Ensembled MP-SAE Approaches

• Ensembled MP-SAE approaches are methods that unroll Matching Pursuit within sparse autoencoders and employ ensembles to robustly recover hierarchical and nonlinear latent features.
• They achieve monotonic reconstruction improvements by iteratively updating residuals, capturing both correlated and hierarchical feature structures beyond single-shot models.
• Ensemble strategies such as bagging, boosting, and modular stacking enhance expressivity, diversity, and downstream performance in tasks like interpretability and classification.

Ensembled MP-SAE approaches are a class of methods that combine Matching Pursuit Sparse Autoencoder (MP-SAE) architectures with ensembling principles, yielding systems that recover hierarchical, nonlinear, and complementary latent features from neural representations. Distinct from conventional single-shot sparse autoencoders, MP-SAE methods unroll the encoder into an iterative sequence of residual-guided feature extractions. This design, when extended with ensemble strategies—such as bagging, boosting, modularization, or aggregation over multiple runs—enhances expressivity, diversity, and robustness in representation learning for interpretability, classification, and downstream analysis.

1. Architectural Principles of MP-SAE

MP-SAE is formulated by unrolling the classical Matching Pursuit (MP) algorithm within a sparse autoencoder framework. The encoder sequentially selects features (atoms) from a learned dictionary by greedily projecting the current residual, updating both reconstruction and residual iteratively:

• The input $x$ and pre-decoding bias $b_{\text{pre}}$ initialize the residual $r^{(0)} = x - b_{\text{pre}}$ and reconstruction $\hat{x}^{(0)} = b_{\text{pre}}$.
• At iteration $t$, the algorithm selects the dictionary index $j^{(t)}$ maximizing $|D_{j}^{\top} r^{(t)}|$ and computes coefficient $\alpha^{(t)} = D_{j^{(t)}}^{\top} r^{(t)}$.
• Reconstruction and residual are updated: $$\hat{x}^{(t+1)} = \hat{x}^{(t)} + D_{j^{(t)}} \alpha^{(t)}$$, $r^{(t+1)} = r^{(t)} - D_{j^{(t)}} \alpha^{(t)}$.
• Each selected atom's contribution ensures the new residual is orthogonal to it: $D_{j^{(t)}}^{\top} r^{(t+1)} = 0$ (Costa et al., 3 Jun 2025, Costa et al., 5 Jun 2025).

This sequential, residual-driven inference differs from traditional SAEs, which perform a single nonlinear projection and rely on near-orthogonal dictionaries, limiting extraction of correlated or hierarchical features.

MP-SAE exhibits monotonic improvement in reconstruction (i.e., $\| r^{(t+1)} \|_2^2 \leq \| r^{(t)} \|_2^2$) and supports adaptive sparsity: the number of iterations (active atoms) can be controlled at inference time to trade off compactness and accuracy (Costa et al., 3 Jun 2025, Costa et al., 5 Jun 2025).

2. Hierarchical and Nonlinear Feature Extraction

Standard SAEs assume that abstract, interpretable features are linearly accessible and nearly orthogonal. Controlled experiments reveal that this quasi-orthogonality assumption inhibits the capture of correlated and hierarchical features. In contrast, MP-SAE:

• Enforces conditional orthogonality at each sequential extraction, allowing correlated atoms in the global dictionary and resolving them locally through residual updates.
• Recovers both intra-level correlations and parent–child hierarchical organization, demonstrated in synthetic data with tree-structured concepts.
• Avoids "feature absorption" observed in shallow SAEs, maintaining distinct parent and child features, and matches ground-truth correlation structure even under increased intra-level interference.

The nonlinear encoding—because each residual depends nonlinearly on the prior reconstruction—extends expressivity beyond what can be achieved with a single projection. This enables MP-SAE to recover higher-order and multimodal features in settings such as vision-LLMing (Costa et al., 3 Jun 2025).

3. Ensemble Strategies in MP-SAE Framework

Ensembling augments MP-SAE by combining multiple decoders, encoders, or inference runs to enhance diversity, stability, and reconstruction fidelity. Major ensemble methods include:

• Naive Bagging: Multiple SAEs (including MP-SAE variants) are trained independently with different initializations. Outputs are averaged:

$$g_{nB}(x; \{\theta^{(j)}\}) = \frac{1}{J}\sum_{j=1}^J g(x; \theta^{(j)})$$

This reduces variance and enhances feature diversity (Gadgil et al., 21 May 2025).

