Maximum Likelihood Matching Pursuit (MLMP)

• Maximum Likelihood Matching Pursuit (MLMP) is a greedy algorithm that selects candidate atoms based on likelihood ratios rather than raw correlation magnitudes.
• It leverages statistical decision theory to improve support recovery in compressed sensing and enhance decoding in high-noise, SIMO, and multi-user SPARC settings.
• By using successive combining instead of cancellation, MLMP attains computational efficiency and robust performance in non-coherent fading channel scenarios.

Maximum Likelihood Matching Pursuit (MLMP) is a family of greedy support-recovery and decoding methods in which candidate atoms or codeword components are selected by maximizing a likelihood-based score rather than a raw correlation magnitude. In the compressed-sensing literature, MLMP appears as the maximum-likelihood specialization of a per-index MAP support detector, obtained by removing prior odds from the selection rule (Lee, 2015). In the sparse regression code (SPARC) literature for non-coherent fading channels, MLMP denotes a practical decoder that approximates the marginalized non-coherent maximum-likelihood rule through greedy support growth and successive combining, first for SIMO channels and then for multi-user multiple-access channels with multiple receive antennas (Kancharana et al., 2024, Sandeep et al., 15 Jul 2025).

1. Terminological scope and research lineages

The acronym “MLMP” is used in at least two closely related but non-identical research lines. In the compressive-sensing setting, the paper “MAP Support Detection for Greedy Sparse Signal Recovery Algorithms in Compressive Sensing” defines MLMP as the maximum-likelihood counterpart of MAP-MP: the per-index score is the log-likelihood ratio, while the MAP version additionally includes prior odds (Lee, 2015). In that formulation, the statistical object is a sparse vector observed through a linear Gaussian measurement model.

A second line of work uses MLMP for non-coherent SPARC decoding. “Sparse Regression Codes for Non-coherent SIMO channels” introduces MLMP as a greedy decoder derived from the marginalized non-coherent likelihood $p(Y\mid s)$, emphasizing successive combining rather than successive cancellation (Kancharana et al., 2024). “Sparse Regression Codes exploit Multi-User Diversity without CSI” extends the same idea to a non-coherent multi-user multiple-access channel with multiple receive antennas, where the decoder greedily grows per-user codeword hypotheses via partial ML metrics (Sandeep et al., 15 Jul 2025).

A further point of comparison is provided by “Bayesian Approach with Extended Support Estimation for Sparse Regression,” which explicitly contrasts BMMP with Lee’s MLMP/MAP-OMP. There, MLMP is described as likelihood-ratio selection based on the normalized OMP correlation, whereas BMMP replaces that correlation with the Davies–Eldar RA-ORMP correlation and adds extended-support growth, multiple candidates, and refinement (Kim et al., 2019).

This terminology suggests that MLMP is best understood as a design principle—greedy matching pursuit driven by likelihood metrics—rather than as one single invariant algorithm.

2. Likelihood-ratio MLMP in sparse recovery

In the compressed-sensing formulation, the measurement model is

$\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$

with $\mathbf{\Phi}\in\mathbb{R}^{M\times N}$, $\mathbf{x}$ $K$-sparse, and $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$. Support detection is posed as binary hypothesis testing for each index $n$ at iteration $k$, using the normalized proxy

$$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$

with hypotheses $H_1(n): n\in \mathrm{supp}(\mathbf{x})$ and $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$0 (Lee, 2015).

The MAP rule compares posterior odds,

$\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$1

where $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$2 is the likelihood ratio. MLMP is the specialization obtained by using uniform or non-informative priors, so the selected index maximizes only

$\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$3

For the binary case treated explicitly in the paper, the MAP score is

$\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$4

and MLMP removes the final prior term $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$5 (Lee, 2015).

In the high-noise regime, the same paper shows

$\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$6

so maximizing the likelihood ratio is equivalent to maximizing the correlation $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$7. This identifies a precise limiting regime in which MLMP, MAP-MP, and conventional correlation-based greedy selection coincide (Lee, 2015).

