Multipath Matching Pursuit Algorithms

• Multipath Matching Pursuit algorithms are greedy sparse recovery techniques that explore multiple candidate supports concurrently to improve performance over traditional single-path methods.
• They use a tree search strategy with branching and pruning to efficiently handle high-coherence and underdetermined measurement scenarios.
• These methods achieve enhanced recovery accuracy and robustness in applications like sparse multipath channel estimation while offering tunable tradeoffs between computational complexity and performance.

Multipath Matching Pursuit algorithms represent a family of greedy sparse recovery procedures that generalize conventional Matching Pursuit (MP) and Orthogonal Matching Pursuit (OMP) by pursuing and evaluating multiple candidate paths per iteration, often formulated as a tree search. These methods were developed to address limitations of single-path greedy approaches in high-coherence and underdetermined sensing scenarios, particularly for sparse multipath channel estimation and compressed sensing. They exploit the combinatorial structure of support selection, explore multiple support hypotheses, and use explicit or implicit pruning strategies to maintain computational feasibility. What distinguishes multipath matching pursuit is the investigation and maintenance of several promising candidates in parallel, rather than irreversible, myopic expansion along a single path.

1. Algorithmic Structure and Core Principles

Multipath Matching Pursuit (MMP) algorithms iteratively construct a set of candidate supports by exploring a tree where each node corresponds to a partial support. At each layer $k$, each current candidate (partial support) is expanded by selecting the $L$ best-scoring atoms (according to the residual correlation or other metrics), yielding up to $L^k$ hypotheses at depth $k$. After reaching the target sparsity $K$, the algorithm selects the final solution as the candidate with the minimal residual norm.

The canonical signal model is: $y = \Phi x + v$ where $y \in \mathbb{R}^M$ is the measurement, $\Phi \in \mathbb{R}^{M \times N}$ is the sensing matrix, $x$ is $K$-sparse, and $v$ is noise. The algorithm tracks singleton and multi-index sets of support candidates, replacing the brute-force ${N \choose K}$ search with a greedy beam search over a smaller tree of promising candidates.

This approach generalizes OMP, which is a single-path, single-candidate greedy pursuit, to the multipath case. Variants such as breadth-first (MMP-BF), depth-first (MMP-DF), hope-tree-based methods, and pruned tree search or multi-branch paradigms have been proposed, differing primarily in tree expansion and pruning mechanisms (Suhyuk et al., 2013, Rossi et al., 2013, Li et al., 2017, Lee et al., 2014).

2. Main Variants and Algorithmic Enhancements

Several multipath matching pursuit algorithms have been developed to improve recovery accuracy and reliability:

• Basic Multipath Matching Pursuit (MMP): Executes a breadth-first beam search, generating $L$ children per candidate node at each of $K$ levels. After $K$ iterations, all length-$K$ candidates are formed and the minimal-residual path is selected (Suhyuk et al., 2013).
• Multi-Branch Matching Pursuit (MBMP): Employs a variable branching vector $d = [d_1, ..., d_K]$ controlling the number of children at each tree level, with explicit support for multiple measurement vectors (MMV) through subspace projections. MBMP generalizes rank-aware and tree-based pursuits, introduces the MB-coherence theoretical condition, and allows performance-complexity tradeoffs (Rossi et al., 2013).
• Hope-Tree Pursuit (GSRA): Uses the intersection of OMP and Subspace Pursuit supports as initialization. Greedily expands a hope-tree beam of width $L$ and depth $N$, with a subsequent decreasing subspace pursuit refinement stage to correct missed atoms. Complexity is managed by restricting the number of explored candidates (Li et al., 2017).
• Tree-Pruned Matching Pursuit (TMP): Incorporates pre-selection (typically with gOMP) to reduce the search space to a small superset, followed by a pruned tree search where each path is expanded by one atom and unpromising paths are aggressively discarded via residual-based pruning. Recursive noncausal set selection enhances efficiency (Lee et al., 2014).

A comparison of several algorithms is provided below:

Algorithm	Candidate Expansion	Pruning	Initialization
MMP-BF/DF	$L$ per node, all layers	Breadth/depth limits	None
MBMP	$d_i$ per level	Branch vector, residual	None
GSRA (Hope-Tree)	$L$ per node, $N$ depth	Max candidate limit	OMP ∩ SP intersection
TMP	$1$ per node, $K$ depth	Residual/metric bounds	gOMP-based pre-selection

These algorithms differ in the number of expanded paths, pruning guards, and initialization strategies, but share the core multipath expansion principle.

3. Theoretical Performance Guarantees

Multipath matching pursuit recovers $K$-sparse solutions under less restrictive conditions than single-path greedy algorithms. The analysis is primarily based on the Restricted Isometry Property (RIP) or coherence bounds:

• MMP (Breadth-First): If $\Phi$ satisfies RIP of order $K+L$ with constant $$\delta_{K+L} < \frac{\sqrt{L}}{\sqrt{K} + 2\sqrt{L}}$$, then exact recovery of every $K$-sparse vector is guaranteed with $L$-branching (Suhyuk et al., 2013).
• TMP: When used with group OMP pre-selection of size $p$ and branch size $L$, TMP guarantees noiseless recovery for $\delta_Q < \frac{\sqrt{L}}{\sqrt{L} + \sqrt{K}}$ (or $\delta_Q < 1/3$ for $K < 4L$, $Q = \max\{L+K, M\}$), with stability guarantees in the noisy case (Lee et al., 2014).
• MBMP: Introduces the MB-coherence condition, which generalizes the exact recovery condition (ERC) by considering the $d$-th largest off-support correlation for $d$-branch expansion. MBMP can succeed with fewer measurements than classical ERC or cumulative-coherence-based methods, especially for large $d$ (Rossi et al., 2013).
• GSRA (Hope-Tree): Under RIP of order $NK$, the hope-tree search combined with backtracking refinement contains the true support in one of its paths, ensuring exact recovery in the noise-free case (Li et al., 2017).

