Hierarchical Block OMP with Prior Support (HiBOMP-P)

• The paper introduces HiBOMP-P, a recursive greedy algorithm that integrates hierarchical block structure and PSI for efficient sparse signal recovery.
• It augments classical OMP by incorporating multi-level block partitions and PSI, enhancing support selection even under noisy conditions.
• Analytical guarantees based on a hierarchical mutual incoherence framework provide precise recovery bounds despite imperfect side information.

Hierarchical Block Orthogonal Matching Pursuit with Prior Support Information (HiBOMP-P) is a recursive greedy algorithm designed to recover hierarchically block-sparse signals from possibly noisy linear measurements, in the setting where side-information about likely signal support may be available. HiBOMP-P generalizes the classical orthogonal matching pursuit (OMP) and block OMP (BOMP) paradigms by imposing a multi-level block partition (“hierarchy”) and incorporating both matched and “mislocated” prior support information (PSI) directly into each selection step. The method is underpinned by new recovery guarantees based on a hierarchical generalization of the mutual incoherence property (MIP), admitting precise and computable performance bounds even when the side information is inexact.

1. Hierarchical Model and Prior Support Information Structure

The measurement model takes the form: $$y = D x + e, \qquad e \in \mathbb{C}^M, \quad \|e\|_2 \leq \varepsilon,$$ where $D \in \mathbb{C}^{M \times (N_1 \cdots N_n d)}$ is a dictionary with unit-norm columns, and the unknown signal $x$ is hierarchically block-structured: it is partitioned into $n$ hierarchical modes, with $N_1$ blocks in mode 1, each further split into $N_2$ sub-blocks at mode 2, and down to mode $n$ with block size $d$.

A signal is termed $(k_1, k_2, \ldots, k_n)$ hierarchically block-sparse if, at each mode $t$, at most $k_t$ blocks (or subblocks) are nonzero. Prior support information (PSI) enters as pairs $(\Theta^t, \Theta^{*\Delta, t})$ per mode:

• $\Theta^t$ contains indices of “suspected” support blocks,
• $\Theta^{*\Delta, t}$ is a possibly disjoint set of additional indices, equipped with weights.

Cardinality overlaps between true support ($\Xi^{*, t}$) and PSI ($\Theta^t, \Theta^{*\Delta, t}$) are tracked with indices such as $\alpha^{*\, t} = |\Xi^{*, t} \cap \Theta^t|$ and other counts for extended and null intersections. This combinatorial structure is crucial for analyzing the efficacy and resilience of PSI in the algorithmic process.

2. PSI-Augmented Recursive Selection Criterion

HiBOMP-P modifies the classical greedy selection by adjusting the residual with PSI-derived components at each iteration. Denoting the current residual at iteration $\ell$ as $r^\ell$, an “augmented” residual is constructed: $$\bar{r}^\ell = r^\ell + D_{\Theta^{*\Delta, t}} x_{*\Delta, t}^\ell,$$ where $x_{*\Delta, t}^\ell$ represents the temporary coefficients assigned to the extra PSI blocks during the current search.

The support block for mode $t$ is chosen as

$$i_\ell = \arg\max_{i\notin\Xi^t_\ell} \left\| \left(P^{\perp}_{D_{\Xi^t_\ell \cup \Theta^t}} D_{[i]}\right)^H \bar{r}^\ell \right\|_2,$$

where $P^{\perp}_{D_S}$ is the projector orthogonal to the span of columns indexed by $S$.

After block selection, the PSI injection is “undone”: $$r^\ell \gets \bar{r}^\ell - D_{\Theta^{*\Delta, t}} x_{*\Delta, t}^\ell.$$ Recursion proceeds down the hierarchy modes, with a least-squares update performed when descent reaches the finest resolution ($t = n$). This approach ensures the algorithm’s selection process is steered by both the data and injected PSI, enabling the exploitation of side information even when imperfect.

3. Algorithmic Structure and Pseudocode

The recursion underlying HiBOMP-P can be summarized as follows:

1. Initialization: Set all hierarchical support containers to empty, $\Xi^t \gets \emptyset$, and the initial residual $r^0 \gets y$.
2. Recursion: For each mode $t$ and search level $\ell$:
   • Inject PSI by augmenting residual,
   • Select the maximal block index using the projected, PSI-tilted residual,
   • Remove the PSI injection and update current support,
   • Recurse to the next subordinate mode or, upon reaching mode $n$, solve a least-squares problem on the accumulated support,
   • Repeat the process while the residual norm exceeds the preset threshold $\epsilon$.

