W-SCOMP: Weighted Sequential OMP

• W-SCOMP is a greedy algorithm for noiseless group testing that incorporates weighted scoring to enhance the identification of defective items.
• It iteratively selects candidates by updating scores based on unresolved positive tests, penalizing ambiguity and improving signal clarity.
• Empirical evaluations show that W-SCOMP offers modest gains in SNR and reduces the number of tests needed compared to traditional SCOMP.

Weighted Sequential Combinatorial Orthogonal Matching Pursuit (W-SCOMP) is a greedy decoding algorithm developed for noiseless non-adaptive group testing. Group testing aims to efficiently identify a small set of defective items from a larger population using pooled test designs. W-SCOMP optimizes the detection of defectives by incorporating a weighted scoring scheme into the classical Sequential Combinatorial Orthogonal Matching Pursuit (SCOMP) decoder, yielding improved signal-to-noise ratio (SNR) and modest but consistent empirical gains, particularly in regimes where recovery is information-limited. The development, structure, and performance guarantees of W-SCOMP are anchored in both combinatorial group testing frameworks and the broader class of weighted greedy methods such as those synthesized from Simultaneous Orthogonal Matching Pursuit with Noise Stabilization (SOMP-NS) (Determe et al., 2015, Franco-Vivo, 12 Jan 2026).

1. Foundation and Problem Setting

The W-SCOMP decoder operates within the classical combinatorial group testing model characterized by the following:

• Population and Defectives: Given $N$ items, a small unknown set $K$ (with $|K|=k \ll N$) is defective. The indicator vector is $s \in \{0,1\}^N$, where $s_j=1$ iff $j \in K$.
• Test Matrix: Tests are defined by $X \in \{0,1\}^{m \times N}$, with $X_{t,j}=1$ if item $j$ is included in test $t$.
• Non-Adaptive Noiseless Model: All tests are run in parallel. The outcome vector $y \in \{0,1\}^m$ is given by

$y_t = \max_{j=1}^N (X_{t,j}s_j)$

so $y_t=1$ if any defective is present in test $t$.

• Priors and Designs: Typically, the defective set is drawn uniformly at random. Standard designs include the Bernoulli design (each $X_{t,j} \sim \mathrm{Bernoulli}(p)$) and constant/near-constant tests-per-item schemes (Franco-Vivo, 12 Jan 2026).

2. Weighted Scoring: The Core Mechanism

The fundamental innovation in W-SCOMP is the replacement of the unit-based scoring function in SCOMP with a weighted contribution reflecting test ambiguity. After initial elimination using COMP or Definite Defectives (DD), the following concepts arise:

• Definite Nondefectives (DND): Items appearing in any negative test.
• Potential Defectives (PD): Set $PD = \{1, \dots, N\} \setminus DND$ minus items identified as definite defectives.
• Unresolved Positive Tests ($T_u$): Positive tests not yet explained by current defective estimates.
• Row-Weight:

$w_t = \sum_{i \in PD} X_{t,i}$

for each unresolved positive test $t$. This quantifies the number of remaining plausible defectives in a pool.

• Per-Test Contribution: For candidate $j$ in test $t$,

$S_{t,j} = \frac{X_{t,j} \, y_t}{w_t^\alpha}$

where typically $\alpha=1$. This yields each item's aggregate score:

$S_j = \sum_{t \in T_u} S_{t,j}$

This weighting penalizes items present in ambiguous (high $w_t$) tests and prioritizes those that better resolve unexplained positives.

3. Algorithmic Structure and Pseudocode

W-SCOMP proceeds as a sequential greedy decoder, iteratively explaining positive tests through candidate selection:

1. Initialization:
   • DND and PD sets identified via negative tests.
   • Definite defectives $K_{DD}$ collected from singleton pools if any.
   • Initial estimate $K_{\mathrm{est}} \gets K_{DD}$.
2. Iterative Selection:
   • While $T_u$ (unexplained positives) and $PD$ are nonempty:
     • For each $t \in T_u$, compute $w_t$.
     • For each $j \in PD$, calculate $S_j$.
     • Add $i^* = \arg\max_{j \in PD} S_j$ to $K_{\mathrm{est}}$.
     • Remove tests in $T_u$ where $X_{t,i^*}=1$.
     • Update $PD \gets PD \setminus \{i^*\}$.
3. Return $K_{\mathrm{est}}$ as the estimated defective set.