A plausible implication is that by increasing the number of simultaneous paths or branches, one relaxes the strictness of the RIP (or coherence) requirements compared to single-path greedy selection.

4. Computational Complexity and Practical Implementation

The intrinsic tradeoff in multipath matching pursuit is between increased recovery reliability and computational complexity:

• Breadth-First Tree Expansion: In MMP-BF, up to $L^K$ candidates can be generated, with each candidate requiring $O(M N)$ (correlation) and $O(K^3)$ (least-squares update) per iteration, leading to $O(L^K (M N + K^3))$ in the worst case. In practice, tree overlap and duplicate pruning can significantly reduce actual costs (Suhyuk et al., 2013).
• Depth-First with Capping (MMP-DF, GSRA): By limiting the total number of explored candidates to $N_{\max}$, computational cost is $O(N_{\max}(MN + K^3))$ and can be tuned to be similar to that of CoSaMP or Subspace Pursuit for moderate $K$ (Suhyuk et al., 2013, Li et al., 2017).
• Pre-selection and Pruning: TMP and GSRA leverage pre-selection and aggressive path pruning to restrict the effective search space; typical pre-selection sizes $p \approx 2K$ and branch limits $N_{\max} \approx 10$--$20$ bring runtimes to within a small constant factor of OMP (Lee et al., 2014, Li et al., 2017).

Multipath approaches are generally more demanding than OMP or CoSaMP for cost but offer substantial performance gains when $K/N$ increases or when sensing matrices exhibit high mutual coherence.

5. Applications in Sparse Multipath Channel Estimation

Multipath matching pursuit methods were originally developed in the context of sparse multipath channel estimation, where the channel is modeled as a superposition of a small number of propagation paths (taps), and the measurement process involves short training sequences relative to the channel length:

• Enhanced Support Recovery: MMP and variants achieve exact-recovery rates for higher $K$ than OMP, notably in settings where standard greedy methods fail due to atom collisions or coherent dictionary structure. For instance, with $M=100, N=256$ and $K=30$, MMP provides substantially better miss/false-alarm rates over OMP, CoSaMP, and SP (Suhyuk et al., 2013).
• Error Performance: Multipath matching pursuit algorithms maintain MSE close to the Oracle-LS lower bound in high-SNR regimes, especially for moderate-to-large $K$ (Suhyuk et al., 2013, Li et al., 2017).
• Tree-Pruned Matching: TMP achieves higher noiseless recovery ratios and lower MSE in both noisy and noiseless environments, approaching the Oracle bound at high SNR (Lee et al., 2014).

A typical application is wideband wireless communications, where accurate estimation of a sparse impulse response enables channel equalization, localization, and multi-user detection.

6. Extensions: Structured, Multi-Branch, and High-Dimensional Pursuits

Multipath Matching Pursuit frameworks have been extended for domain-specific structures and complex recovery settings:

• Structured Matching Pursuit (SMP): For dynamic sparse channels with temporally correlated support (common vs. dynamic taps), SMP first detects persistent common taps across time slots and then refines slot-specific dynamic taps via a modified greedy selection, enabling efficient reconstruction under slowly time-varying support (Zhu et al., 2015).
• Multi-Branch and Rank-Aware Pursuits: MBMP exploits not only the multipath tree but also simultaneous multi-snapshot (MMV) structure, ranking candidates via subspace projections (Rossi et al., 2013).
• Multidimensional Matching Pursuit (MOMP): In high-dimensional parametric settings such as mmWave MIMO channel estimation, multidimensional pursuits avoid forming an exponentially-large global dictionary by using independent lower-dimensional dictionaries along each channel parameter axis (e.g., angle, delay), thus rendering tractable the estimation and localization of multipath parameters in 3D scenarios (Palacios et al., 2022, Palacios et al., 2022).

These extensions demonstrate the adaptability of the core multipath pursuit paradigm to models with inherent structure (temporal, spatial, or parametric), further expanding the algorithmic scope.

7. Significance, Limitations, and Comparative Position

Multipath Matching Pursuit achieves near-optimal support recovery under significantly milder conditions than classical greedy methods, at the expense of increased (but tunable) complexity. Empirical findings consistently show:

• Superior Sparsity Range: Substantially higher exact-recovery rates as sparsity increases, for both synthetic and real sparse signals.
• Robustness to Noise: Improved stability in the presence of observation noise, tracking the Oracle-LS bound at moderate-to-high SNR.
• Flexible Complexity-Performance Tradeoff: User-controllable parameters (e.g., branch counts, search depth) allow adaptation of runtime and recovery reliability to application constraints.
• Benefit in Coherent/Challenging Sensing Regimes: Particularly advantageous in underdetermined, coherent, or highly-structured sensing architectures where OMP and traditional greedy methods are inadequate.

Limitations include potentially exponential complexity in naive tree expansion, which must be controlled by pre-selection, pruning, or beam-capping strategies.

In summary, multipath matching pursuit algorithms constitute an important class of sparse recovery techniques that interpolate between single-path greedy selection and exhaustive search, providing substantial gains in support recovery for compressed sensing, sparse multipath channel estimation, and related applications (Suhyuk et al., 2013, Rossi et al., 2013, Lee et al., 2014, Li et al., 2017, Zhu et al., 2015, Palacios et al., 2022, Palacios et al., 2022).