Layer Contribution in Multilayer Systems

• Layer contribution is the quantitative assessment of how individual layers, through methods like ablation and gradient analysis, impact overall system performance and interpretability.
• It employs metrics such as Shapley values, information-theoretic measures, and occlusion scores to assign clear contributions in both neural networks and physical systems.
• Empirical studies across domains like deep learning, vision models, and material science demonstrate that optimizing high-contribution layers can enhance efficiency while maintaining accuracy.

Layer contribution refers to the quantitative assessment of how individual layers in a multi-layer system—such as deep neural networks, multilayer physical materials, or stratified physical domains—impact global processes, model predictions, or system-level observables. The concept has been formalized and measured in architecture design, interpretability, pruning, data selection, condensed matter physics, and theoretical transport. Across disciplines, rigorous metrics and experimental protocols have been developed to attribute outcome changes, feature transformations, or response enhancements directly to distinct layers.

1. Formalizations of Layer Contribution in Deep Learning

Layer contribution in neural networks encompasses several perspectives:

• Performance attribution: Measuring how altering, updating, or ablating a layer changes task-specific performance, e.g., through freezing, pruning, or single-layer fine-tuning (Zhang et al., 1 Jul 2026, Ding et al., 8 Feb 2026).
• Information–theoretic and geometric metrics: Quantifying how much task-relevant information a layer’s representation preserves, for example via mutual information, entropy, or separation statistics (Skean et al., 4 Feb 2025, Waagen et al., 2019).
• Parameter movement and adaptation: Assessing contribution based on the degree and nature of weight changes during task adaptation (Park et al., 2024).
• Feature and concept tracing: Decomposing predictions into contributions of input features per layer or spatial location (He et al., 2023, Wang et al., 2022).

Formal definitions include:

• Fractional RL contribution: For RL adaptation, $$\mathcal{C}(k)=\frac{S_k-S_\mathrm{base}}{S_\mathrm{full}-S_\mathrm{base}}$$, capturing the share of improvement achieved by tuning only layer $k$ (Zhang et al., 1 Jul 2026).
• Shapley value–based contribution: For pruning or cooperative-game analysis, the Shapley value $\phi_i$ quantifies the expected marginal benefit of keeping layer $i$ relative to all coalitions (Ding et al., 8 Feb 2026).
• Occlusion, gradient, and ablation scores: These attribute saliency or influence to hidden layers or tissue strata by systematic masking or differential analysis (Zeng et al., 25 Nov 2025, Wang et al., 2022).
• Information and representation metrics: Effective rank, matrix-based entropy, curvature, and InfoNCE loss quantify the quality and diversity of layer representations (Skean et al., 4 Feb 2025).

2. Quantifying Layer Contribution: Methodologies

2.1 Neural Network Protocols

• Single-layer training or ablation: Systematically retraining or pruning individual layers to empirically measure performance deltas (Zhang et al., 1 Jul 2026).
• Iterative/pruned aggregations: In deep layer aggregation (DLA), iterative and hierarchical structures are used to study the compounding effects of fusing multiple layer outputs (Yu et al., 2017).
• Filter / gradient-based attribution: FAIG (Filter Attribution with Integrated Gradients) quantifies each layer’s “movement” and saliency during adaptation, e.g., for image restoration (Park et al., 2024).
• HP statistic permutation tests: The statistical significance of class separation improvements across layers is determined via the Henze–Penrose statistic and permutation tests, identifying which layers most increase discriminative power (Waagen et al., 2019).

2.2 Interpretability and Explainability

• Occlusion analysis: Removing or perturbing layer features and measuring output impact; includes volume-normalized saliency for 3D image layers (Zeng et al., 25 Nov 2025), and concept-based contribution via TCAV/SACV for spatially localized explanations (Wang et al., 2022).
• Layer-wise decomposition in non-DL models: In deep forests, estimation plus calibration steps propagate feature importance and contributions transparently across tree-derived layers (He et al., 2023), enabling both local (per-instance) and global (MDI) attributions.

2.3 Data and Source Contribution

• Layer Value Contribution (LVC): In data selection for neural net training, LVC decomposes each sample’s influence into quality, relevance (gradient alignment), and diversity for every layer, and aggregates these to steer adaptive, budgeted data inclusion (Zhang et al., 24 Oct 2025).
• Cooperative-game contributions: For LLMs, layer-wise (and data-source) Shapley contributions guide optimal pruning, balancing efficiency and accuracy (Ding et al., 8 Feb 2026).

