DecompX: Multi-Domain Decomposition Framework

• DecompX is a multi-domain framework that systematically decomposes complex mathematical and computational objects while preserving structure and semantic information.
• It employs advanced strategies such as nested commutator inclusion for quantum simulation, per-token attribution in transformers, and tensor operation optimizations for parallel computation.
• Applications span quantum physics, deep learning interpretability, topological data analysis, and hardware acceleration, thereby improving performance and enabling deeper analysis.

DecompX refers to a class of methods, algorithms, and interpretability frameworks across multiple research domains in which a complex object—be it an operator, model, tensor computation, or deep learning representation—is systematically decomposed into more elementary, interpretable, or computationally efficient components. The unifying principle is the preservation of structure, utility, or semantic information during the decomposition process, enabling downstream efficiency, analysis, or explainability. The term has been adopted in quantum physics for exponential operator splittings, in topological data analysis for algebraic module decompositions, in deep neural network interpretability, and in scalable machine learning systems, among other areas.

1. Exponential Operator Decomposition in Quantum Simulation

In operator theory and quantum simulation, DecompX denotes systematic and generalizable strategies to approximate the exponential of a sum of noncommuting operators by a product of exponentials, potentially incorporating exponentials of nested commutators for error cancellation beyond the standard Trotter and Suzuki formulas. Key features include:

• Direct Construction: Rather than relying exclusively on recursive or fractal-based schemas (e.g., Suzuki Trotterization), DecompX encompasses decompositions that directly optimize the order and coefficients of individual exponentials and nested commutators to more effectively match the Baker–Campbell–Hausdorff (BCH) expansion up to a desired precision.
• Nested Commutator Inclusion: By explicitly including exponentials of commutators such as [A, B], [A,[A,B]], and higher-order terms, DecompX improves on the error scaling of classical formulas, yielding more accurate simulation with fewer steps for a given error budget.
• General Applicability: The framework is systematic and not limited by the structure of the underlying operator algebra, making it viable for a wide range of quantum control and simulation tasks, including those with complex Hamiltonians.
• Illustrative Formula:

$$e^{t(A+B)} \approx \prod_{j=1}^{N} \exp(\alpha_j t A) \exp(\beta_j t B) \exp(\gamma_j t^2 [A,B]) \exp(\delta_j t^3 C_j) \cdots$$

with coefficients $\{\alpha_j, \beta_j, \gamma_j, \delta_j, \ldots\}$ chosen to cancel error terms in the BCH expansion as desired.

• Application Domains: Quantum control, quantum circuit synthesis, time evolution in quantum computation, and condensed matter simulation all benefit from these decompositions via improved fidelity and reduced circuit depth, thereby supporting both experimental and theoretical advancements (Sefi et al., 2011).

2. Deep Learning Interpretability via Transformer Decomposition

In the context of model explainability, DecompX identifies a framework for decomposing the information flow within Transformer-based models by attributing the contribution of each input token to the final class prediction in an additive, per-token, per-class manner (Modarressi et al., 2023). Principal aspects include:

• Vector-Based Token Decomposition: At each layer, the representation of a token $i$ is decomposed as $x_i^\ell = \sum_k x_{i \leftarrow k}^\ell$, allocating each component to the influence of a distinct input token $k$.
• Propagation through All Model Components: The decomposition propagates through self-attention, residual connections, feed-forward networks (with activation functions approximated per input), and the classification head, ensuring fidelity to the model's entire computational graph.
• Faithfulness and Per-Class Attribution: By tracing through all nonlinearities and class-specific heads, the framework yields both positive and negative attributions for every input–output token–class pair, with empirical results demonstrating superior performance over both gradient- and attention-based attribution baselines.
• Formal Decomposition at Prediction: For a final logit $y_c$,

$y_c = \sum_k y_{c \leftarrow k}$

allows direct assessment of each token’s role in the decision.

• Research Utility: DecompX supports transparency, debuggability, and auditing in applications ranging from sentiment analysis through to high-stakes domains such as hate speech detection, and forms a key component in prompt compression systems (see §6).

3. Decomposition in Parallel and Efficient Tensor Computation

DecompX is used to describe tensor operation decomposition strategies, particularly those that enable highly parallel computation by exploiting explicit structure in the Einstein summation specification of computational graphs (Bourgeois et al., 3 Oct 2024). Salient characteristics are:

• Extended Einstein Summation: By labeling tensor indices and decoupling aggregation and elementwise operations, arbitrary tensor contractions can be split into “tensor-relational” forms, exposing independent sub-tensors for parallel execution.
• Generalization of Parallelism: The decomposition generalizes classic data- and model-parallel techniques, allowing arbitrary hybrid sharding by selecting partitioning vectors $d$ for each index.
• Automated Scheduling: The EinDecomp algorithm uses dynamic programming to globally optimize partitionings across the computation DAG, minimizing communication and maximizing concurrency.
• Performance Impact: Experiments demonstrate that the approach can outperform domain-specific libraries such as ScaLAPACK and DASK in distributed settings, providing scalability for both training and inference in large model deployments.