The BMMP paper recasts this line in a more general sparse-regression language. There, Lee’s MLMP uses the normalized OMP correlation

$\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$8

and the per-index score

$\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$9

with the likelihood under $\mathbf{\Phi}\in\mathbb{R}^{M\times N}$0 marginalized over the amplitude prior $\mathbf{\Phi}\in\mathbb{R}^{M\times N}$1. BMMP keeps the likelihood-ratio principle but changes the correlation statistic and the search strategy (Kim et al., 2019).

3. Non-coherent SIMO SPARCs and successive combining

In the non-coherent SIMO SPARC model, over a codeword of length $\mathbf{\Phi}\in\mathbb{R}^{M\times N}$2 complex channel uses, the $\mathbf{\Phi}\in\mathbb{R}^{M\times N}$3-th receive antenna observes

$\mathbf{\Phi}\in\mathbb{R}^{M\times N}$4

where $\mathbf{\Phi}\in\mathbb{R}^{M\times N}$5 is the transmitted SPARC codeword, $\mathbf{\Phi}\in\mathbb{R}^{M\times N}$6 is an unknown flat-fading scalar constant over the codeword, and $\mathbf{\Phi}\in\mathbb{R}^{M\times N}$7. Stacking the antenna observations gives

$\mathbf{\Phi}\in\mathbb{R}^{M\times N}$8

The SPARC dictionary $\mathbf{\Phi}\in\mathbb{R}^{M\times N}$9 is partitioned into $\mathbf{x}$0 disjoint sections, one column is chosen per section, and the codeword is

$\mathbf{x}$1

with $\mathbf{x}$2 having exactly one nonzero per section (Kancharana et al., 2024).

Marginalizing the unknown fading yields a Gaussian likelihood with covariance

$\mathbf{x}$3

Using Woodbury,

$\mathbf{x}$4

For $\mathbf{x}$5 antennas, the marginalized ML score becomes

$\mathbf{x}$6

with

$\mathbf{x}$7

The full ML search is exponential in $\mathbf{x}$8, so MLMP replaces it with greedy section-wise support recovery. If $\mathbf{x}$9 is the current combined hypothesis, the $K$0-th selection is

$K$1

where

$K$2

and the effective noise variance is approximated by

$K$3

The update is purely additive,

$K$4

which is why the method is described as successive combining rather than successive cancellation (Kancharana et al., 2024).

At $K$5, all columns have equal norm, so the first decision reduces to

$K$6

The same paper also introduces Parallel-MLMP (P-MLMP): the top $K$7 indices from the first iteration define $K$8 decoding paths, each path runs MLMP, and the final output maximizes the global ML score $K$9 over the $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$0 completed hypotheses.

A noiseless perfect recovery condition is derived for the successive-combining rule. If the dictionary has mutual coherence $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$1 and

$$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$2

then the first selected index is correct, and induction yields exact support recovery in $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$3 steps. For the MUB dictionaries used in the paper, $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$4, so a sufficient condition is $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$5; for $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$6, $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$7 satisfies the bound (Kancharana et al., 2024).

4. Multi-user MLMP without CSI

The multi-user generalization studies $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$8 single-antenna users communicating with a base station having $$\mathbf{w}\sim\mathcal{N}(0,\sigma_w^2\mathbf{I}_M)$$9 receive antennas over a non-coherent flat Rayleigh fading channel constant over the $n$0-length codeword. If $n$1 is user $n$2’s SPARC codeword and $n$3, the received vector at antenna $n$4 is

$n$5

or, in matrix form,

$n$6

The receiver knows the variances $n$7 and $n$8 but not the channel realizations, so the model is non-coherent (Sandeep et al., 15 Jul 2025).

Conditioned on the users’ codewords, the per-antenna covariance is

$n$9

and the exact multi-antenna ML metric is

$k$0

The joint ML estimate is the maximizer of this expression over all users’ codewords. If $k$1, then $k$2, and Woodbury reduces $k$3 and $k$4 to operations on the $k$5 matrix $k$6.