The only modification from standard SCOMP is the use of the $1/w_t$ weighted score versus the count of unresolved positives (Franco-Vivo, 12 Jan 2026).

4. Theoretical Guarantees and SNR Analysis

Performance analysis leverages the framework of additive scoring rules to separate defectives from nondefectives:

• Per-Test SNR:

$$\mathsf{SNR}_{\mathrm{per}} = \frac{(\mu_D - \mu_{ND})^2}{\sigma_D^2 + \sigma_{ND}^2}$$

with $\mu_c$, $\sigma_c^2$ computed for $c \in \{\mathrm{D},\mathrm{ND}\}$ (defective, nondefective). Aggregate SNR scales as $\mathsf{SNR} = m \, \mathsf{SNR}_{\mathrm{per}}$.

• Error Bounds:
  • Chebyshev: $P_{\mathrm{err}} \leq \frac{2}{\mathsf{SNR}}$.
  • Gaussian approximation: $$P_{\mathrm{err}} \leq 2\exp\left(-\frac{\mathsf{SNR}}{4}\right)$$.
• Per-Test Design: In Bernoulli design (with $p=1/(k+1)$), closed-form $\mu_c, \sigma_c^2$ are provided in Propositions 2–5 (Franco-Vivo, 12 Jan 2026). The key result is:

$$\mathsf{SNR}_{\mathrm{w}}(N,k) \geq \mathsf{SNR}_{\mathrm{u}}(k) \quad \forall N>k$$

so W-SCOMP is never dominated by SCOMP on SNR grounds.

• Sample Complexity: To guarantee vanishing error rate ($P_{\mathrm{err}}\leq 1/N$),

$$m = O\left(\frac{1}{\mathsf{SNR}_{\mathrm{per}}}\log N\right)$$

and with $\mathsf{SNR}_{\mathrm{per}} = \Theta(1/k)$ under Bernoulli design,

$m = O(k\log N)$

5. Computational Complexity

W-SCOMP retains the computational simplicity of greedy decoders:

• Initial DND and DD: $O(mN)$.
• Per Iteration: Computing $w_t$ and $S_j$ each takes $O(m|PD|)$; with up to $k$ selections, the practical runtime is $O(k m N)$, which can be reduced to $O((m+N)k)$ with sparse data structures (Franco-Vivo, 12 Jan 2026).

6. Comparative Performance and Empirical Validation

Theoretical and empirical evaluations highlight the following:

Algorithm	Tests for $P_{\text{success}} \approx 0.95$
COMP	$\approx 120$
DD	$\approx 115$
SCOMP	$\approx 105$
W-SCOMP	$\approx 100$

• W-SCOMP consistently achieves the same probability of exact recovery as SCOMP but with approximately 5 fewer tests (for $N=500$, $k=10$, Bernoulli design).
• Both false-positive and false-negative rates decline more rapidly for W-SCOMP (Franco-Vivo, 12 Jan 2026).
• Set-similarity metrics (Jaccard, $F_1$-score) exhibit a small but robust edge for W-SCOMP.
• The average misclassification difference is nonnegative, confirming uniform noninferiority.
• All empirical findings are robust to variations in test design and number of defectives.

7. Broader Context and Extensions

W-SCOMP exemplifies a class of algorithms leveraging weighted joint selection rules to optimize combinatorial recovery under uncertainty. The foundational analysis for such weighted sequential/combinatorial OMP variants, including noise-stabilized frameworks like SOMP-NS (Determe et al., 2015), demonstrates a universal template:

1. Introduce nonnegative per-measurement or per-test weights in the selection score.
2. Propagate these weights into subsequent updates.
3. Redo theoretical analysis (e.g., via restricted isometry property or exact recovery criterion) using corresponding weighted quantities.
4. Choose weights according to statistical or information-theoretic optimality criteria (e.g., inverse-variance for MMV, ambiguity-based for group testing).

A plausible implication is that similar weighted greedy principles can be fruitfully extended to new combinatorial recovery settings by adapting the weight selection to the specifics of the underlying statistical model and measurement process (Determe et al., 2015, Franco-Vivo, 12 Jan 2026).