A brief pseudocode schematic:

Step	Operation	Purpose
1	Initialize supports $\Xi^t$, residual	Preparation for recursive steps
2	Recursively call SelectBlock from mode 1	Begin hierarchical block selection process
3	Inject PSI in the residual	Incorporate prior information to tilt selection toward suspected blocks
4	Select block index maximizing objective	Greedy search using current augmented residual
5	Remove PSI injection from residual	Ensure unbiased descent after selection
6	Update support and recurse to finer modes	Hierarchical enforcement
7	Upon base mode, perform LS update	Coefficient estimation on gathered supports
8	Check residual norm for termination	Stopping condition

At each recursive level, steps 3–5 guarantee that the influence of PSI is local to the selection phase, offering both guidance and immunity from dominance by erroneous support beliefs.

4. Analytical Guarantees and Hierarchical Mutual Incoherence

Precise recovery analysis is anchored by a hierarchy of mutual incoherence parameters—generalizations of classic notions such as column and block coherence. For mode $t$ of minimum block size $d_t$: $$\mu_{d_t} = \max_{I \neq J} \frac{\|D_I^H D_J\|_2}{d_t}, \qquad \nu_{d_t} = \max_{i \neq j,\, i,j \in I} |\langle D_i, D_j \rangle|,$$ with $D_I$ the submatrix corresponding to block index set $I$.

Exact recovery at each mode is governed by the sum: $G_* + G_\circ < 1,$ where $G_*$, $G_\circ$ quantify multi-block interference and depend on MIP quantities and signal-to-residual ratios. Sufficient conditions are given in terms of block lengths, sparsity, and hierarchy parameters, for instance: $$k_t (d^* + d^{*\Delta}) < \frac{1 - (d^* + d^{*\Delta} - 1)\nu - (d^* + d^{*\Delta})\mu}{\mu} + (d^* + d^{*\Delta}) = K^*,$$ establishing the maximum reconstructible sparsity per mode and, transitively, for the global problem. This analytical form reveals the explicit role of both true and side-information block counts at each level, enabling direct tuning of sparsity levels to available incoherence.

5. Robustness in Noisy Settings and Hierarchy Optimization

In presence of noise ($\|e\|_2 \leq \varepsilon$), HiBOMP-P maintains its selection guarantees if, for each mode,

$$\delta_{\min}\|x_{\rm new}\| - \sqrt{\delta_{\rm int}}\|x_{\rm outside}\| > \frac{2\|(A_{\overline{\Xi} \setminus \Theta})^H e\|_{(\bar{d})2,\infty}}{1 - (\overline{G}_* + \overline{G}_\circ)},$$

with upper bounds such as

$$\|(A_{\overline{\Xi}\setminus\Theta})^H e\|_{(\bar{d})2,\infty} \leq \sqrt{\bar{d}}\,\varepsilon.$$

This leads to a halting criterion immediately after all true blocks have been selected, provided the noise amplification through the dictionary remains sufficiently small.

A notable conclusion is that an “optimal” hierarchical configuration—where $k_1 = \cdots = k_{n-1} = 1$, concentrating nonzeroes in the last mode—eliminates inter-mode interference, simplifying the recovery conditions and permitting higher sparsity thresholds. This structural insight enables optimized signal model design when applicable.

6. Empirical Findings and PSI Mismatch

Empirical results in regimes such as noiseless (2-PAM) and noisy (Gaussian) scenarios demonstrate that HiBOMP-P consistently outperforms standard OMP and BOMP (and their hierarchical variants) across support-recovery error and MSE metrics. Notably, a critical qualitative result is obtained: even with completely non-overlapping side information ($\Theta \cap \Xi^* = \emptyset$), the inclusion of “extra” suspected blocks ($\Theta^{*\Delta}$) improves the algorithm’s ability to “park” the residual in subspaces containing the true support. The derived bound on reconstructible sparsity $K^*$ increases monotonically with the cardinality of $\Theta^{*\Delta}$ so long as these blocks possess sufficient column coherence mass with the dictionary $D$.

This suggests a pronounced resilience of HiBOMP-P to side information–support mismatch, in contrast to classical approaches where non-overlapping PSI can degrade performance. When purely additional PSI is used ($\Theta^{*\Delta}$ only), HiBOMP-P3 achieves the lowest observed support-recovery and MSE errors among tested algorithms.

7. Significance, Applications, and Methodological Summary

HiBOMP-P constitutes an integration of hierarchical compressed sensing and side-information-driven pursuit, yielding a procedure characterized by:

• Recursive selection fusing hierarchical block structure and PSI in each greedy step,
• Explicit, closed-form mutual incoherence–based exact-recovery conditions,
• Direct extension to bounded-noise settings with error control scaling as $\sqrt{\bar{d}}\,\varepsilon$,
• Hierarchical model design guidance for maximizing recoverable sparsity,
• Robust performance when prior support knowledge is incomplete or even disjoint from the true support.

A plausible implication is that HiBOMP-P is suited to large-scale, structured sparse recovery applications where side information is heterogeneous—characteristic of modern multi-resolution sensing, multi-user communications, and joint inference tasks where auxiliary sources or historical estimates inform the search for signal support. The theoretical framework and algorithmic mechanisms outlined ensure principled and tunable exploitation of side information, even under significant model mismatch.