Decomposition Sampling (DECOMP)

• Decomposition Sampling (DECOMP) is a framework that divides complex models, signals, or combinatorial patterns into smaller, manageable subcomponents to enhance sampling efficiency and eliminate bias.
• It employs domain-specific methods—such as parallel MCMC, motif sampling in graphs, stepwise signal decompositions, and tensor factor selections—to optimize computational and memory performance.
• By leveraging structured decompositions, DECOMP methods enforce causality and accelerate convergence, making them invaluable in time-series forecasting, active learning, and high-dimensional inference.

Decomposition Sampling (DECOMP) refers to a class of sampling, algorithmic, and pre-processing frameworks unified by the principle of breaking up a complex object—sample space, model, signal, or combinatorial pattern—into sub-components amenable to independent or conditional sampling. The unifying goal of DECOMP methods is to exploit structured decompositions (spatial, combinatorial, algebraic, or class-based) to accelerate sampling, eliminate bias, enforce causality, or reduce computational/memory complexity. Methodologies are domain-specific and include subspace overlaps, motif factorizations, sequential window decompositions, tensor-network factor sampling, and class-conditioned region selection.

1. Conceptual Foundations and Variants of Decomposition Sampling

DECOMP frameworks are defined by their partitioning or decomposing of the space of interest—be it probability measures, graphs, time series, tensors, or image regions—so that sampling can be carried out within or across components, often in parallel or with improved efficiency. Prominent DECOMP designs have appeared in:

• Monte Carlo and Markov chain Monte Carlo (MCMC) as parallelization via overlapping covers, supporting independent subchain sampling and subsequent recombination (Hallgren et al., 2014).
• Graph motif sampling/counting, where a target motif $H$ is optimally decomposed into collections of odd cycles $O_k$ and stars $S_p$, with sampling routines tailored per component (Biswas et al., 2021).
• Signal processing, via stepwise or causal decomposition schemes in hybrid time series forecasting, preventing future-data leakage by stepwise, window-contingent decompositions (Zhang et al., 2023).
• Tensor methods, through sampling-based factor selection within alternating least squares, informed by leverage scores derived from the current tensor decomposition (Malik et al., 2022).
• Bayesian graphical models, where decomposable graphs are explored using Markov chains on junction trees, and permissible multi-edge updates are classified by explicit decomposition partitions of the clique structure (Elmasri, 2022).
• Active learning in dense prediction tasks, where pseudo-label-driven class-based decomposition of candidate annotation regions ensures diversity and minor-class coverage (Qiu et al., 8 Dec 2025).

2. Algorithmic Structures and Mathematical Formulations

Core algorithmic elements are specific to the domain but recurrent aspects include:

Markov Chain & Parallelization Frameworks

In parallel MCMC (Metropolis–Hastings), DECOMP constructs a “linked cover” $X = \cup_{i=1}^K X_i$ with overlapping regions. Separate MCMC chains are run on $\widetilde\pi_i(x) = \pi(x)\mathbf{1}_{X_i}(x)/Z_i$ per subset, followed by downsampling and merging to recover samples from the target $\pi$. The method guarantees unbiased estimates under correct normalization, and enables near-ideal speedup proportional to $K$ (Hallgren et al., 2014).

Motif Sampling in Graphs

For motif $H$ and graph $G$, the canonical decomposition $D^*(H)$ consists of odd cycles and stars obtained via edge fractional covering. DECOMP samples one representative subgraph per component (using StarSampler, CycleSampler) and tests for a successful embedding. The expected query cost is controlled by the decomposition-cost metric, which depends on component counts $o_k$, $s_p$ and target count $h$ (Biswas et al., 2021).

Stepwise/Sequential Decomposition in Signals

For a time series $S = \{S_1,\ldots,S_N\}$, the fully stepwise decomposition-based (FSDB) sampling method grows an explanatory window and applies a decomposition operator (e.g., SSA or VMD) at each time, extracting inputs and targets without reference to future data. This strictly preserves real-time causality, avoiding optimistic bias in downstream machine learning models (Zhang et al., 2023).

