---
title: Maximum Likelihood Matching Pursuit (MLMP)
url: https://www.emergentmind.com/topics/maximum-likelihood-matching-pursuit-mlmp
type: topic
---

# Maximum Likelihood Matching Pursuit (MLMP)

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 [1508.00964]. 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 [2405.09915], [2507.11383].

## 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 [1508.00964]. 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 [2405.09915]. “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 [2507.11383].

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 [1904.01818].

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 \(H_0(n): n\notin \mathrm{supp}(\mathbf{x})\) [1508.00964].

The MAP rule compares posterior odds,
\[
\frac{P(H_1(n)\mid \mathbf{y})}{P(H_0(n)\mid \mathbf{y})}
=\Lambda_n(\mathbf{y})\cdot \frac{P(H_1(n))}{P(H_0(n))},
\]
where \(\Lambda_n(\mathbf{y})\) is the likelihood ratio. MLMP is the specialization obtained by using uniform or non-informative priors, so the selected index maximizes only
\[
\ln\!\left(\frac{p(z_n^{(k)}\mid H_1)}{p(z_n^{(k)}\mid H_0)}\right).
\]
For the binary case treated explicitly in the paper, the MAP score is
\[
\Lambda\!\left(z_n^{(k)}\right)=\frac{(z_n^{(k)})^2}{2\left(\tfrac{K-k+1}{M}+\sigma_w^2\right)}
-\frac{(z_n^{(k)}-1)^2}{2\left(\tfrac{K-k}{M}+\sigma_w^2\right)}
+\frac{1}{2}\ln\!\left(\frac{(K-k+1)+M\sigma_w^2}{(K-k)+M\sigma_w^2}\right)
+\ln\!\left(\frac{K}{N-K}\right),
\]
and MLMP removes the final prior term \(\ln(K/(N-K))\) [1508.00964].

In the high-noise regime, the same paper shows
\[
\Lambda\!\left(z_n^{(k)}\right)\simeq \frac{2z_n^{(k)}-1}{2\sigma_w^2},
\]
so maximizing the likelihood ratio is equivalent to maximizing the correlation \(z_n^{(k)}\). This identifies a precise limiting regime in which MLMP, MAP-MP, and conventional correlation-based greedy selection coincide [1508.00964].

The BMMP paper recasts this line in a more general sparse-regression language. There, Lee’s MLMP uses the normalized OMP correlation
\[
h_i(\Delta)=r(\Delta)^{\top}\dot{\phi}_i,
\]
and the per-index score
\[
\Theta_{\mathrm{MLMP}}(z_i)=\log \frac{p(z_i\mid \mathcal{T}_1)}{p(z_i\mid \mathcal{T}_0)},
\]
with the likelihood under \(\mathcal{T}_1\) marginalized over the amplitude prior \(f_{x^*}(u)\). BMMP keeps the likelihood-ratio principle but changes the correlation statistic and the search strategy [1904.01818].

## 3. Non-coherent SIMO SPARCs and successive combining

In the non-coherent SIMO SPARC model, over a codeword of length \(N\) complex channel uses, the \(i\)-th receive antenna observes
\[
y_i=h_i s+v_i,\qquad i\in\{1,\ldots,R\},
\]
where \(s\in\mathbb{C}^N\) is the transmitted SPARC codeword, \(h_i\sim \mathcal{CN}(0,\sigma_h^2)\) is an unknown flat-fading scalar constant over the codeword, and \(v_i\sim \mathcal{CN}(0,\sigma_v^2 I_N)\). Stacking the antenna observations gives
\[
Y=s h^T+V.
\]
The SPARC dictionary \(A\in\mathbb{C}^{N\times L}\) is partitioned into \(K\) disjoint sections, one column is chosen per section, and the codeword is
\[
s=\sum_{k=1}^K a_{m_k}=Ax,
\]
with \(x\in\{0,1\}^L\) having exactly one nonzero per section [2405.09915].

Marginalizing the unknown fading yields a Gaussian likelihood with covariance
\[
C=\sigma_h^2 s s^*+\sigma_v^2 I_N.
\]
Using Woodbury,
\[
C^{-1}=\frac{1}{\sigma_v^2}\!\left[I_N-\frac{\sigma_h^2}{\sigma_v^2+\sigma_h^2\|s\|^2}ss^*\right],
\qquad
|C|=\sigma_v^{2(N-1)}(\sigma_v^2+\sigma_h^2\|s\|^2).
\]
For \(R\) antennas, the marginalized ML score becomes
\[
\hat{s}=\arg\max_{s\in \mathcal{C}}\ \beta_s\sum_{i=1}^R |\langle y_i,s\rangle|^2-R\gamma_s,
\]
with
\[
\beta_s=\frac{\sigma_h^2/\sigma_v^2}{\sigma_v^2+\sigma_h^2\|s\|^2},
\qquad
\gamma_s=\log\!\left(1+\frac{\sigma_h^2}{\sigma_v^2}\|s\|^2\right).
\]

