Spectral-Aligned Pruning (SAP) Methods

• Spectral-Aligned Pruning (SAP) is a neural network pruning strategy that preserves eigenvalue and singular value spectra to maintain model convergence and generalization.
• SAP employs techniques such as NTK-SAP, SVD-based layer-wise pruning, and graph-spectrum matching for code transformers, ensuring accuracy with high sparsity.
• Empirical results show that SAP variants achieve minimal performance degradation even at extreme pruning ratios, outperforming traditional pruning methods.

Spectral-Aligned Pruning (SAP) denotes a family of methodologies for neural network pruning that explicitly aim to preserve or align the spectral properties—eigenvalues or singular values—of certain matrices associated with network layers, architectures, or application domains. Across dense layers, convolutional networks, transformers, and specialized decoders, SAP provides a principled path for reducing parameter counts and computational cost while maintaining post-pruning performance and, crucially, the dynamics or representational power of the original model. Distinct implementations include NTK-SAP (aligning the Neural Tangent Kernel spectrum), layer-wise spectral pruning via SVD, structured pruning for code-graph transformers based on graph spectra, and spectrum-preserving matrix sparsification. The unifying rationale is that spectral alignment yields pruned networks that faithfully retain convergence and generalization behavior of the unpruned models.

1. Theoretical Foundations of Spectral Alignment in Network Pruning

The central motivation for SAP is the insight that key aspects of neural network function and trainability are governed by spectral properties of matrices, including (but not limited to) layer weight matrices, neural tangent kernels (NTKs), and, in specialized application domains, code-related graphs (Wang et al., 2023, Yao et al., 2023, Buffoni et al., 2021, Cho et al., 2 Feb 2026). For a weight matrix $A$, the singular value spectrum $\{\sigma_i(A)\}$ or eigenvalue spectrum $\{\lambda_i(A)\}$ encapsulate essential information about the transformation properties and sensitivity of the network.

In NTK-SAP, the analytically tractable NTK Gram matrix $\widehat{\Theta}(\theta) = J(\theta)J(\theta)^T$ (with $J(\theta)$ the input-output Jacobian) determines the dynamics of gradient descent. Preserving its spectrum, or minimizing perturbations thereof when pruning, ensures that each eigenmode of the output evolves under learning at rates matching those of the dense model. The dominant strategy thus seeks to minimize changes in a spectral norm (e.g., the trace or nuclear norm) when connections are removed (Wang et al., 2023).

In error-correcting code transformers, SAP utilizes the spectra of adjacency matrices corresponding to bipartite Tanner graphs, capturing second-order code connectivity. Similarity in this spectrum guides mask re-use across codes, enabling shared model pruning and cross-label generalizability (Cho et al., 2 Feb 2026).

Layer-wise SAP for standard neural networks typically relies on preserving the Frobenius or spectral norm (linked to the singular value decomposition), reflecting the intuition that prediction and generalization robustness depend on the preservation of dominant subspaces spanned by leading singular vectors (Yao et al., 2023).

2. SAP Methodologies Across Architectures

The specific implementation of SAP varies according to context and architectural constraints.

• NTK-SAP (Neural Tangent Kernel Spectral-Aligned Pruning): For foresight (pre-training) pruning, SAP assigns each weight a saliency score proportional to the perturbation in the NTK trace norm caused by its removal. This is estimated via a finite-difference approximation of the Jacobian Frobenius norm, typically using data-agnostic Gaussian inputs and weight-agnostic multisampling. Weights with the smallest saliency are iteratively pruned until the desired sparsity is achieved (Wang et al., 2023).
• SVD-based Layer-wise SAP: For dense and convolutional layers, SAP computes a truncated SVD of each layer’s weight matrix, distinctly preserving the top $K$ singular components. Quantile-based hard truncation is followed by random sampling for remaining entries, with sampling probabilities reflecting principal component magnitudes. This two-stage process guarantees minimization of deviation in the Frobenius norm given a fixed sparsity, enabling explicit control over spectral distortion (Yao et al., 2023).
• Spectral Node Pruning: In spectral parametrizations, neuron importance is linked directly to the magnitudes of eigenvalues in the transfer matrix. SAP can operate in both pre-training (train eigenvalues, prune, then retrain eigenvectors) and post-training (prune as per learned eigenvalues) modes (Buffoni et al., 2021).
• Structured SAP in Code Transformers: For universal code transformers, SAP constructs a “spectral signature” of each code from the leading eigenvalues of its adjacency matrix. Pruning masks for heads and FFN channels are either reused from the library for spectrally similar codes (as defined by an RBF kernel of signature distances) or constructed afresh when no close match exists. Per-code adaptation is then achieved via parameter-efficient low-rank updates (LoRA), allowing a single pruned backbone to serve many codes effectively (Cho et al., 2 Feb 2026).

3. Algorithms, Pseudocode, and Key Hyperparameters

Algorithmic realization of SAP can be summarized as follows, with distinct nuances for different architectures:

• NTK-SAP (Foresight Pruning):

1. Initialize mask to all ones.
2. For a specified number of rounds, compute per-weight saliency scores via the finite-difference method over random Gaussian inputs and initializations.
3. Aggregate scores and prune the lowest percentile to the target density.
4. Repeat until final sparsity is reached.
5. Hyperparameters: rounds $T$, batch size $B$, noise scale $\epsilon$ (Wang et al., 2023).

