---
title: Channel Charting (Self-Supervised Radio Geometry)
url: https://www.emergentmind.com/topics/channel-charting-cc
type: topic
---

# Channel Charting (Self-Supervised Radio Geometry)

Channel charting (CC) is a self-supervised geometry-learning framework that maps high-dimensional channel state information (CSI) collected at network infrastructure into a low-dimensional latent space whose coordinates act as pseudo-positions. Its defining objective is not exact coordinate recovery, but preservation of local spatial and radio geometry: users that are close in physical space should remain close in the chart, while the chart itself may be rotated, translated, scaled, or nonlinearly warped without violating the core CC criterion [1807.05247][2312.04514].

## 1. Foundations and problem formulation

The canonical CC setting begins with CSI measurements acquired at a base station or access point from many user equipments over time. In the original formulation, a multi-antenna receiver observes high-dimensional CSI that lives in a nonlinear “radio geometry,” and the task is to learn a low-dimensional chart from CSI alone, without GNSS, triangulation, or labeled position data [1807.05247]. In a common neural formulation, each CSI matrix $\bH^{(n)} \in \mathbb{C}^{B \times W}$ is converted into a real-valued feature vector $\vec f^{(n)} \in \mathbb{R}^{D'}$, and a learned mapping $g_{\boldsymbol\theta} : \mathbb{R}^{D'} \to \mathbb{R}^{D}$, typically with $D=2$ or another small dimension, produces the chart coordinate $\hat{\vec x}^{(n)} = g_{\boldsymbol\theta}(\vec f^{(n)})$ [2312.04514].

What distinguishes CC from supervised positioning is the source of supervision. Standard positioning networks regress from CSI to known coordinates, whereas CC learns from relationships internal to the radio measurements themselves. A widely used objective is to preserve pairwise dissimilarities derived from CSI, for example
$$
L(\boldsymbol\theta)=\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\Big(d_{i,j}-\|g_{\boldsymbol\theta}(\vec f^{(i)})-g_{\boldsymbol\theta}(\vec f^{(j)})\|\Big)^2,
$$
where $d_{i,j}$ is computed from CSI rather than labels [2312.04514]. Earlier Siamese formulations made the same idea explicit as a parametric version of Sammon’s mapping, replacing the latent points by outputs of a shared network $f_{\bm\theta}$ [1909.13355].

A central consequence is that conventional CC recovers relative geometry rather than absolute physical coordinates. The chart is useful for neighborhood reasoning, trajectory structure, grouping, and manifold-aware control, but by itself it does not anchor axes to the world. Later work therefore distinguishes between pseudo-positioning and real-world-coordinate charting, often by injecting additional weak or self-supervised structure such as anchor constraints, access-point locations, model-based geometry, or digital twins [2308.14498][2511.09227].

## 2. Feature design and channel dissimilarity

CC quality depends heavily on the feature representation and the dissimilarity measure used to approximate spatial proximity. The original pipeline emphasized large-scale channel properties rather than instantaneous small-scale fading. A central construction was the raw second moment,
$$
\bar{\bH} = \frac{1}{T}\sum_{t=1}^T \bmh_t \bmh_t^H,
$$
optionally combined with CSI scaling and a discrete Fourier transform into beamspace; empirically, the absolute value of the raw second moment in the angular domain was reported as the most robust feature in the original study [1807.05247].

Later work specialized the feature space to different propagation regimes. For streaming CSI, a measured-CSI pipeline used IDFT to the delay domain, truncation to the first $C$ taps, vectorization, entrywise magnitude, and unit-norm normalization, then defined an angle-delay-profile dissimilarity
$$
\tilde d_{i,j}=\sum_{\tau=1}^{C}\left(1-\frac{|(\underline{\vech}_\tau^{(i)})^H\underline{\vech}_\tau^{(j)}|^2}{\|\underline{\vech}_\tau^{(i)}\|^2\|\underline{\vech}_\tau^{(j)}\|^2}\right),
$$
followed by a $K$-nearest-neighbor graph and shortest-path geodesic distances [2312.04514]. This explicitly shifts the learning target from raw pairwise similarity to local manifold geometry.

