OGSBI: Off-grid Sparse Bayesian Inference

• Off-grid Sparse Bayesian Inference is a technique that uses first-order Taylor expansion and hierarchical Bayesian modeling to correct discretization biases in sparse signal estimation.
• The method employs closed-form EM updates to jointly estimate continuous-valued sparse support, amplitudes, and nuisance parameters such as mutual coupling and noise precision.
• Validated in direction finding and communications, OGSBI demonstrates super-resolution performance and robustness against model mismatches and coupling errors.

Off-grid Sparse Bayesian Inference (OGSBI) is a unified methodology for high-resolution parameter estimation in physical systems where signal sources are sparse but the underlying dictionary is discretized, resulting in “off-grid” model mismatch. OGSBI frameworks combine first-order Taylor expansion of the parametric measurement model with hierarchical Bayesian inference, enabling closed-form expectation-maximization (EM) estimation of continuous-valued sparse support, amplitudes, and any nuisance parameters such as mutual coupling or noise precision. The approach, validated in direction finding under unknown antenna coupling as well as broader array and communication systems, directly addresses discretization-induced bias and achieves super-resolution accuracy, even under challenging scenarios of model imperfection and moderate SNR (Chen et al., 2018).

1. Measurement Model and Off-grid Linearization

OGSBI models define the observed data $\mathbf{y}$ as a noisy linear transformation of a sparse signal, where discretization of the continuous parameter space induces a misalignment between the actual signal and the predefined dictionary. The canonical example is uniform linear array (ULA) direction finding, where true direction-of-arrival (DoA) values $\boldsymbol\theta = [\theta_1, ..., \theta_K]^T$ do not align with a discretized grid $\{\zeta_u\}$; each actual signal direction differs by an off-grid perturbation $\nu$. The steering vector for each source can be represented as $$a(\theta_k) \approx a(\zeta_{u_k}) + \nu_{u_k} a'(\zeta_{u_k})$$. This first-order Taylor linearization yields the measurement model:

$$\mathbf{y} = C \cdot A(\boldsymbol\theta + \boldsymbol\nu) \cdot \mathbf{x} + \mathbf{n},$$

where $C$ is a coupling matrix (e.g., unknown symmetric Toeplitz in array systems), $A(\cdot)$ is the steering-vector dictionary, $\mathbf{x}$ is the sparse signal, and $\mathbf{n}$ is additive Gaussian noise (Chen et al., 2018).

The generalized model stacks all dictionary atoms and derivatives to construct an “off-grid dictionary” $\Psi(\nu) = D + (\Gamma \otimes I_N)\Xi$, with $D$ and $\Xi$ comprising steering vectors and their derivatives, and $\Gamma$ encoding the off-grid shifts. The measurement equations can thus accommodate imperfections and unknown coupling vectors, facilitating robust formulation.

2. Hierarchical Bayesian Priors and Model Structure

OGSBI places conjugate priors on all unknowns, yielding a hierarchical structure for Bayesian inference. Signal amplitudes $x_m$ are assigned complex Gaussian priors parameterized by unknown variances $\gamma_i$, which themselves have Gamma priors. Noise precision $\alpha_n$ (inverse variance) follows a Gamma prior, as do any mutual coupling coefficients and their variances. The theoretical model is:

• $$x_m | \gamma \sim \mathcal{CN}(0, \mathrm{diag}(\gamma))$$
• $\gamma_i \sim \mathrm{Gamma}(a_\gamma, b_\gamma)$
• $n_m | \sigma^2 \sim \mathcal{CN}(0, \sigma^2 I_N)$
• $$\sigma^{-2} \equiv \alpha_n \sim \mathrm{Gamma}(a_n, b_n)$$
• $$c | \theta_c \sim \mathcal{CN}(0, \mathrm{diag}(\theta_c))$$
• $\theta_{c,n} \sim \mathrm{Gamma}(a_c, b_c)$

These hierarchical priors promote sparsity, allow robust estimation of nuisance parameters, and facilitate a fully Bayesian propagation of uncertainty (Chen et al., 2018).