The full ML search is exponential in \(K\), so MLMP replaces it with greedy section-wise support recovery. If \(s_{k-1}\) is the current combined hypothesis, the \(k\)-th selection is
\[
a_{\hat{m}_k}
=
\arg\max_{m\in \text{section }k}
\ \beta_{k,m}\sum_{i=1}^R |\langle y_i,s_{k-1}+a_m\rangle|^2-R\gamma_{k,m},
\]
where
\[
\beta_{k,m}=\frac{\sigma_h^2/\tilde{\sigma}_{v,k}^2}{\tilde{\sigma}_{v,k}^2+\sigma_h^2\|s_{k-1}+a_m\|^2},
\qquad
\gamma_{k,m}=\log\!\left(1+\frac{\sigma_h^2}{\tilde{\sigma}_{v,k}^2}\|s_{k-1}+a_m\|^2\right),
\]
and the effective noise variance is approximated by
\[
\tilde{\sigma}_{v,k}^2 \approx \sigma_v^2+\sigma_h^2 (K-k)/N.
\]
The update is purely additive,
\[
s_k=s_{k-1}+a_{\hat{m}_k},
\]
which is why the method is described as successive combining rather than successive cancellation [2405.09915].

At \(k=1\), all columns have equal norm, so the first decision reduces to
\[
a_{\hat{m}_1}=\arg\max_{m\in \text{section }1}\sum_{i=1}^R |\langle y_i,a_m\rangle|^2.
\]
The same paper also introduces Parallel-MLMP (P-MLMP): the top \(P\) indices from the first iteration define \(P\) decoding paths, each path runs MLMP, and the final output maximizes the global ML score \(p(Y\mid \hat{s}_n)\) over the \(P\) completed hypotheses.

A noiseless perfect recovery condition is derived for the successive-combining rule. If the dictionary has mutual coherence \(\mu(A)\) and
\[
\mu(A)<\frac{1}{2K-1},
\]
then the first selected index is correct, and induction yields exact support recovery in \(K\) steps. For the MUB dictionaries used in the paper, \(\mu(A)=1/\sqrt{N}\), so a sufficient condition is \(N>(2K-1)^2\); for \(K=4\), \(N\ge 64\) satisfies the bound [2405.09915].

## 4. Multi-user MLMP without CSI

The multi-user generalization studies \(U\) single-antenna users communicating with a base station having \(M\) receive antennas over a non-coherent flat Rayleigh fading channel constant over the \(N\)-length codeword. If \(s_j\in\mathbb{C}^N\) is user \(j\)’s SPARC codeword and \(h_{i,j}\sim \mathcal{CN}(0,\sigma_j^2)\), the received vector at antenna \(i\) is
\[
y_i=\sum_{j=1}^U h_{i,j}s_j+v_i,
\]
or, in matrix form,
\[
Y=\sum_{j=1}^U s_j h_j^T+V.
\]
The receiver knows the variances \(\sigma_j^2\) and \(\sigma_v^2\) but not the channel realizations, so the model is non-coherent [2507.11383].

Conditioned on the users’ codewords, the per-antenna covariance is
\[
F=\sum_{j=1}^U \sigma_j^2 s_j s_j^*+\sigma_v^2 I_N,
\]
and the exact multi-antenna ML metric is
\[
\mathrm{MLM}(y_1,\ldots,y_M; s_1,\ldots,s_U,\sigma_v^2)
=
-M\log |F|-\sum_{i=1}^M y_i^* F^{-1} y_i.
\]
The joint ML estimate is the maximizer of this expression over all users’ codewords. If \(B=[\sigma_1 s_1,\ldots,\sigma_U s_U]\), then \(F=\sigma_v^2 I_N + B B^*\), and Woodbury reduces \(F^{-1}\) and \(|F|\) to operations on the \(U\times U\) matrix \(B^*B\).

MLMP approximates this infeasible search by greedily building partial codeword hypotheses. At iteration \(t\), user \(j\) has hypothesis \(\hat{s}_j^{(t)}\), and the true codeword is split as
\[
s_j=\hat{s}_j^{(t)}+\tilde{s}_j^{(t)}.
\]
The undetected components are modeled as effective Gaussian noise with variance
\[
\sigma_u^2
=
\frac{1}{N}\mathbb{E}\!\left\|\sum_{j=1}^U h_{i,j}\tilde{s}_j^{(t)}\right\|^2
=
\frac{1}{N}\sum_{j=1}^U \sigma_j^2 \mathbb{E}\|\tilde{s}_j^{(t)}\|^2.
\]
Under the assumption \(E\{\langle a_i,a_j\rangle\}=0\) for randomly chosen columns,
\[
E\|\tilde{s}_j^{(t)}\|^2 = K_j-K_j^{(t)},
\]
so
\[
\sigma_u^2 = \frac{1}{N}\sum_{j=1}^U \sigma_j^2 (K_j-K_j^{(t)}).
\]