3. Empirical Patterns and Insights Across Domains

The contribution of layers is not distributed uniformly:

• Transformers and RL: RL-induced gains concentrate in a narrow band of mid-stack layers; tuning only a single high-contribution layer can match or surpass full-parameter RL improvements, whereas input/output-near layers contribute minimally (Zhang et al., 1 Jul 2026).
• Hidden representation optimality: Intermediate layers typically optimize the balance between information compression and task-relevant signal, peaking in representation quality and transferability before the final layer; proxy metrics (entropy, curvature, invariance) reliably locate these optima (Skean et al., 4 Feb 2025).
• Vision models: In deep image restoration architectures, encoder/decoder blocks dominate adaptation (high FAIG), with bottleneck blocks showing sparse contribution despite parameter mass (Park et al., 2024).
• Non-neural settings: Critical- and buffer-layer contributions play dominant roles, e.g., the buffer layer in graphene/SiC systems uniquely shapes the Raman spectrum (Fromm et al., 2012), and the critical layer in quadratic acoustic boundary layers dominates downstream Green’s function response (King et al., 2021).

4. Representative Algorithms and Layer Contribution Applications

Domain	Main Contribution Metric	Core Method/Framework
Transformer RL	Fractional improvement $\mathcal{C}(k)$	Frozen-layer RL sweep (Zhang et al., 1 Jul 2026)
LLM pruning	Shapley value $\phi_i$	Surrogate + stratified Monte Carlo (Ding et al., 8 Feb 2026)
Image restoration	FAIG (filter-integrated gradient)	Rank-allocating LoRA (Park et al., 2024)
ML interpretability	HP statistic, occlusion saliency, TCAV	MST, occlusion, concept gradients (Waagen et al., 2019, Zeng et al., 25 Nov 2025, Wang et al., 2022)
Data selection	LVC (quality, relevance, diversity)	Online data-value with LSH/UCB (Zhang et al., 24 Oct 2025)
Nonlinear materials	Layer-resolved polarizability	Projector integral/curvature (Radhakrishnan et al., 5 Dec 2025)

Selecting high-contribution layers for training, adaptive adaptation, or explanation enables substantial reductions in parameter flops, memory, and data usage, while often achieving or exceeding full-model accuracy.

5. Advanced Interpretability and Extension to Physical and Biological Systems

Layer contribution frameworks have been adapted beyond classic DNNs:

• Tissue layer attribution: In 3D ultrasound, LAYER sharply resolves which anatomical tissue strata drive disease prediction; e.g., the deep fascial membrane (DFM) exhibits the highest directional and absolute saliency for myofascial pain (Zeng et al., 25 Nov 2025).
• Feature tracing in forests: Calibration and tracing in deep forests propagate and allocate feature importances through cascaded, non-differentiable models, ensuring local and global interpretability for each layer (He et al., 2023).
• Concept attribution: SACV localizes not only the global concept influence but also its spatial footprint within each hidden layer, pinpointing which layer maps attend to meaningful, context-sensitive concepts (Wang et al., 2022).
• Layered condensed matter systems: Layer-resolved magnetoelectric polarizability in antiferromagnetic bilayers is partitioned via projector integrals, revealing both geometric and topological contributions that are layer-specific and symmetry-protected (Radhakrishnan et al., 5 Dec 2025).

6. Physical and Mathematical Theories: Boundary-Layer and Critical-Layer Contributions

• Linear transport and hydrodynamics: Layer contributions in physical theory quantify how boundary, buffer, or critical strata generate distinct solution regimes (e.g., anisotropic fluxes, singular boundary corrections, or algebraically decaying/acoustic modes), with composite expansions connecting dominant and residual terms (Gaggioli et al., 2021, Gungor et al., 2024).
• Electrodynamics and friction: The electrical double layer at metal surfaces enhances quantum and electrostatic friction forces by 8–10 orders of magnitude, shifting detection regimes and mediating interaction with evanescent fields. The EDL’s “layer contribution” is thus pivotal for both macroscopic effects and reconciliation of theory with experiment (Volokitin, 2021).

7. Open Challenges and Practical Implications

While layer contribution metrics and protocols have proven invaluable for interpretability, efficiency, and empirical gains, several challenges remain:

• Combinatorial explosion: Accurate game-theoretic contribution estimates (e.g., Shapley) remain infeasible for very large L (number of layers) without sophisticated surrogates (Ding et al., 8 Feb 2026).
• Task and architecture dependence: The optimal contributing layer(s) can shift under transfer learning, domain adaptation, or architectural change; in LLMs, compressive bottlenecks and fine-tuning can re-shape the depth at which contribution peaks (Skean et al., 4 Feb 2025).
• Dynamic adaptivity and online reallocation: Layer importance can change during training or across data regimes, motivating adaptive strategies such as dynamic LoRA rank, online LVC, or cross-task layer re-ranking (Park et al., 2024, Zhang et al., 24 Oct 2025).
• Limitations in non-differentiable or highly coupled systems: Some protocols are restricted to differentiable or specific architectures; tracing contributions in non-differentiable, highly coupled, or non-stationary systems remains a challenging frontier.

Ultimately, layer contribution serves as a foundational principle guiding architecture design, resource allocation, interpretability, and analysis across the full spectrum of layered systems, from neural networks to quantum and biological multilayers. The advancement of both theory and scalable algorithms for quantifying contribution at arbitrary depths and modalities continues to inform leading-edge research and practical deployment (Yu et al., 2017, Zhang et al., 1 Jul 2026, Skean et al., 4 Feb 2025).