Decomposition Architecture in Complex Systems

• Decomposition architecture is a modular design paradigm that divides complex systems into manageable subsystems to enhance scalability, interpretability, and parallel processing.
• It leverages formal models, optimization methods, and tensor or domain decompositions to streamline processes in software, neural networks, and distributed systems.
• Applications include microservices design, CNN acceleration, time-series anomaly detection, and resource optimization in fog/edge computing with measurable efficiency gains.

Decomposition architecture encompasses a broad spectrum of modular design principles and algorithmic strategies for partitioning complex systems into smaller, more manageable subcomponents. These approaches are foundational across domains including software engineering, neural network design, optimization, distributed systems, and computational biology. Decomposition serves to enhance scalability, interpretability, parallelism, and resource efficiency by isolating distinct functionalities or data flows in a structured manner.

1. Formal Models of Decomposition and Modularization

Most decomposition architectures derive from a formal task, data, or topology model. In distributed systems such as virtual network embedding, the optimization problem is defined as a constrained integer program involving resource discovery, mapping, and allocation, with linking constraints that couple these phases; decompositional strategies (e.g., primal/dual methods) partition this complexity across modules or agents (Esposito et al., 2014). In regulatory networks, decomposition segments large graphs into functional modules on the basis of connectivity and clustering coefficients, with mathematical cutoffs delineating “hierarchical” and “modular” nodes (Freyre-González et al., 2014). Domain decomposition in neural networks partitions the input space into spatial or logical blocks, each processed in parallel by dedicated subnetworks, and then fused via high-level aggregators (Klawonn et al., 2023).

In formal software architecture, decomposition is modeled as coalition-forming games among requirements, where each coalition is scored for cohesion and expansion-freedom using pairwise utility functions and constraint satisfaction; algorithms extract partitions guaranteed to exist and interpretable as architectural modules (Liu et al., 2015).

2. Decomposition Mechanisms Across Paradigms

2.1. Software Systems and Microservices

Structured approaches for decomposing monolithic applications—such as Feature Table analysis, Jackson Problem Frames, and process-mining frameworks—systematically identify functional, environmental, and interaction-based “Feature Cards” or “Problem Diagrams”. Subsequent merging algorithms leverage formalized metrics (e.g., semantic overlap, hardware facility sharing) and outcome-based rules to synthesize candidate microservices—each exposing a cohesive set of functionalities and interfaces (Li et al., 2022, Taibi et al., 2019). Quantitative evaluation of these decompositions focuses on coupling metrics, service size, code duplication, and boundary definition.

2.2. Neural Network Decomposition

Neural architectures apply domain decomposition to split inputs (e.g., images, volumes) into subdomains for parallel CNN/DNN processing (Klawonn et al., 2023), and tensor decomposition such as Tucker or Block-Term for factorizing convolutional kernels, which reduces parameter count and accelerates training (Elhoushi et al., 2019, Howard-Jenkins et al., 2019). Subband decomposition architectures—filter-bank-based or adaptive subband decomposition—partition feature maps spectrally, enforcing structural regularization and robustness to quantization noise while dramatically decreasing computational cost (Sinha et al., 2023).

Group-size Series (GroSS) decomposition generalizes grouped convolutions, embedding the entire group-size search space in a differentiable series of factorization terms to enable concurrent optimization and efficient architecture search (Howard-Jenkins et al., 2019).

2.3. Data and Signal Decomposition

In time-series and image analysis, decomposition exploits both time-domain and frequency-domain methods. Frequency band cascades (e.g., TimeKAN: Cascaded Frequency Decomposition + Multi-order KAN blocks) extract multiscale representations attuned to distinct signal complexities, while architecture-specific mixing blocks recombine them with high fidelity (Huang et al., 10 Feb 2025). In anomaly detection (TFAD), decomposition into trend and residual components, each encoded via temporal and spectral branches, enables more precise anomaly localization and interpretability (Zhang et al., 2022). Blind image decomposition architectures utilize parameter-free channel slicing and recombination blocks inserted in bottleneck layers, allowing controllable restoration according to user intent, at negligible computational overhead (Zhang et al., 2024).

2.4. Optimization and Distributed Systems

Decomposition-based architectures for resource allocation and distributed scheduling (e.g., virtual network embedding in cloud infrastructure) utilize primal or dual decomposition to split global integer programs into tractable subproblems solved in parallel. Resource splits or dual pricing variables mediate coordination among agents, and branch-and-bound algorithms with subgraph isomorphism enable synthesis of optimal system topologies under stringent energy and performance constraints (Esposito et al., 2014, 0710.4707).

