowledge map construction, and downstream inference from sparse or low-cost measurements. In its most specific usage, CSG is the framework that uses only beam-level reference signal received power (RSRP) to estimate Channel Angle Power Spectra (CAPS) and partition samples into channel-homogeneous grids, unifying channel estimation and gridization in a single optimization problem (Wang et al., 21 Jul 2025). In the broader CKM literature, closely related ideas appear as spatial discretization of service areas into cells, voxelized channel gain maps, point-cloud-conditioned continuous predictors, adaptive octree-based channel representations, and virtual-scatterer latent models (Jiang et al., 2024, Wang et al., 17 Apr 2025, Wang et al., 26 Jun 2025, Zhang et al., 21 May 2026, Sun et al., 13 Feb 2026). This suggests that CSG is best understood as a family of methods for organizing wireless environments by channel structure rather than by geometry alone.

1. Definition, scope, and motivation

CSG is introduced as a response to a specific limitation of large-scale wireless optimization: optimizing directly over individual user samples is unstable and expensive, whereas optimization over a smaller set of grids can be more reusable if each grid contains users with similar communication characteristics (Wang et al., 21 Jul 2025). The paper formalizes this motivation by contrasting per-user optimization,

$$\Psi_{1}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_1} u_i(\Psi)$$

and

$$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$

with grid-level optimization,

$$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$

where $|\set{K}| \ll |\set{I}|$ (Wang et al., 21 Jul 2025). The central claim is that the right similarity notion for this reduction is similarity of the underlying wireless channel structure, not merely spatial proximity or similarity of beam-level powers.

Two earlier gridization paradigms motivate this position. Geographical Space Gridization (GSG) clusters by physical location, but depends on location data that may be unavailable, privacy-sensitive, or noisy, and often requires expensive drive tests (Wang et al., 21 Jul 2025). Beam Space Gridization (BSG) clusters directly in the space of beam-level RSRP vectors, using readily available measurement reports, but assumes that similar RSRP implies similar channel structure; the CSG paper explicitly rejects that assumption as flawed because different multipath configurations can produce similar beam-level received powers (Wang et al., 21 Jul 2025).

Within the wider CKM literature, the same motivation recurs in different technical forms. A CKM is defined as a geo-tagged, site-specific database representing channel knowledge across an area of interest, and multiple papers argue that channel fields should be structured using environment-aware and propagation-aware representations rather than naive interpolation over measurement points (Wang et al., 9 Oct 2025, Wang et al., 26 Jun 2025, Jiang et al., 2024). A plausible implication is that CSG is not limited to one architecture or one discretization scheme; rather, it names the general move from geometry-only partitioning toward channel-structure-aware partitioning.

2. Mathematical formulation of channel-space gridization

The most explicit formalization appears in the CSG paper through CAPS-based latent clustering (Wang et al., 21 Jul 2025). The observable inputs are beam-level RSRP measurements,

$$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$

and each sample is assumed to arise from an unobserved CAPS vector $\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$ through the Localized Statistical Channel Model (LSCM),

$\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$

where $M$ is the number of beams, $N=N_VN_H$ is the number of angular bins, and $\mat{A}\in\mathbb{R}^{M\times N}$ is the beam pattern matrix determined by base-station configuration (Wang et al., 21 Jul 2025).

The latent CAPS is written as

$$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$0

and the grid centers are a finite set

$$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$1

Sample assignment is nearest-neighbor in CAPS space: $$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$2 with membership set

$$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$3

The framework assumes sparse, nonnegative grid centers,

$$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$4

and models each sample as

$$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$5

with a zero-mean perturbation assumption on dominant-path components (Wang et al., 21 Jul 2025).

The resulting joint optimization problem combines data fidelity and grid-center representativeness: $$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$6 Here $$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$7 is a dB-domain reconstruction loss and $$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$8 encourages each center to match the projected average of the assigned CAPS samples (Wang et al., 21 Jul 2025).

