Adaptive K-Means for Beamforming

• The paper demonstrates an adaptive K-means clustering approach that compresses high-dimensional CSI into compact codebooks for efficient beamforming design.
• It tailors clustering structure and similarity metrics, using techniques like k-means++ and dynamic K adaptation, to balance quantization distortion and feedback overhead.
• Simulations reveal significant performance gains, such as a 54% goodput improvement and near-optimal beamforming gains, across WLAN, massive MIMO, and UAV-assisted networks.

Adaptive K-means clustering for beamforming is a class of methodologies in which K-means–type unsupervised algorithms are leveraged to generate compact feedback codebooks or cluster user groups for advanced multi-antenna wireless communications. In this paradigm, the structure and cardinality of the clustering, as well as the metric of similarity, are tailored to the channel or system characteristics to maximize beamforming efficiency while minimizing feedback overhead or transmit power, thereby achieving near-optimal system capacity and robustness under resource constraints.

1. Problem Statement and Clustering Formalisms

The central objective is to compress high-dimensional channel state information (CSI) or beamforming matrix feedback into minimal representations, or to partition users for joint transmission, without compromising link performance. In feedback reduction for WLANs, the approach transforms the original matrix-valued beamforming problem into the construction of a K-codeword codebook of candidate precoding matrices, where K is tuned for the trade-off between quantization distortion and feedback overhead (Deshmukh et al., 2023). Analogous clustering-driven codebook design is pursued on the Grassmann manifold for massive MIMO CSI quantization, where codewords correspond to the centroids minimizing the average squared chordal distance to principal channel singular vectors (Bhogi et al., 2020). In robust UAV-NOMA, clustering is applied directly to user coordinates to minimize intra-cluster channel disparity, improving the tractability and efficacy of downstream beamforming design (Xu et al., 2020).

2. Clustering Algorithms and Distance Metrics

K-means and its variants (notably k-means++) are deployed as the core unsupervised clustering mechanism. For codebook design in WLANs, the training set consists of serialized unitary beamforming matrices, $X = \{x_i\}$, each $x_i$ obtained by stacking entries of the precoding matrix from channel SVD; the objective is to minimize within-cluster Euclidean or angular (cosine) distances. Initialization via k-means++ probabilistically seeds centroids for improved convergence (Deshmukh et al., 2023, Xu et al., 2020). For product codebooks in massive MIMO, centroids are optimized on the Grassmannian $\mathcal{G}(M,1)$ using the squared chordal distance $d^2(f_1,f_2)=1-|f_1^Hf_2|^2$, and the update step computes leading eigenvectors of cluster sample covariances (Bhogi et al., 2020).

The choice of distance metric is pivotal. In WLAN feedback, cosine distance (CD), $d_{\rm CD}(v_a,v_b)=1-\frac{|v_a^Hv_b|}{N_c^2}$ (for unitary matrices), yields superior codebook quality and link metrics compared to squared Euclidean distance. In Grassmannian k-means, the metric aligns with maximum beamforming gain objectives, ensuring that centroid directions capture the statistical structure of principal transmit directions.

3. Adaptive Codebook Construction and System Integration

Adaptive selection of the cluster count K is supported via two rules: the elbow method on quantization distortion (minimize K subject to variance threshold matching the PER target) and channel-statistics-driven scaling (proportional to the condition number $\kappa$ of the transmit covariance $R=\mathbb{E}[HH^H]$) (Deshmukh et al., 2023). After clustering, centroids are reshaped to matrices and orthonormalized (e.g., via Gram–Schmidt) to ensure valid unitary beamforming, and indexed for index-based feedback (Deshmukh et al., 2023). In Grassmannian product codebooks, codeword complexity is reduced by partitioning the codebook along array dimensions, independently clustering vertical and horizontal singular vectors (Bhogi et al., 2020).

In UAV-NOMA, user clustering is performed using k-means++ on user positions to form spatially compact clusters, followed by robust beamformer design using semidefinite programming; the clustering operates independently of subsequent CSI uncertainty models and QoS requirements (Xu et al., 2020).

4. Feedback Overhead Reduction and System Performance

Quantization via codebook indices provides a dramatic reduction in multibit feedback compared to element-wise or compressed feedback schemes. For instance, in the IEEE 802.11be context, index-based k-means clustering reduces per-group feedback from $B_0=130$ bits to $b=10$ bits for $K=2^{10}$ codewords, yielding net overhead savings of $G\cdot(B_0-b)$ bits across $G$ subcarrier groups (Deshmukh et al., 2023). Simulations demonstrate up to 54% goodput improvement at 32 dB SNR despite only a $\sim$2 dB packet error rate degradation relative to baseline, unlocking higher-order MCS operation at lower SNR and outperforming previous index-based methods by 4 dB at PER 0.01 for MCS 11.

In massive MIMO, Grassmannian k-means codebooks achieve normalized average beamforming gain $\Gamma_{\rm av}\approx0.97$ (4×4 UPA, B=8 bits), consistently within 1–2 dB of ideal MRT and outperforming DFT or random line-packing codebooks by 4–6 dB at identical feedback rates (Bhogi et al., 2020).

5. Algorithmic Complexity, Adaptivity, and Practical Considerations

Standard k-means complexity is $\mathcal{O}(NKD)$ per iteration (N: samples, K: clusters, D: dimension); Grassmannian variants incur extra cost from eigen-decomposition but remain tractable for moderate K. Product codebooks ameliorate the curse of dimensionality by independently clustering lower-dimensional marginals (Bhogi et al., 2020). Adaptive elements include:

• Dynamic K adaptation: K tuned to instantaneous channel statistics or application requirements (feedback–distortion trade-off) (Deshmukh et al., 2023).
• Metric switching: SED for diversity in offline clustering, CD for deployment/selection. Mixed-metric strategies empirically maximize NMSE and cosine-similarity performance (Deshmukh et al., 2023).
• Online clustering: Mini-batch or periodically updated codebooks permit tracking of non-stationary environments.
• Distributed optimization: In UAV-NOMA, ADMM enables decentralized, scalable robust beamforming once clustering is complete (Xu et al., 2020).

6. Domain-Specific Extensions and Simulation Evidence

Extensive validation spans WLANs (multi-antenna, OFDM with subcarrier grouping), indoor and LOS/NLOS massive MIMO (full-dimension UPA), and UAV-assisted NOMA (3-D user clustering). Simulations in each case demonstrate the tight linkage between adaptive cluster/codebook design and system Pareto-optimality in overhead, throughput, and power minimization. Notably, clustering-centric feedback schemes retain near-optimal PER, with only minimal cost in SNR, and robustly accommodate evolving device and propagation statistics (Deshmukh et al., 2023, Bhogi et al., 2020, Xu et al., 2020).

7. Limitations and Future Directions

Current schemes rely on adequate offline sampling of channel conditions to ensure representative training sets. The non-convexity of K-means induces local minima, although initialization heuristics (k-means++) ameliorate this to some extent (Xu et al., 2020). K-means and Grassmannian algorithms are not universally optimal in the presence of colored noise, highly-correlated antennas, or heterogeneous user densities. Future directions may include joint clustering and codebook learning with temporal adaptation, cross-layer codebook inference leveraging data traffic models, and scalable non-Euclidean clustering for extreme-scale antenna systems, as suggested by simulation constraints and the need for rapid feedback/codebook updates (Deshmukh et al., 2023, Bhogi et al., 2020).