Rank-Aware Block Orthogonal Matching Pursuit

• RA-BOMP is a rank-aware, iterative algorithm that recovers block-sparse signals in time-varying mmWave MIMO channels by dynamically enforcing exact sparsity constraints.
• It leverages robust rank-one matrix completion to determine the precise number of iterations and support size, improving estimation accuracy and reducing computational complexity.
• The adaptive measurement matrix design optimizes energy allocation and angular resolution, ensuring reliable performance in high-mobility, dynamic wireless environments.

Rank-Aware Block Orthogonal Matching Pursuit (RA-BOMP) is an iterative greedy algorithm designed for efficient sparse recovery in block-sparse signal models, with a specific focus on time-varying millimeter-wave (mmWave) multiple-input multiple-output (MIMO) channel estimation. It extends classical Block Orthogonal Matching Pursuit by incorporating dynamically estimated rank information to enforce exact sparsity constraints, improving estimation accuracy, computational efficiency, and robustness in rapidly changing wireless scenarios.

1. Problem Setting and Block-Sparse Model

RA-BOMP addresses the recovery of a time-varying mmWave MIMO channel in its virtual (beamspace) representation. The virtual channel matrix $\bar{H}_t \in \mathbb{C}^{L_1 \times L_2}$ exhibits block sparsity: only a small number $R_t = \text{rank}(H_t) \ll \min(L_1, L_2)$ of its rows or columns contain nonzero blocks corresponding to active Angle-of-Arrival (AoA) and Angle-of-Departure (AoD) pairs. The matrix is partitioned into $N$ blocks $\{\bar{H}_t^{[b]}\}_{b=1}^B$, each covering a grid region in the angular domain.

Observations are obtained via an over-complete dictionary (measurement matrix) $$\mathcal{D}_t = \Theta_{\text{MS}} \otimes \Theta_{\text{BS}}^H$$, where $\Theta_{\text{MS}}, \Theta_{\text{BS}}$ are the receive/transmit steering dictionaries. After initial matrix completion, the observation can be vectorized as

$\mathbf{y}_t = \Phi_t \mathbf{x}_t + \mathbf{n}_t,$

where:

• $$\mathbf{x}_t = \operatorname{vec}(\bar{H}_t) \in \mathbb{C}^N$$ is block-sparse, with $|S_t| = R_t$ active blocks,
• $\Phi_t = \mathcal{X}_t = F_t^T \otimes W_t^H$ is the known sensing matrix (hybrid precoder $F_t$, combiner $W_t$),
• $\mathbf{y}_t \in \mathbb{C}^M$ is the completed observation,
• $\mathbf{n}_t \sim \mathcal{N}(0, \sigma^2 I)$.

This casting enables the application of block-sparse signal recovery methods, with block support corresponding to angular clusters.

2. RA-BOMP Algorithmic Framework

RA-BOMP integrates a dynamically estimated rank $\widehat{R}_t$, obtained from robust rank-one matrix completion (R1MC), to enforce exact iteration count and support size in block selection. This eliminates heuristic stopping rules and prevents over-selection, which is especially crucial for low-rank physical channels.

At each iteration $n$:

• The current residual is updated as $$\mathbf{r}^{(n)} = \mathbf{y}_t - \Phi_t \hat{\mathbf{x}}^{(n)}$$.
• Block selection proceeds by maximizing the block-correlation:

$$d^{(n)} = \underset{b \notin J^{(n)}}{\arg\max} \; \| [\Phi_t^{[b]}]^H \mathbf{r}^{(n)} \|_F,$$

where $\Phi_t^{[b]}$ is the submatrix for block $b$ and $J^{(n)}$ is the set of selected block indices.

• The support is augmented: $J^{(n)} = J^{(n-1)} \cup \{ d^{(n)} \}$.
• Coefficient update: solve the least-squares problem for the selected active blocks.
• Update residual; repeat for exactly $n = 1, ..., \widehat{R}_t$ iterations.

A gain-only update (least squares only) is optionally performed if the estimated rank and support remain unchanged across time steps, exploiting the temporal coherence of AoA/AoD.

Pseudocode

Algorithm 1: Rank-Aware Block Orthogonal Matching Pursuit (RA-BOMP) Inputs: y_t ∈ ℂ^M, Φ_t ∈ ℂ^{M×N}, block partition {Φ_t^[b]}_{b=1}^B, rank estimate 𝑅̂ₜ. Initialize: r^(0) ← yₜ, J^(0) ← ∅, x̂^(0) ← 0. For n = 1 to 𝑅̂ₜ do  1. d^(n) = arg max_{b∉J^(n−1)} ‖Φ_t^[b]^H r^(n−1)‖_F.  2. J^(n) = J^(n−1) ∪ {d^(n)}.  3. x̂_{J^(n)} = arg min_{u} ‖yₜ − Φ_t_{J^(n)} u‖₂².  4. r^(n) = yₜ − Φ_t_{J^(n)} x̂_{J^(n)}. End For Output: x̂^(𝑅̂ₜ), support J^(𝑅̂ₜ).