For line-of-sight multicarrier multiantenna systems, the phase-insensitive distance
$$
d^\star(\mathbf h_i,\mathbf h_j)=\sqrt{2-2s^\star(\mathbf h_i,\mathbf h_j)}, \qquad
s^\star(\mathbf h_i,\mathbf h_j)=\frac{|\mathbf h_i^H\mathbf h_j|}{\|\mathbf h_i\|_2\,\|\mathbf h_j\|_2},
$$
was shown to have useful properties but also periodic and oscillatory ambiguities. In that analysis, far users can appear artificially close because the similarity factorizes into radial and angular terms with aliasing and sidelobe structure; the proposed mitigation was to restrict operation to identifiable regions, prefer a uniform circular array over a uniform linear array, tune bandwidth and array radius for “round” neighborhoods, and threshold the similarity to suppress sidelobes [2310.03762].

Other application-specific metrics were built from second-order statistics or synchronized time-domain structure. In pilot reuse for spatially correlated massive MIMO, CC used covariance overlap
$$
\delta(R_n, R_j) = \frac{\operatorname{tr}(R_n R_j)}{\|R_n\|_F \|R_j\|_F}
$$
to build an interference map [2203.06651]. In synchronized multi-anchor SISO indoor localization, a time-distance-based metric derived from aligned channel impulse responses was designed to be approximately linear in Euclidean distance locally, then lifted to global geodesic distances [2210.06294]. In smart-radio-environment work, channel covariance matrices were compared through a Log-Euclidean distance, again using large-scale statistics rather than raw CSI snapshots [2508.07305].

A recurring implication is that CC is not a single algorithm so much as a geometry-preservation principle coupled to a representation choice. Different regimes—wideband delay structure, covariance overlap, phase-insensitive LoS similarity, synchronized TDoA structure, or universal embeddings—induce different notions of “nearness,” and chart quality tracks the fidelity of that induced geometry.

## 3. Learning algorithms and geometric regularization

CC has been implemented with both classical dimensionality reduction and neural parametric models. Linear PCA provides a simple baseline when the feature engineering is strong [1807.05247]. Nonlinear manifold methods such as Sammon’s mapping and Isomap were early reference points: Sammon’s mapping emphasizes preserving small feature-space distances, while Isomap replaces local distances by graph geodesics and is therefore suited to globally unfolded charts [1807.05247][2210.06294].

Neural parametric CC introduced out-of-sample inference and unified training across supervision regimes. Siamese networks map paired inputs through shared weights and minimize mismatch between embedding-space distances and feature-space or CSI-derived distances. In one formulation, the loss is
$$
L(\bmtheta) = \sum_{n=1}^{N}\sum_{m=n+1}^{N}
w_{n,m} \big (\|\bmx_n-\bmx_m \| - \|f_\bmtheta(\bmx_n)-f_\bmtheta(\bmx_m) \| \big)^2,
$$
with weights such as $w_{n,m} = \|\bmx_n-\bmx_m\|^{-1}$ to emphasize local geometry [1909.13355]. Such networks also support semisupervised positioning by combining pairwise metric preservation with anchor-point regression.

Autoencoder-based CC instead learns a bottleneck representation through reconstruction. Its main advantage is scalable parametric inference, but plain reconstruction does not explicitly preserve geometry. Representation-constrained autoencoders addressed that limitation by adding pairwise latent constraints such as fixed or maximum relative distance and anchor-based absolute constraints, which substantially improved Kruskal stress and recovered more meaningful global geometry for approximate positioning [1908.02878].

Triplet-based self-supervision later became a major design pattern. Timestamp-based triplets impose that a sample should be closer to a temporally nearby sample than to a temporally distant one. In real-world-coordinate CC, triplet loss was combined with a bilateration loss derived from known access-point locations and relative received-power comparisons across AP pairs [2308.14498]. More universal representation learning followed the same principle at a broader scale: CSI2Vec first learns a compact embedding with triplet loss across scenarios, antenna counts, and subcarrier selections, then feeds those embeddings to downstream CC via a Siamese mapper or PCA [2506.05237].

These developments also exposed a methodological misconception: minimizing CSI reconstruction is not equivalent to preserving radio geometry. CSI2Vec explicitly argued that autoencoders optimized for reconstruction are often unsuitable for CC because positioning and charting do not require raw CSI reconstruction, and preserving small-scale fading can be counterproductive when the downstream objective is neighborhood preservation [2506.05237].

## 4. Temporal, streaming, and robustness extensions

Early CC formulations largely assumed a static dataset and a fixed input structure. Subsequent work broadened the framework to continual, dynamic, and failure-prone deployments.