MLMP approximates this infeasible search by greedily building partial codeword hypotheses. At iteration $k$7, user $k$8 has hypothesis $k$9, and the true codeword is split as

$$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$0

The undetected components are modeled as effective Gaussian noise with variance

$$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$1

Under the assumption $$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$2 for randomly chosen columns,

$$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$3

so

$$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$4

When a candidate column $$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$5 is tentatively added to user $$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$6, the candidate-specific effective noise variance becomes

$$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$7

and the score is the partial ML metric

$$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$8

The algorithm evaluates this metric for every user and every candidate column in an unidentified section, chooses

$$z_n^{(k)} \triangleq \frac{\mathbf{a}_n^\top \mathbf{r}^{(k-1)}}{\|\mathbf{a}_n\|_2},$$9

updates only the winning user’s hypothesis, and repeats until all sections are filled.

The paper characterizes this decoder as a successive-combining energy detector. Unlike OMP-like and SIC-type decoders, it does not subtract a detected component from the received signal. Instead, it recomputes a likelihood for the enlarged joint hypothesis, thereby avoiding error propagation due to incorrect early subtractions and coherently combining energy across antennas and users in the ML metric (Sandeep et al., 15 Jul 2025).

Two high-rate modifications are added. MLMP-R first runs standard MLMP for $H_1(n): n\in \mathrm{supp}(\mathbf{x})$0 iterations, then revisits each identified section and rescans that section while keeping all other selected columns fixed; if another column yields a higher ML metric, it replaces the current one. P-MLMP selects the top $H_1(n): n\in \mathrm{supp}(\mathbf{x})$1 first-step candidates, spawns $H_1(n): n\in \mathrm{supp}(\mathbf{x})$2 MLMP paths, and retains the final support with the largest overall ML metric. The motivation is explicit: when $H_1(n): n\in \mathrm{supp}(\mathbf{x})$3 is large, the residual interference represented by $H_1(n): n\in \mathrm{supp}(\mathbf{x})$4 makes the early selections less reliable, so replacement and parallel path exploration improve robustness (Sandeep et al., 15 Jul 2025).

5. Complexity, empirical behavior, and comparative performance

In the sparse-recovery line, the dominant computational work remains proxy computation and residual or least-squares updates. The likelihood-ratio evaluation adds only $H_1(n): n\in \mathrm{supp}(\mathbf{x})$5 scalar operations per iteration beyond the $H_1(n): n\in \mathrm{supp}(\mathbf{x})$6 inner products, and the main theorem in that literature is stated for MAP-MP rather than MLMP: with i.i.d. Gaussian $H_1(n): n\in \mathrm{supp}(\mathbf{x})$7, binary $H_1(n): n\in \mathrm{supp}(\mathbf{x})$8-sparse $H_1(n): n\in \mathrm{supp}(\mathbf{x})$9, and Gaussian noise, MAP-MP perfectly recovers $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$00 within $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$01 iterations almost surely as $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$02 provided

$\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$03

That paper also reports that MAP-based greedy methods outperform conventional greedy methods and often outperform basis pursuit in exact recovery probability and NMSE. Since MLMP is the likelihood-only specialization, a plausible implication is that the sparse-recovery literature treats prior information as the principal distinction between MLMP and MAP-MP rather than altering the greedy architecture itself (Lee, 2015).

BMMP is positioned as a more elaborate alternative to Lee’s MLMP/MAP-OMP. It uses the Davies–Eldar correlation $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$04, grows an extended support up to size $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$05, builds multiple support candidates, and refines them by subset replacement. Empirically, BMMP approaches the ideal $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$06 sparsity limit $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$07 in the noiseless Gaussian setting and outperforms MAP-OMP variants in both noiseless and noisy regimes, at higher algorithmic complexity (Kim et al., 2019).

In the SIMO SPARC setting, MLMP has linear complexity after a one-time precomputation. The paper gives a precompute cost $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$08 for the per-antenna correlation vectors $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$09, followed by $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$10 total scoring cost across $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$11 sections of size $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$12. With $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$13 parallel paths, the path-dependent cost scales by $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$14. Empirically, the paper reports that MLMP outperforms AMP and other greedy decoders, and that SPARC with MLMP outperforms polar codes employing pilot-based channel estimation and polar codes with non-coherent decoders (Kancharana et al., 2024).