3. Expectation-Maximization Inference and Posterior Update

OGSBI employs a closed-form EM algorithm, optimizing the complete-data joint log-likelihood over both the sparse signal support and all hyperparameters. In the E-step, the posterior mean and covariance of signal amplitudes are computed; the distribution is multivariate complex Gaussian, parameterized by the current estimates of grid shifts, coupling, and noise precision. In the M-step:

• Noise precision is updated via evidence maximization.
• Signal variances follow a closed-form update derived from the expected posterior.
• Coupling vector and variances are refined by solving linear systems.
• Off-grid shifts are updated by minimizing the Q-function (expected complete-data log-likelihood) with respect to $\nu$, admitting efficient closed-form or constrained quadratic solutions.

The algorithm repeats these E/M steps until parameter changes fall below a convergence threshold or a maximum number of iterations is reached, guaranteeing monotonic increase in the marginal likelihood (Chen et al., 2018).

4. Algorithmic Structure, Convergence, and Complexity Analysis

The OGSBI algorithm initializes all hyperparameters and iteratively refines the measurement model and sparse support via the following cycle:

1. Form the off-grid (Taylor-augmented) dictionary $\Psi(\nu)$ with the current coupling and grid shift estimates.
2. E-step: compute posterior mean and covariance of the sparse signal.
3. M-step: update all hyperparameters in closed form, including noise, signal, coupling, and grid shifts.
4. Project updated grid shifts and coupling parameters back into their domains if necessary.

Each iteration primarily costs $O(M U^2 N + U^3 + N^3)$ flops (where $U$ is grid size, $N$ is array size, $M$ is snapshots) due to matrix inversions and block operations (Chen et al., 2018). Convergence is well-behaved: the EM updates monotonically increase the evidence lower bound, and practical implementations require on the order of $100$–$500$ iterations.

5. Off-grid Correction, Mutual Coupling, and Super-resolution Performance

OGSBI exploits off-grid correction to surpass the fundamental error bounds of discretized (on-grid) approaches. At coarse grid spacing (e.g., $\delta = 1^\circ$), off-grid SBL yields root mean square error (RMSE) much lower than classical methods:

• On-grid SBL RMSE $\approx 0.3^\circ$
• OGSBI RMSE $\approx 0.35^\circ$
• OGSBI with coupling-awareness $\approx 0.08^\circ$ at SNR $= 20$ dB

Error vs. SNR curves demonstrate 5–10 dB performance gain over classical SBL and MUSIC under strong mutual coupling and grid mismatch (Chen et al., 2018). Coupling-blind methods degrade rapidly as true signals deviate from grid, whereas joint OGSBI approaches remain robust. The architecture enables accurate estimation even when array perturbations, model errors, or noise impede conventional techniques.

6. Comparative Analysis, Applicability, and Limitations

OGSBI stands out by fusing first-order continuous-angle modeling with full Bayesian hierarchical inference. It generalizes efficiently to arbitrary array geometries, diverse coupling structures, corrupted measurements, and broader classes of sparse inverse problems. Comparative studies illustrate major reductions in modeling bias and improvements in resolution—superseding on-grid SBL, OGSBI without coupling correction, and classical subspace methods.

Limitations emerge when the local linearization breaks down (i.e., grid offsets exceed the validity domain of Taylor approximation), or when extremely large dictionary sizes induce computational bottlenecks in practical implementations. Nevertheless, performance remains stable in moderate regimes, with documented subdegree estimation error and fast convergence under realistic SNR (Chen et al., 2018).

7. Significance and Research Directions

OGSBI enables robust super-resolution sparse estimation under practical impairments such as mutual coupling and discretization errors. The closed-form EM updates, hierarchical Bayesian priors, and direct accounting of off-grid and coupling effects mark a significant advancement in array signal processing. Extensions may address generalization to 2-D/3-D parametric models, incorporation of structured sparsity priors, or acceleration via low-rank and active-set subspace methods. The OGSBI paradigm continues to be influential in radar, communications, and array processing theory, providing a foundation for next-generation high-fidelity sparse inference algorithms (Chen et al., 2018).