• Layer-wise SVD SAP:

1. For each weight matrix $A$, compute truncated SVD ($K$ largest singular components).
2. Hard threshold: keep entries above specified quantile exactly.
3. Random sample remaining entries according to normalized squared values.
4. Optional: drop columns/rows with very low sampling probability.
5. Hyperparameters: SVD rank $K$, quantile $\tau$, sampling floor $c$ (Yao et al., 2023).

• Code Transformer SAP:

1. Compute spectral signature $\phi(H)$ from the code's adjacency matrix.
2. Search mask library for closest match in spectral signature; compute similarity $\kappa$ via RBF kernel.
3. If above threshold $\tau$, reuse stored mask; else, generate new structured mask.
4. Train LoRA adapters per code atop shared backbone.
5. Hyperparameters: length $K$ of spectral signature, threshold $\tau$, kernel scaling $\beta$, LoRA rank $r$ (Cho et al., 2 Feb 2026).

4. Empirical Results and Comparative Performance

Comprehensive experiments demonstrate the efficacy of SAP variants:

• NTK-SAP consistently outperforms SNIP, GraSP, SynFlow, magnitude, and random pruning on CIFAR-10, CIFAR-100, Tiny-ImageNet, and ImageNet. For example, at 95.60% sparsity on ResNet-50 (ImageNet), NTK-SAP achieves top-1 accuracy 60.79%, compared to 58.88% (SynFlow), 59.73% (GraSP), and 43.79% (magnitude). The benefit is most pronounced at extreme sparsity and in large-scale settings (Wang et al., 2023).
• Layer-wise SAP maintains near-baseline accuracy up to high pruning ratios. On Fashion-MNIST, a three-layer MLP with pre-training SAP retains >92% accuracy even after 90% hidden node removal, compared to benchmarks which fail beyond 60%. On CIFAR-10 using MobileNetV2, post-training SAP keeps accuracy within 1–2% of baseline at 80–90% neuron pruning (Buffoni et al., 2021).
• SVD-based SAP for dense and convolutional layers in LeNet and VGG19 tightly controls test accuracy degradation, with the critical observation that accuracy loss tracks spectrum deviation of the pruned vs. original weight matrices almost monotonically. Custom SAP algorithms outperform pure magnitude thresholding at every sparsity level (Yao et al., 2023).
• Code Transformer SAP enables up to 40% FLOPs and parameter reduction while preserving bit error rate (BER) and frame error rate (FER). For all BCH, LDPC, and polar codes tested, the loss in decoding accuracy from reusing spectrally aligned masks is negligible ($\Delta \ge -0.15$). Empirically, spectral similarity ($\kappa$) correlates strongly with actual mask overlap (Jaccard $\rho=0.94$), supporting the utility of spectra as a structural proxy (Cho et al., 2 Feb 2026).
• Per-code adapter (LoRA) overhead is minimal (≈7% of model size) and does not degrade accuracy in the tested code families.

5. Limitations, Guidelines, and Extensions

Limitations of SAP approaches depend on the exact instantiation:

• The effectiveness of eigenvalue-based neuron pruning relies on appropriate initialization and model parameterization; networks trained and pruned in direct (weight) space cannot be post-hoc pruned spectrally if transfer matrices lack appropriate spectrum (Buffoni et al., 2021).
• For SVD-based SAP, selection of truncation quantile $\tau$, principal component count $K$, and sampling floor $c$ affect the tradeoff between sparsity and performance, and poorly calibrated settings may underperform thresholding for certain architectures (Yao et al., 2023).
• In code-graph SAP, naive mask transfer for spectrally dissimilar codes ($\kappa < \tau$) leads to performance collapse; thresholded acceptance is crucial (Cho et al., 2 Feb 2026).

Best practices include mask-guided retraining for iterative pruning, application of regularization (e.g., $\ell_1$ on eigenvalues), and, for convolutional networks, reshaping kernels for direct matrix operation. The framework is most robust when spectral properties are well defined and well separated.

6. SAP Beyond Canonical Network Compression

SAP’s core principle—that spectrum preservation aligns pruned models with desired learning and representational dynamics—has led to several proposed extensions:

• Integration with quantization or unstructured sparsity to further improve efficiency beyond what structured SAP alone enables (Cho et al., 2 Feb 2026).
• Application of spectrum-guided mask selection in other structured-pruning regimes, such as convolutional filter or transformer weight pruning in LLMs or beyond.
• Adaptive thresholds or richer spectral fingerprints (e.g., including eigenvectors, subspace angles) for finer alignment or for reuse across more heterogeneous domains.

A plausible implication is that as network sizes and performance demands grow, SAP enables transfer and compression strategies grounded in the structural essence of network computations rather than data or weight heuristics alone.

7. Summary and Outlook

Spectral-Aligned Pruning approaches, including NTK-based, eigenvalue-ranked, SVD-informed, and graph-spectrum-matching variants, establish spectrum preservation as a unifying foundation for neural network pruning. Across tasks, preserving the spectrum under pruning enables high sparsity with minimal loss in accuracy and maintains the original model’s training and inference characteristics. SAP extends naturally to dense, convolutional, and transformer architectures, as well as highly structured domains such as error-correcting code decoding, with documented advantages in both theoretical justification and empirical performance (Wang et al., 2023, Cho et al., 2 Feb 2026, Buffoni et al., 2021, Yao et al., 2023). Further development is likely to focus on hybrid efficiency schemes, domain transferability, and advanced spectral metrics for mask selection and transfer.