Leverage-Score-Based Sampling in Tensor Networks

ALS-based tensor decomposition leverages leverage score distributions over the design matrices to sample constraints when solving least-squares subproblems at sublinear cost. Each factor update in the network is performed using a small, adaptively sampled batch whose rows are selected according to the exact leverage scores, ensuring $(1+\epsilon)$-approximation error (Malik et al., 2022).

Partition-Aware Junction Tree Proposals in Graphical Models

In decomposable-graph inference, DECOMP partitions permissible edge updates into compatible sets (neighboring cliques or leaf cliques) enabling simultaneous and independent (parallel) MCMC moves within a junction tree. This retains decomposability while increasing mixing relative to single-move proposals (Elmasri, 2022).

Class-Wise Region Sampling in Active Learning

For dense prediction, DECOMP decomposes each sample’s region set according to pseudo-predicted classes, samples class-specific regions weighted by class-level predictive uncertainty, and thus ensures that hard and underrepresented classes are prioritized for annotation (Qiu et al., 8 Dec 2025).

3. Application Domains and Method Integration

Time Series Forecasting

FSDB sampling has been successfully deployed for water-level forecasting. Integrating FSDB with SSA or VMD produces $K$ components per window, on which $K$ regressors are trained, and their outputs are aggregated to recover the signal forecast. This avoids information leakage and systemic overfitting observed in previous (ODB, KN, SDB) schemes. Empirical results on Chinese river basins demonstrated Nash-Sutcliffe Efficiency (NSE) gains of 1.1–28.8% over previous state-of-the-art samplers (Zhang et al., 2023).

Subgraph Counting and Motif Sampling

DECOMP algorithms enable sublinear-time uniform sampling/counting for arbitrary graph motifs. For motifs with favorable decompositions (many star components), polynomial improvements in sample complexity are possible—especially in sparse or low-arboricity graphs. Tight lower bounds, matching algorithmic upper complexity, are established for decompositions containing odd cycles (Biswas et al., 2021).

Tensor Decompositions

DECOMP achieves input-sublinear per-iteration complexity in ALS for CP and tensor-ring decompositions, using exact leverage scores on the matricized network’s design matrix. Empirical feature extraction experiments show DECOMP-CP and DECOMP-TR outperform classical deterministic and randomized ALS methods in runtime while maintaining comparably low relative errors and high classification accuracy (Malik et al., 2022).

Parallel MCMC and Bayesian Graphical Models

Within MCMC, DECOMP enables independent parallel simulation across overlapping subregions, with unbiased expectation estimation and accelerated decorrelation, as demonstrated in both synthetic and real calibration problems, including particle marginal MH for stochastic volatility (Hallgren et al., 2014). In graphical model structure learning, the parallelDG Python package implements DECOMP-based partitioned parallel MCMC over high-dimensional junction trees, outperforming prior art in mixing and ROC accuracy (Elmasri, 2022).

Active Learning for Dense Prediction

Region-based DECOMP AL provides state-of-the-art annotation efficiency in ROI classification, 2D, and 3D segmentation. By decomposing regions by class and proportionally allocating annotation budget, it achieves superior coverage of minority and hard classes. For BRACS, Cityscapes, and KiTS23, DECOMP reached 95% of full-annotation performance with 40%, 2.4%, and 0.15% of the annotation budget respectively—substantially surpassing random, uncertainty, and clustering-based baselines (Qiu et al., 8 Dec 2025).

4. Theoretical Guarantees and Performance Analysis

DECOMP frameworks generally preserve the target distribution (or estimation objective) provided that overlap correction, sufficient sample size, or proper window selection is enforced.