The broader CKM literature employs alternative mathematical objects but the same structural idea: a finite spatial partition endowed with compact channel descriptors. One paper partitions a base-station coverage area into square grids of size $$\Psi_{2}^{*} \triangleq \argmax_{\Psi \in \mathcal{C}(\Psi)} \sum_{i \in \set{I}_2} u_i(\Psi),$$9 m$$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$0 and assigns each grid $$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$1 a set of effective path delays, angles, and powers,

$$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$2

derived from uplink received signals under a spatial-consistency assumption (Jiang et al., 2024). Another paper defines a channel gain map as average channel power $$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$3 over grid $$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$4,

$$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$5

with virtual-scatterer parameterization

$$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$6

thereby making each grid value a function of scatterer geometry and directional scatterer response coefficients (Sun et al., 13 Feb 2026). These formulations differ in latent variables but share the same CSG pattern: discretize space, retain structured channel descriptors, and infer them from limited observations.

3. CSG-AE and the joint learning of channel estimation and gridization

The neural realization of CSG is the CSG Autoencoder (CSG-AE), composed of a trainable RSRP-to-CAPS encoder, a learnable sparse codebook quantizer, and a physics-informed decoder based on the LSCM (Wang et al., 21 Jul 2025). The encoder maps beam-level RSRP in dBm to a nonnegative CAPS estimate,

$$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$7

where $$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$8 (Wang et al., 21 Jul 2025). In experiments it is implemented as a 6-layer MLP with 256 hidden units per layer and skip connections at layers 2 and 4 (Wang et al., 21 Jul 2025).

The quantizer uses a learnable codebook

$$\max_{\Psi \in \mathcal{C}(\Psi)} \sum_{k \in \set{K}} \tilde{u}_k(\Psi),$$9

with the $|\set{K}| \ll |\set{I}|$0-th center processed as

$|\set{K}| \ll |\set{I}|$1

where $|\set{K}| \ll |\set{I}|$2 keeps the $|\set{K}| \ll |\set{I}|$3 largest elements and zeros out the rest (Wang et al., 21 Jul 2025). Assignment is nearest-neighbor: $|\set{K}| \ll |\set{I}|$4 This makes grid centers explicitly sparse and nonnegative CAPS prototypes.

The decoder is fixed by physics rather than learned: $|\set{K}| \ll |\set{I}|$5 This is the operational use of the LSCM,

$|\set{K}| \ll |\set{I}|$6

which itself is derived from a URA-based wideband channel model under random phase averaging (Wang et al., 21 Jul 2025). The fixed decoder constrains the latent space to remain interpretable as CAPS rather than an arbitrary embedding.

Training optimizes

$|\set{K}| \ll |\set{I}|$7

where the first term reconstructs observed RSRP and the second matches each codeword to the projected average of the assigned embeddings (Wang et al., 21 Jul 2025).

The paper argues that naive end-to-end training is unstable because of codebook collapse, embedding drift, assignment hysteresis, and gradient conflicts between reconstruction and quantization terms (Wang et al., 21 Jul 2025). To address this it proposes the Pretraining-Initialization-Detached-Asynchronous (PIDA) scheme. Pretraining first optimizes only $|\set{K}| \ll |\set{I}|$8 to stabilize the embedding manifold. Initialization then applies $|\set{K}| \ll |\set{I}|$9-means to pretrained embeddings to seed the codebook. Detached update uses

$$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$0

so that $$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$1 updates only the codebook, not the encoder. Asynchronous update recomputes fresh detached embeddings after each encoder step, reducing stale-assignment effects (Wang et al., 21 Jul 2025). The paper reports that pretraining alone or $$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$2-means initialization alone is insufficient, whereas together they maintain over 95% codebook utilization (Wang et al., 21 Jul 2025).

A plausible implication is that CSG-AE should be viewed less as a generic autoencoder and more as a constrained latent-variable solver: inference of CAPS, vector quantization into sparse channel prototypes, and reconstruction through a fixed forward model.

4. Spatial representations beyond CAPS codebooks

Although the term CSG is explicitly introduced in the CAPS-based framework, related papers show that channel-space structuring can be implemented through several distinct spatial representations.

One line of work adopts explicit voxelization. A 3D channel gain map for urban low-altitude communications partitions a rectangular region of size $$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$3 into cubic cells of side length $$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$4, with

$$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$5

and voxel center

$$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$6

Each voxel stores a scalar channel gain in dB, with occupied building voxels assigned

$$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$7

in simulation (Wang et al., 17 Apr 2025). In that work, a 3D-CGAN learns the coordinate-to-volume mapping

$$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$8

from base-station position $$\set{Y_{I}} \triangleq \{\vec{y}_i\}_{i=1}^{I}, \qquad \vec{y}_i \in \mathbb{R}^{M},$$9 to the full 3D channel tensor (Wang et al., 17 Apr 2025). This is a literal Cartesian form of gridization.