In streaming CC, the principal challenge is memory rather than only training. Practical access points may estimate CSI tens to hundreds of times per second over long periods, so storing all samples is infeasible, while naive sliding windows induce catastrophic forgetting. A streaming architecture addressed this by maintaining a fixed-capacity core CSI memory $\setM$ and training the chart only on that curated subset. Its main contribution was a greedy min-max-similarity policy, Similarity-Based Subset (SimS), which replaces highly redundant stored samples with new, more diverse samples according to cosine similarity of features. On the DICHASUS dataset, SimS with only $M=1000$ stored samples approached the performance of training on all $17{,}516$ streaming samples; the worst reported gap to the full-data baseline was only $0.015$ in any metric, whereas random replacement suffered catastrophic forgetting [2312.04514].

Dynamic CC addressed a different assumption: that spatial geometry alone is sufficient. An LSTM-AE-based approach constructed windows
$$
X_t = [x_{t-T+1}, \ldots, x_t] \in \mathbb{R}^{T \times D}
$$
and jointly optimized reconstruction and topology-preserving losses so that the latent chart captured temporal dependence as well as spatial structure. Reported gains were especially pronounced in stability and long-term predictability, with CT $=0.9998$ versus $0.9966$, TW $=0.9998$ versus $0.9965$, KS $=0.0115$ versus $0.0835$, and MAE $=0.0619$ m versus $0.4847$ m relative to a static baseline on the reported dataset [2601.18473].

A third extension concerned missing measurements. Most CC architectures assume fixed-size CSI inputs, but practical systems experience antenna failures, blockage, reconfiguration, and handovers. AdaPos addressed this with a set-based architecture: each available CIR is encoded by a shared 1D-ResNet, enriched with a learnable antenna identifier, and fused by a transformer encoder with self-attention. Missing inputs are simply omitted rather than zero-padded. On public SISO and MIMO datasets, AdaPos maintained state-of-the-art accuracy under missing-antenna conditions and replaced roughly 57 configuration-specific models with a single unified model [2602.04704].

| Issue | Mechanism | Representative result |
|---|---|---|
| Streaming CSI and forgetting | Fixed-capacity core memory with SimS curation | Near-full-data quality with $M=1000$; worst gap $0.015$ [2312.04514] |
| Temporal drift and instability | LSTM encoder plus AE reconstruction and topology loss | Lower KS and MAE than static CC [2601.18473] |
| Variable antenna availability | Set-based transformer fusion with learnable antenna IDs | One model replaces roughly 57 subset-specific models [2602.04704] |

Taken together, these works indicate that CC has evolved from an offline manifold-learning tool into a deployment-oriented family of models for continual, temporally structured, and partially observed radio environments.

## 5. Grounding charts in real-world coordinates and extending beyond position

Standard CC yields pseudo-coordinates in an arbitrary reference frame. Several research directions have therefore sought to embed charts directly in physical coordinates without reverting to fully supervised fingerprinting.

One line uses weak geometric anchors. Real-world-coordinate CC introduced a bilateration loss for multipoint systems in which AP positions are known but UE labels are not. For a sample $n$, the method compares received powers $P^{(n,a)}$ across AP pairs and penalizes embeddings that place the user closer to the weaker AP than to the stronger one:
$$
L_b = \frac{1}{|\mathcal{N}|\,|\mathcal{P}^{(n)}|} \sum_{n\in\mathcal{N}} \sum_{(a_c,a_f)\in\mathcal{P}^{(n)}}
\Big( \|g_{\boldsymbol\theta}(\mathbf{f}^{(n)})-\mathbf{a}_c\| - \|g_{\boldsymbol\theta}(\mathbf{f}^{(n)})-\mathbf{a}_f\| + M_b \Big)^+ .
$$
Combined with standard triplet loss, this produced charts directly in real-world coordinates without any UE labels, improving the mean Euclidean error from $15.02$ m for bilateration alone to $7.64$ m for the joint triplet-plus-bilateration objective in the reported simulation [2308.14498].

A second line grounds CC through external geometric priors. Digital-twin-aided CC replaced direct coordinate regression by a probability distribution over predefined DT grid points, then combined a conventional timestamp-based CC loss with a cosine-similarity DT loss between measured large-scale CSI features and DT-generated features. In the reported indoor simulation, the best “Power” feature achieved MDE $=0.81$ m versus $1.15$ m for the referenced state of the art, corresponding to a relative mean distance error reduction of about 29%; the method was also reported as robust to moderate DT mismatch and a test distribution shift in UE height [2511.09227].