When a candidate column \(a_h\) is tentatively added to user \(j\), the candidate-specific effective noise variance becomes
\[
\sigma_{e,j}^2=\sigma_v^2+\sigma_u^2-\frac{1}{N}\sigma_j^2,
\]
and the score is the partial ML metric
\[
\mathrm{MLM}(y_1,\ldots,y_M; \hat{s}_1^{(t)},\ldots,\hat{s}_j^{(t)}+a_h,\ldots,\hat{s}_U^{(t)},\sigma_{e,j}^2).
\]
The algorithm evaluates this metric for every user and every candidate column in an unidentified section, chooses
\[
(\hat{\jmath},\hat{a}_h)=\arg\max_{j,h}\ \mathrm{score}(j,h),
\]
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 [2507.11383].

Two high-rate modifications are added. MLMP-R first runs standard MLMP for \(K\) 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 \(P\) first-step candidates, spawns \(P\) MLMP paths, and retains the final support with the largest overall ML metric. The motivation is explicit: when \(K\) is large, the residual interference represented by \(\sigma_u^2\) makes the early selections less reliable, so replacement and parallel path exploration improve robustness [2507.11383].

## 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 \(O(N)\) scalar operations per iteration beyond the \(O(MN)\) inner products, and the main theorem in that literature is stated for MAP-MP rather than MLMP: with i.i.d. Gaussian \(\mathbf{\Phi}\), binary \(K\)-sparse \(\mathbf{x}\), and Gaussian noise, MAP-MP perfectly recovers \(\mathbf{x}\) within \(K\) iterations almost surely as \(N,K\to\infty\) provided
\[
M=\mathcal{O}\big((K+\tilde{\sigma}_w^2)\ln N\big),\qquad \tilde{\sigma}_w^2\triangleq \sigma_w^2/M.
\]
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 [1508.00964].

BMMP is positioned as a more elaborate alternative to Lee’s MLMP/MAP-OMP. It uses the Davies–Eldar correlation \(g_i(\Delta)=r(\Delta)^\top \dot{\phi}_i(\Delta)\), grows an extended support up to size \(m\), builds multiple support candidates, and refines them by subset replacement. Empirically, BMMP approaches the ideal \(\ell_0\) sparsity limit \(K\approx m/2\) in the noiseless Gaussian setting and outperforms MAP-OMP variants in both noiseless and noisy regimes, at higher algorithmic complexity [1904.01818].

In the SIMO SPARC setting, MLMP has linear complexity after a one-time precomputation. The paper gives a precompute cost \(O(NLR)\) for the per-antenna correlation vectors \(u_i=A^*y_i\), followed by \(O(KMR)\) total scoring cost across \(K\) sections of size \(M\). With \(P\) parallel paths, the path-dependent cost scales by \(P\). 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 [2405.09915].

The multi-user SPARC paper provides explicit metric-evaluation counts. To recover \(K\) sections, MLMP evaluates \(((K+1)L)/2\) partial ML metrics, MLMP-R evaluates \(((K+3)L)/2\), and P-MLMP with \(P\) paths evaluates \(P(K-1)L/2 + L\). The same comparison gives OMP as \(KL-K(K-1)/2 \approx KL\), CoSaMP as \(2KL-2K^2+K\approx 2KL\), and ideal joint ML as \((L/K)^K\). For \(K=4\) and \(L=256\), MLMP evaluates \(640\) metrics versus \(1018\) for OMP and \(2020\) for CoSaMP [2507.11383].

The performance findings in the same multi-user study are more specific. Using a MUB dictionary with \(N=128\), \(L=N^2=16384\), \(M=4\) 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 \(U=1\) slightly outperforms \(U=3\) 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 \(U=3\) is about \(4\) dB at target BLER [2507.11383].

The same study also reports an asymmetric-channel result: with \(\sigma_1^2:\sigma_2^2:\sigma_3^2=3:2:1\) at high rate, allocating sections as \(K_1:K_2:K_3=3:2:1\) 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 [2507.11383].

## 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 \(f_{x_n}(u)\). 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 [1508.00964], [1904.01818].

In the SPARC literature, the channel model is flat block fading, constant over the \(N\)-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
\[
\tilde{\sigma}_{v,k}^2 \approx \sigma_v^2+\sigma_h^2(K-k)/N,
\]
derived from mutual-coherence bounds rather than an exact interference law. It also notes that very short \(N\), which increases \(\mu(A)\), weakens both guarantees and empirical margins [2405.09915].

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 \(E\{\langle a_i,a_j\rangle\}=0\) for randomly chosen columns, approximately true in expectation for MUB dictionaries. The same paper identifies several limitations: the large MUB dictionary size \(L=N^2\) can stress memory and compute; at very high \(K\) 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 [2507.11383].

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 [1508.00964], [1904.01818]. 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 [2405.09915], [2507.11383].

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.

Source: https://www.emergentmind.com/topics/maximum-likelihood-matching-pursuit-mlmp