3. Rank-Aware Measurement Matrix Design

The measurement matrix $\Phi_{t+1}$ is adaptively designed, leveraging prior channel support:

1. Energy-based angular clustering: For each quantized angle, compute $\eta_{[\ell]} = \| \hat{H}_t^H D_{t,[:,\ell]} \|_2$, then select the top $\widehat{R}_t$ angles as cluster centroids $\{ \hat{\theta}_i \}$.
2. Beam-steering matrix construction: For each angular cluster, angular samples $\tilde{\theta}_\ell$ are taken uniformly within the interval $[\hat{\theta}_i^{\min}, \hat{\theta}_i^{\max}]$. The receive steering submatrix

$$\Theta_{\text{MS},[i]} = [ a_{\text{MS}}(\tilde{\theta}_1), \ldots, a_{\text{MS}}(\tilde{\theta}_{L_1}) ] \in \mathbb{C}^{N_{\text{MS}}\times L_1 }$$

is used to update the dictionary slice.

1. The sensing matrix includes all selected angular sectors $$[ D_{t+1,(:,1)}, \ldots, D_{t+1,(:,\widehat{R}_t)} ]$$ and a set of randomized beams to avoid missed paths.

This adaptive construction enhances the accuracy of angle estimation, steers sensing energy efficiently, and enables rapid updating in dynamic environments.

4. Computational Complexity

Let $L_1 \cdot L_2 = N$, support size $k$, sparsity $R = \widehat{R}_t$, and $M = N_{\text{MS}} \cdot N_{\text{BS}}$.

• Per iteration:
  • Block-correlation over $B$ blocks: $O(NM)$.
  • Least squares over $k$ active blocks: $O(k^2 M + k^3)$.
• Total cost for $R$ iterations:

$O(RNM + R^3 + R^2 M)$

Because $R \ll N$ and is enforced exactly, RA-BOMP achieves efficiency by minimizing both the number of iterations and the subproblem sizes, unlike standard BOMP which must use conservative sparsity bounds $s_{\max} \gg R$.

Algorithm	Iterations	LS Subproblem Size	Total Complexity
RA-BOMP	$R$	$O(R)$	$O(RNM + R^3 + R^2 M)$
Classical BOMP	$s_{\max}$	$O(s_{\max})$	$O(s_{\max}NM + s_{\max}^3)$

This complexity advantage is particularly significant in large-scale arrays or highly dynamic situations.

5. Numerical Performance and Robustness

Comprehensive simulation studies demonstrate the practical benefits of RA-BOMP under a variety of conditions (with $N_{\text{BS}} = N_{\text{MS}} = 8$, mobility $v = 120$ km/h, and up to $20\%$ missing/corrupted samples):

• Reconstruction probability vs. SNR: RA-BOMP plus R1MC achieves $>90\%$ successful recovery even at $0$ dB SNR, a 10–20 dB improvement over SPC-TDCS and standard SOMP.
• NMSE vs. pilot overhead: Achieves an NMSE floor of $-20$ dB using only $\sim6\%$ pilot symbols, whereas benchmarks require $\geq 10\%$.
• NMSE vs. SNR and array size: As $N_{\text{BS}}$ increases from $8$ to $64$, RA-BOMP's NMSE improves by $5$–$10$ dB over baselines at moderate SNR.
• Robustness to angle spread: Maintains a $5$–$8$ dB NMSE advantage as angle spread $\Delta \theta$ increases to $10^\circ$.
• Ablation (no rank-restraint): Disabling rank-aware stopping degrades NMSE by $>6$ dB at high SNR.
• BER performance: In QPSK link-level tests, RA-BOMP achieves BER $<10^{-3}$ at $15$ dB SNR, versus $>10^{-2}$ for SOMP or CNN-based methods.

These results evidence gains not only in estimation quality and robustness but also in system-level metrics such as reduced training overhead and support for high-mobility scenarios.

6. Significance and Applications

RA-BOMP exemplifies the integration of low-rank and sparse priors in compressed sensing, providing an efficient solution for fast time-varying mmWave MIMO channel estimation where physical rank varies rapidly. By using the exact, dynamically-estimated rank to bound iterations and guide support selection, it eliminates ad hoc parameter tuning and enhances both accuracy and computational tractability. Its adaptive measurement matrix design further allows for energy-efficient and robust angle-resolution in changing environments.

The RA-BOMP framework is particularly suited to large-scale and high-mobility wireless systems, where traditional compressed sensing and convex-relaxation approaches may incur excessive computation or pilot overhead. A plausible implication is that similar rank-aware methods can be extended to other structured signal recovery contexts, where low-rank and block-sparse structures co-exist.