A second line uses point clouds and nonuniform propagation-aware partitioning rather than regular Euclidean grids. In point-cloud-based CKM construction, the channel object at receiver position $\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$0 is the power delay profile

$\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$1

and the environment is represented as

$\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$2

The core “Point Selector” constructs co-focal ellipsoidal shells tied to ToA bins. For bin $\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$3,

$\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$4

with $\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$5 the shell between two ellipsoids defined by path-length bounds (Wang et al., 26 Jun 2025). The same idea is described as partitioning the environment into mutually exclusive regions between confocal ellipsoids with Tx and Rx as foci, each region corresponding to one ToA bin (Wang et al., 9 Oct 2025). This is not voxelization, but it is a physically informed discretization of propagation space.

A third line uses adaptive spatial hierarchies. OctCGS partitions the 3D environment into a full octree, anchors one anisotropic Gaussian primitive to each occupied leaf, and constrains its center by

$\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$6

where $\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$7 is the leaf-cell center (Zhang et al., 21 May 2026). The octree node features are

$\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$8

and are recursively aggregated upward by

$\vec{x}_i \in \mathbb{R}_{\ge 0}^{N}$9

This yields a sparse, hierarchical, geometry-aware partition rather than a uniform lattice (Zhang et al., 21 May 2026).

A fourth line replaces per-cell free parameters by latent scatterers. In the virtual-scatterer model, a 2D map embedded in 3D is discretized into regular grids $\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$0, but each grid value is generated from a set of scatterers with positions $\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$1 and directional scatterer response coefficients,

$\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$2

The number of virtual scatterers is increased progressively, and missing directional SRCs are inferred by GPR with kernel

$\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$3

(Sun et al., 13 Feb 2026).

Taken together, these works show that CSG can denote several mathematically distinct choices: uniform Cartesian gridding, propagation-domain shell partitioning, adaptive octrees, and latent-scatterer grid generation. What they share is the replacement of raw sample clouds by a structured channel field indexed by a compact spatial substrate.

5. CKM construction, interference-aware extraction, and downstream estimation

A major extension of CSG is the inference of per-grid channel structure directly from received base-station signals rather than from pre-extracted channel labels. In the interference-cancellation-based CKM framework, the coverage area is partitioned into $\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$4 square grids, with location map

$\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$5

and each grid $\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$6 stores

$\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$7

(Jiang et al., 2024). The user channel inside the grid is approximated as

$\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$8

The received BS signal includes both target-user and inter-cell interferer contributions, and the paper formulates per-grid CKM construction as Bayesian inference with a Bernoulli-Gaussian block-sparsity prior over interferer coefficient blocks,

$\vec{y}_i = \mat{A}\vec{x}_i, \quad \forall i \in \set{I},$9

The posterior is approximated using hybrid message passing over target and interference variables (Jiang et al., 2024). This directly addresses a practical problem that some CSG formulations leave implicit: gridization is only useful if cell-level descriptors can be extracted robustly under interference.

Once constructed, the gridized map induces a structured covariance model for channel estimation. The CKM-derived covariance is

$M$0

and the MMSE-IRC estimator is

$M$1

Using Woodbury and Khatri-Rao structure, the inversion burden is reduced from $M$2 to

$M$3

(Jiang et al., 2024). This demonstrates a key systems-level virtue of CSG: a gridized map is not merely a visualization object, but a reusable prior that reduces downstream inference complexity.

A related but more abstract point appears in the mixture-of-experts channel cartography literature. There, channel gain between transmitter and receiver is modeled either in geographic pair space or in pilot-feature pair space, with adaptive blending

$M$4

where $M$5 depends on localization uncertainty (Lopez-Ramos et al., 2020). This suggests that CSG can also be interpreted as adaptive choice of the coordinate system in which gridization is performed: physical space when location is reliable, feature space when it is not.

6. Empirical results, limitations, and terminological boundaries

The CAPS-based CSG paper reports strong gains on both synthetic and real-world data. On real-world datasets, CSG-AE reduces Active MAE by 30% and Overall MAE by 65% on RSRP prediction accuracy compared to salient baselines using the same data (Wang et al., 21 Jul 2025). The reported numbers are approximately 4.6 dB Active MAE for CSG-AE with PIDA, versus about 6.5 dB for the best BSG baseline and about 6.7 dB for the best GSG baseline; for Overall MAE, 12.3 dB for CSG-AE with PIDA versus 35.2 dB for the best BSG baseline (Wang et al., 21 Jul 2025). The same study reports improved channel consistency, more balanced cluster sizes, and active ratio around 90% (Wang et al., 21 Jul 2025).