3. Decomposition Algorithms and Training Workflows

Most deep learning applications of decomposition unfold mathematical solvers (e.g., scaled alternating direction method of multipliers—ADMM, ISTA, PCA, RPCA) into explicit staged networks. ℓ₁DecNet+ implements a multi-block scaled-ADMM solver where each iteration is a network layer, with trainable thresholds, penalty parameters, and convolutional operators learned end-to-end; the sparse feature output is routed to a lightweight segmentation module, offering principled integration of physical image priors and data-driven segmentation (Ren et al., 2022). In video decomposition, robust PCA unrolling provides global structured representations, which are further refined by patch-recurrent ConvLSTM with backprojection modules into orthogonal control layers, achieving state-of-the-art segmentation and restoration in noisy, dynamic backgrounds (Qin et al., 2022).

Tensor decomposition strategies for CNN acceleration apply Tucker/CP/BTT factorization mid-training, yielding substantial reductions in model size and training time while maintaining accuracy; automated rank selection via VBMF ensures optimal compression-speedup trade-offs (Elhoushi et al., 2019).

4. Architecture Fusion, Resource Allocation, and Parallelism

Decomposition architectures frequently blend local (fine-scale) and global (coarse/cross-domain) modules. Domain decomposition networks aggregate local subnetwork decisions via a global DNN, achieving embarrassingly parallel model training and deployment. In overlapping domain decomposition for PINNs, solution continuity is maintained using partition-of-unity window functions, allowing subdomains to overlap without explicit interface losses (Huang et al., 14 Nov 2025). In microservice and modular system design, domain ownership and communication boundaries are assigned via quantitative rules over resource usage and semantic correlation, resulting in well-bounded, scalable architectures (Li et al., 2022, Taibi et al., 2019).

Linked-microservices and service-oriented cloud/fog architectures align decomposition and computational distribution: edge nodes execute lightweight preprocessing, fog nodes run moderate analytics, and cloud nodes perform heavy inference. Explicit pseudocode (see procedure DecomposeAndDeploy in (Alturki et al., 2019)) codifies resource-matching, dynamic deployment, and objective function–driven optimization.

5. Applications, Impact, and Quantitative Outcomes

Decomposition-based architectures yield measurable benefits in efficiency, scalability, and interpretability:

• Fog/Edge reductions: 10–70% network data savings; hybrid models realize up to 97.6% reduction on numerical datasets, with latency and energy gains balanced against minor accuracy loss (Alturki et al., 2019).
• Deep learning acceleration: up to 2× training speedup and 20× parameter reduction with ≤2% accuracy penalty, hardware-agnostic (Elhoushi et al., 2019).
• Image and shape modeling: learnable convex decomposition enables unsupervised part segmentation and interpretable mesh extraction; on ShapeNet, CvxNet matches or outperforms state-of-the-art volumetric methods (Deng et al., 2019).
• Microservice slicing: objective quantification of coupling, duplication, and service size reduces decomposition subjectivity and aligns outcomes to business constraints (Taibi et al., 2019).
• NoC topology synthesis: decomposition—using communication primitives—yields 36% throughput increase and 51% energy reduction over mesh architectures, with implied design guidelines on decomposition depth and primitive selection (0710.4707).
• Biological network inference: natural decomposition in E. coli recovers 62 functional modules, five global chains of command, and identifies intermodular “multiplexer genes” (Freyre-González et al., 2014).

6. Theoretical Foundations and Design Principles

Decomposition architectures are grounded in a mix of combinatorial optimization, algebraic factorization, coalition game theory, and information-theoretic principles. Fundamental solution concepts include:

• Cohesion and expansion-freedom: partitions must be locally optimal (no subset is better) and globally stable (no merge increases utility) (Liu et al., 2015).
• Partition-of-unity fusion: overlapping subdomains maintain smoothness and adaptivity via analytic window functions (Huang et al., 14 Nov 2025).
• Structural regularization: subband decomposition architectures restrict cross-subband co-adaptation to improve generalization and noise robustness (Sinha et al., 2023).
• Semantic correlation and environment alignment: microservice decomposition integrates requirements, topology, and environmental interactions to generate clear domain-driven boundaries (Li et al., 2022).

7. Guidelines, Best Practices, and Future Directions

Designing effective decomposition architectures involves:

• Profiling resource footprints and communication costs.
• Dynamically allocating modules to match capabilities and runtime constraints.
• Automated rank/band/group-size selection via analytic or data-driven methods.
• Quantitative evaluation of trade-offs between modularity, coupling, duplication, and performance.
• Iterative refinement via user feedback and data analysis.

Prospective research spans automated, Pareto-optimal decomposition search (e.g., evolutionary algorithm–driven microservice slicing), integration of dynamic system metrics into architecture selection, and domain-specific extensions (e.g., physics-informed models, multi-objective decomposition in heterogeneous systems).

Decomposition architecture thus represents a universal paradigm for structuring complex systems, algorithmically or semantically, across a wide array of computational disciplines.