A third line uses model-based geometry and synchronization structure. TDoA-based self-supervised CC incorporated known transmission-reception-point locations, TDoA consistency losses, short-interval displacement constraints, and binary NLoS masking based on normalized CIR peaks. On a real 5G O-RAN testbed, the reported localization accuracy was 2–4 m in 90% of cases across different LoS ratios [2510.08001]. UNILocPro likewise combined model-based localization and CC for mixed LoS/NLoS settings, using model-based path geometry to define pairwise dissimilarities, LoS-anchor losses, and optimal-transport map alignment; with timestamps, its MAE of $4.17$ m was reported as comparable to fully supervised fingerprinting at $4.46$ m [2510.27394].

The notion of what can be charted has also expanded beyond position. “Channel Charting for Position and Orientation” extended real-world-coordinate CC to joint estimation of 2D position and yaw from single CSI snapshots. Orientation was represented by an unnormalized 2D vector and trained with a cosine-based triplet loss respecting angular periodicity, plus a self-supervised alignment loss tying orientation to motion-direction chords derived from estimated real-world positions. On measured 5G NR CSI, the proposed self-supervised method achieved orientation performance close to supervised learning, with mean/median/95th-percentile orientation errors of $22^\circ/11^\circ/108^\circ$ on the reported test set [2606.18151].

## 6. Applications, evaluation, and limitations

Because CC learns radio geometry rather than only coordinates, its application range extends beyond localization. In beamforming and channel mapping, CC has been used as a latent pseudo-location fed into a location-based beamforming network to predict downlink precoders from uplink channels, including cross-frequency and cross-base-station settings; in that study, CC-based location-based beamforming with chart dimension $D=5$ performed better than classical location-based beamforming using true 3D spatial locations [2212.12340]. In pilot reuse and scheduling, CC has been used to build interference maps from covariance similarity in spatially correlated massive MIMO [2203.06651] and to embed near-field polar-domain covariance features for non-orthogonal pilot allocation in XL-MIMO, where near-field-aware CC-assisted scheduling outperformed far-field CC-assisted baselines in MSE and sum rate [2412.20920]. Semi-supervised mmWave IoT localization has likewise used CC as an unsupervised radio-geometry pretraining stage prior to labeled refinement [2108.08241].

Evaluation conventions are comparatively standardized. Across much of the literature, chart quality is assessed with trustworthiness (TW), continuity (CT), Kruskal stress (KS), and sometimes Rajski distance (RD), while downstream positioning uses mean or percentile Euclidean error after affine or analogous alignment when the chart is not already in world coordinates [2312.04514][2308.14498]. Representative reported values illustrate the span of current practice: streaming SimS achieved TW/CT/KS/RD of $0.963/0.963/0.212/0.814$ versus $0.975/0.975/0.197/0.799$ for training on all stored CSI [2312.04514], while CSI2Vec-based CC produced competitive charts from 16-dimensional embeddings rather than raw 1024-dimensional features across simulated and measured datasets [2506.05237].

Several limitations recur across the literature. Standard CC preserves local geometry but not global coordinates, motivating anchor-based, AP-based, model-based, or DT-based grounding [2308.14498][2511.09227]. Feature and metric choice remains decisive: the LoS analysis of phase-insensitive distance showed that inappropriate system design can create artificial neighbors through periodicity and sidelobes [2310.03762]. Temporal consistency over long horizons remains unresolved in streaming settings; the streaming CC paper explicitly identified chart consistency over time as an open problem [2312.04514]. Robustness to missing links, sensor failures, and scenario shift is nontrivial, which motivated set-based architectures and universal embeddings [2602.04704][2506.05237]. Some advances remain simulation-based, while others are validated on measured 5G or CSI datasets; the degree of real-world maturity therefore varies across subtopics [2212.12340][2606.18151].

A common misconception is that CC is merely unsupervised localization. More precisely, CC is a family of representation-learning methods for radio geometry. Localization is one outcome, but the same geometry-preserving latent spaces support beamforming, pilot scheduling, interference-aware control, handover reasoning, and trajectory-aware inference. The field’s recent trajectory—toward streaming memories, temporal models, variable-input architectures, weakly grounded coordinates, and environment shaping through electromagnetic skins—suggests that CC is increasingly being treated not only as a manifold-visualization tool, but as a systems primitive for structure-aware wireless intelligence [2508.07305][2511.00919].

Source: https://www.emergentmind.com/topics/channel-charting-cc