The multi-user SPARC paper provides explicit metric-evaluation counts. To recover $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$15 sections, MLMP evaluates $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$16 partial ML metrics, MLMP-R evaluates $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$17, and P-MLMP with $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$18 paths evaluates $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$19. The same comparison gives OMP as $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$20, CoSaMP as $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$21, and ideal joint ML as $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$22. For $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$23 and $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$24, MLMP evaluates $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$25 metrics versus $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$26 for OMP and $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$27 for CoSaMP (Sandeep et al., 15 Jul 2025).

The performance findings in the same multi-user study are more specific. Using a MUB dictionary with $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$28, $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$29, $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$30 receive antennas, and short-block regimes, MLMP consistently outperforms OMP and CoSaMP in BLER at both low and high rates. At high rate, MLMP-R and P-MLMP further improve over plain MLMP. Against pilot-aided transmissions with CRC-aided list polar decoding, the paper reports that at low rate $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$31 slightly outperforms $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$32 for both SPARC and PAT, while at high rate three-user SPARC with MLMP achieves a marked multi-user diversity gain without CSI and outperforms both single-user operation and PAT; the reported gain over PAT with polar codes for $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$33 is about $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$34 dB at target BLER (Sandeep et al., 15 Jul 2025).

The same study also reports an asymmetric-channel result: with $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$35 at high rate, allocating sections as $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$36 yields the best BLER, equal allocation is second-best, and reverse-proportional allocation is worst. This supports adaptive section allocation to match long-term user strengths (Sandeep et al., 15 Jul 2025).

6. Assumptions, limitations, and interpretive points

In sparse recovery, the likelihood-ratio derivations rely on i.i.d. Gaussian sensing matrices and Gaussian noise, with either binary nonzeros or an amplitude prior $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$37. The likelihoods used for per-index decisions are exact only under those modeling assumptions or their moment-matched approximations. The BMMP analysis makes the same dependence explicit and notes that for non-Gaussian sensing matrices or noise, the likelihood ratio can still be used as an approximation, but guarantees need not hold (Lee, 2015, Kim et al., 2019).

In the SPARC literature, the channel model is flat block fading, constant over the $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$38-symbol block, with unknown fading realizations and known variances. The SIMO paper assumes one active column per section, unit-norm dictionary columns, and an effective-noise approximation

$\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$39

derived from mutual-coherence bounds rather than an exact interference law. It also notes that very short $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$40, which increases $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$41, weakens both guarantees and empirical margins (Kancharana et al., 2024).

The multi-user paper imposes additional assumptions: users are synchronous and perfectly time/frequency aligned across the block, each user has a single antenna, and the interference-as-noise step uses $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$42 for randomly chosen columns, approximately true in expectation for MUB dictionaries. The same paper identifies several limitations: the large MUB dictionary size $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$43 can stress memory and compute; at very high $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{w},$44 or heavy user loading, early-selection errors can still occur; very strong users can dominate; and sensitivity to non-Gaussian noise, channel correlation across antennas, or time selectivity within the block is not analyzed (Sandeep et al., 15 Jul 2025).

Several recurrent misconceptions can therefore be resolved precisely. First, MLMP is not a synonym for OMP with a different threshold; in all cited uses, the defining feature is likelihood-based scoring. Second, MLMP is not always a strict MAP rule: in the original sparse-recovery formulation, MAP-MP adds prior odds, whereas MLMP drops them; with uniform index priors, the per-index prior factor is constant and does not affect ranking (Lee, 2015, Kim et al., 2019). Third, in non-coherent SPARC decoding, MLMP is not successive cancellation. Its defining operation is successive combining: previously selected columns remain in the hypothesis, and candidate columns are evaluated jointly through a marginalized ML metric rather than through residual subtraction (Kancharana et al., 2024, Sandeep et al., 15 Jul 2025).

Taken together, these works place MLMP at the intersection of greedy pursuit and statistical decision theory. Across compressed sensing and non-coherent SPARC decoding, the common structural idea is to preserve greedy tractability while replacing purely geometric selection rules by likelihood metrics that incorporate signal, noise, and nuisance-parameter statistics.