Other channel-space structuring methods report complementary empirical evidence. The point-cloud-based CKM achieves RMSE $M$6 dB and $M$7 dB for PDP reconstruction in two areas of interest, compared with 7.32 dB and 8.11 dB for Wireless Insite ray tracing and 6.79 dB and 8.43 dB for point-cloud ray tracing (Wang et al., 26 Jun 2025). For radio-map construction it reports RMSE $M$8 dB and $M$9 dB, versus 2.88 dB and 1.92 dB for Wireless Insite and 1.68 dB and 0.99 dB for Kriging (Wang et al., 26 Jun 2025). OctCGS reports average MAE $N=N_VN_H$0 dB and NMAE $N=N_VN_H$1, outperforming BiWGS by 0.88 dB MAE and 0.021 NMAE (Zhang et al., 21 May 2026). The virtual-scatterer model reports NMSE 0.67 and 0.42 under two sampling schemes with $N=N_VN_H$2 measurements, compared with 2.16/5.92 for KPSM and 10.96/7.65 for Kriging (Sun et al., 13 Feb 2026). The interference-aware CKM reports about $N=N_VN_H$3 dB CKM accuracy at $N=N_VN_H$4 m, $N=N_VN_H$5, and SINR $N=N_VN_H$6 dB, far better than OMP-based and interference-non-cognitive baselines (Jiang et al., 2024).

These results support three recurrent conclusions. First, channel-aware partitioning is consistently stronger than geometry-only or observation-only clustering when the target is downstream wireless optimization or prediction. Second, the representation chosen for each grid matters as much as the partition itself: CAPS prototypes, multipath parameter lists, virtual scatterers, and octree Gaussians all outperform weaker surrogates in their own settings. Third, physics-aware priors and environment-aware structure reduce the number of measurements needed to build useful maps.

The limitations are equally consistent. Current methods are site-specific and generally assume static or quasi-static environments (Wang et al., 21 Jul 2025, Jiang et al., 2024, Zhang et al., 21 May 2026). Several require known or reconstructable geometry, whether in the form of beam matrix $N=N_VN_H$7, point clouds, octrees, scatterer locations, or delay-angle path models (Wang et al., 21 Jul 2025, Wang et al., 26 Jun 2025, Zhang et al., 21 May 2026, Sun et al., 13 Feb 2026). Uniform-grid methods face the standard resolution-storage tradeoff, since smaller cells improve homogeneity but increase map size (Jiang et al., 2024, Wang et al., 17 Apr 2025). Feature-learning methods depend on stable training and sufficient measurement diversity, which is why PIDA is necessary in CSG-AE (Wang et al., 21 Jul 2025). Point-cloud and octree methods avoid some voxel inefficiency but introduce their own complexity in selection, rendering, and hierarchical attention (Wang et al., 26 Jun 2025, Zhang et al., 21 May 2026).

The term itself also has notable ambiguities. In one unrelated paper, “CSG” refers to constructive solid geometry, specifically the enumeration of CSG trees from fitted primitives and point clouds (Friedrich et al., 2021). In another, “CSG” refers to the Chklovskii–Shklovskii–Glazman picture of compressible stripes in the integer quantum Hall effect (Oswald, 2021). These usages are unrelated to Channel Space Gridization. Within wireless communications, however, the term is tied to gridization by channel structure and is most precisely instantiated by the CAPS-based framework that jointly estimates latent channel structure and partitions samples into channel-homogeneous grids (Wang et al., 21 Jul 2025).

A plausible synthesis of the current literature is that Channel Space Gridization is evolving from a single clustering idea into a broader design principle: represent wireless environments through structured channel fields whose indexing geometry is chosen to preserve propagation similarity. In this reading, CAPS codebooks, grid-wise multipath parameter sets, voxelized gain maps, virtual-scatterer models, propagation-aware point-cloud shells, and octree-contextual Gaussian splatting are not competing definitions of CSG so much as different operational answers to the same question: how to partition space so that each cell is meaningful in channel space rather than only in physical space.