• Parallel MCMC: Guarantees unbiasedness and TV-convergence as long as block-weights converge and subchain kernels are irreducible/aperiodic (Hallgren et al., 2014).
• Graph motifs: Delivers uniform random samples at a cost scaling with the maximum per-component sampling cost and target count, with optimality characterized by decomposition structure (Biswas et al., 2021).
• Stepwise signal decomposition: Empirically shown to avoid over-optimistic validation metrics, eliminate future-data bias, and outperform less-causal alternatives (Zhang et al., 2023).
• Tensor networks: (1+$\epsilon$)-relative error is achieved with high probability with a sample size proportional to the rank (Malik et al., 2022).
• Active learning: DECOMP’s annotation efficiency and class-coverage are tied to the decomposition’s ability to reflect class distribution and model uncertainty; ablation studies confirm distinct contributions from both image and region selection phases (Qiu et al., 8 Dec 2025).

5. Implementation Considerations and Practical Guidelines

• Cover and overlap selection are fundamental; for MCMC, quantile-based splitting informed by pilot runs is recommended, with overlap sizes tuned to ensure all regions are well-sampled (Hallgren et al., 2014).
• In time-series prediction, lag $m$ must reflect system memory, and decomposition parameters (window length, mode count, penalty) are chosen via hyperparameter search on held-out data (Zhang et al., 2023).
• For motif-sampling, computing exact counts of stars/cycles and optimal decompositions of $H$ is necessary for optimality; in practice, this automation is feasible for moderate-size motifs (Biswas et al., 2021).
• In tensorized settings, the TN must admit fast contraction and affordable Gram inverses; leverage-score sampling is then implemented using sequential conditional evaluations on the TN graph (Malik et al., 2022).
• For active learning, class confidence thresholds are robust; efficient implementation leverages integral images and avoids high-dimensional region feature indexing (Qiu et al., 8 Dec 2025).
• Graphical model samplers exploit parallel execution across partitions; empirical performance degrades mildly with increasing dimensionality but is mitigated by adjusting prior penalties and skeleton-update frequencies (Elmasri, 2022).

6. Extensions, Open Problems, and Comparative Evaluation

• For graph motif DECOMP, open problems include (i) removal of suboptimal polylogarithmic factors, (ii) achieving tight lower bounds for star-dominated decompositions, and (iii) extending to non-canonical motif covers (Biswas et al., 2021).
• In parallel MCMC, the efficacy depends on block choice and the dimensionality regime; high-dimensional spaces or those with complex geometry may challenge naive covers (Hallgren et al., 2014).
• Weaknesses of decomposition-based signal forecasting arise in highly non-stationary scenarios, where ML models without decomposition may outperform all hybrids (Zhang et al., 2023).
• In AL, region-level DECOMP outcompetes feature-clustering and uncertainty-only strategies especially in minority and poorly predicted classes, but total annotation gain is bounded by the accuracy of pseudo-labels and class granularity (Qiu et al., 8 Dec 2025).

Application	Core Decomposition Mechanism	Key Theoretical Guarantee
MCMC Parallelization	Overlapping subspace covers	TV-convergent, unbiased expectations
Motif Sampling	Star/cycle LP-motif decomposition	Query cost tied to decomp-cost
Signal Forecasting	Stepwise sliding window decomps	Strict causal/no-leakage learning
Tensor ALS	Leverage-score constrained samps	Sublinear, $(1+\epsilon)$ error
Graphical Models	Clique-partition parallelism	Correct, fast-mixing junction tree
Active Learning	Class-conditioned region sets	Optimal budget-diversity tradeoff

7. Summary and Impact

Decomposition Sampling methods leverage structural decomposability to improve sampling fidelity, computational efficiency, and downstream statistical or learning performance. By aligning the sampling methodology with inherent substructure—be it topological, algebraic, sequential, or semantic—DECOMP enables rigorous guarantees and it drives advances in high-dimensional inference, combinatorial optimization, time series prediction, tensor learning, and annotation-efficient active learning. Open directions involve dynamic or recursive decompositions, extension to latent-variable models, and formal analysis of performance gains in non-canonical or adversarial settings (Hallgren et al., 2014, Biswas et al., 2021, Malik et al., 2022, Zhang et al., 2023, Elmasri, 2022, Qiu et al., 8 Dec 2025).