4. Decomposition in Algebraic Topology and Persistence Modules

In topological data analysis, DecompX designates algorithms for decomposing multiparameter persistence modules into indecomposable summands, a task fundamental to understanding high-dimensional persistent homology (Dey et al., 10 Apr 2025). The framework is characterized by:

• Matrix Reduction Algorithms: The presentation matrix of a module is systematically block-reduced, with groupings and operations guided by the degrees of the generators and relations. The algorithm is applicable even if relations are not distinctly graded.
• Fixed-Parameter Tractability: Complexity is parameterized by $k$, the maximum number of relations per multidegree, yielding efficient computation in practice ($O(n^3)$ in the interval-decomposable case).
• Software Implementation: The aida library operationalizes these theoretical advances, demonstrating fast decomposition for modules with large presentations (hundreds of thousands of relations).
• Theoretical Advances: The supporting theory introduces parameter restriction functors and generalizes homological algebra constructions (e.g., $U$-projectivity), providing foundational tools for further research in multiparameter persistence modules.

5. Interpretability-Informed Prompt Compression in NLP

DecompX provides fine-grained token attribution scores used to efficiently frugalize prompts for LLMs (Raiyan et al., 18 Oct 2025). In this setting:

• Token Attribution: By propagating decomposed token vectors throughout the model, each token's contribution to the eventual output is quantified (“salience scoring”).
• Prompt Reduction: Tokens with highest DecompX-based scores are retained in the compressed prompt, drastically reducing context length with minimal loss in NLP tasks tolerant to semantic summarization (e.g., sentiment, QA).
• Trade-offs and Task Sensitivity: While aggressive prompt frugalization yields only marginal performance degradation in summarization or classification, performance on tasks that require strict token-level continuity (e.g., mathematical reasoning) degrades substantially, underlining the method's limits and informing the design of adaptive prompt engineering systems.

6. Computational Hardware Acceleration for Decomposition

For the real-time application of input (activation) decomposition in LLMs, system-level advances termed DecompX have been realized in the form of hardware/software co-design (Tahmasebi et al., 15 Oct 2025):

• Iterative Decomposition Algorithms: Progressive and Lanczos-based algorithms efficiently compute low-rank approximations of activations within tight inference constraints.
• Co-Accelerator Architecture: Specialized hardware with compute replication shifts the decomposition operation from a memory-bound to a compute-bound regime, leveraging parallel FP16 clusters and optimized on-chip reductions.
• Shape-Preserving Pipeline: An output shape-preserving computation strategy minimizes the need for repeated decompositions between layers, preserving both efficiency and functional equivalence across model layers.
• Multi-track and Outlier-Aware Decomposition: High-magnitude output channels are handled separately to maintain accuracy, suppressing approximation-induced degradation—demonstrated to yield a 22% reduction in latency versus A100 GPU at accuracy cost of approximately 3% on standard tasks.

7. Cross-Domain Variants and Future Directions

Across operator theory, deep learning, algebraic topology, and system-level optimization, DecompX typifies a strategy of decomposing global, complex objects into substantiated elements that preserve analytic properties, inform system design, or enhance interpretability:

• In quantum mechanics and numerical analysis, DecompX-type operator splittings shape time-evolution simulation strategies for both classical and quantum systems.
• In deep neural networks, DecompX frameworks provide a foundation for token-level interpretability, which in turn enables principled prompt compression.
• In computational topology, DecompX advances facilitate the scalable decomposition of algebraic structures, key to analyzing high-dimensional persistent homology data.
• In hardware architectures, DecompX reflects a convergence of software algorithm design and tailored accelerator engineering, essential for feasible deployment of next-generation large models.

Research continues into optimizing decomposition for adaptivity (context-aware token retention), extending algebraic theory for more general modules, and refining hardware/software interfaces to close the gap between algorithmic sophistication and real-world system constraints.

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DecompX, as a term, encapsulates a rigorous, multi-domain methodology for decomposing, analyzing, and exploiting structure in complex mathematical, computational, and data representations, with broad implications for both theory and large-scale systems practice across contemporary machine learning and scientific computing.