• Boosting: SAEs are sequentially trained such that each reconstructs the residual error left by its predecessors:

$$g_{\text{Boost}}(x; \{\theta^{(j)}\}) = \sum_{j=1}^J g(x^{(*, j)}; \theta^{(j)}),$$

where $x^{(*, j)}$ denotes the input to the $j^\text{th}$ SAE, possibly preconditioned by prior reconstructions. Boosting more aggressively reduces bias and uncovers more specialized, complementary features (Gadgil et al., 21 May 2025).

• Modular/Stacked Ensembles: Incorporating modular autoencoder blocks or stacking hierarchically organized autoencoders allows tuning diversity–accuracy trade-offs, as parameterized in the modular autoencoder (MAE) framework via the diversity parameter $\lambda$ (Reeve et al., 2015), or building multilayer paired-sample ensembles (Zhou et al., 2022).
• Aggregation over Random Masking: As shown in multiple random masking autoencoder ensembles, ensembles can be constructed implicitly by aggregating reconstructions from repeated inference under different random masks, promoting robustness in multimodal and missing-data scenarios (Todoran et al., 12 Feb 2024).

4. Expressivity, Reconstruction Fidelity, and Interpretability

Empirical results indicate that MP-SAE and its ensemble extensions outperform shallow SAEs on expressivity, reconstruction fidelity, and feature diversity, particularly in hierarchical or high-correlation representational regimes:

• On synthetic hierarchical trees, MP-SAE preserves ground-truth hierarchical structure and intra-level correlation better than conventional SAEs.
• In natural data (e.g., CLIP, DINOv2, SigLIP, ViT models), MP-SAE yields higher Pareto frontiers for reconstruction $R^2$ at similar sparsity levels, with adaptive inference enabling tailored reconstructions.
• For downstream tasks—such as concept detection and spurious correlation removal—ensembled SAEs (especially boosting) yield improved classification accuracy and greater ability to ablate undesired correlations (Gadgil et al., 21 May 2025).
• Aggregated outputs from MP-SAE’s sequential inference (or from ensembles over masks) function as an ensemble of local feature selections, leveraging conditional orthogonality and reconstructing complementary information (Costa et al., 3 Jun 2025, Todoran et al., 12 Feb 2024).

5. Ensemble Construction and Theoretical Justification

Theoretical results justify ensemble MP-SAE designs:

• Ensemble averaging (bagging) of SAE outputs is formally equivalent to concatenating the decoded feature vectors and coefficients from multiple SAEs (Proposition 1) (Gadgil et al., 21 May 2025).
• Naive bagging reduces variance; boosting reduces bias and specializes in fitting residuals. As the number of ensemble members increases, the variance contribution vanishes by law of large numbers.
• In MP-SAE, stepwise orthogonality guarantees monotonic reduction of reconstruction error, and in the limit, full orthogonal projection onto the dictionary’s span is achieved (Costa et al., 5 Jun 2025).
• Modular architectures support tuning the trade-off between diversity and reconstruction accuracy, where intermediate diversity parameter values optimize ensemble performance (Reeve et al., 2015).

6. Applications and Practical Considerations

Ensembled MP-SAE approaches have been validated on a wide range of benchmarks:

• Interpretability: Enhanced recovery of hierarchical, nonlinear, and shared features across modalities and tasks, offering finer control for sparse representation analysis (Costa et al., 3 Jun 2025).
• Classification and Concept Detection: Improved accuracy and stability in classification tasks over multiple domains (e.g., Amazon Reviews, GitHub Code, AG News, language detection) (Gadgil et al., 21 May 2025).
• Spurious Correlation Mitigation: Ensembled architectures enable the targeted removal of features associated with spurious or domain-specific correlations, improving fairness and robustness (Gadgil et al., 21 May 2025).
• Multimodal and Semi-supervised: Aggregation over multiple masking patterns enables robust handling of missing modalities and generates high-quality pseudo-labels for semi-supervised training, as shown in Earth observation and climate modeling applications (Todoran et al., 12 Feb 2024).

Limitations include increased computational cost for ensemble training (especially boosting), possible redundancy in bagging, or overspecialization in boosting. Adaptive inference and modular design mitigate some trade-offs by supporting dynamic sparsity and scalable ensemble construction.

7. Connections to Broader Ensemble and Modular Methods

Ensembled MP-SAE methods are conceptually linked to other ensemble and modular architectures:

• The modular autoencoder (MAE) (Reeve et al., 2015) and multilayer envelope embedded stack autoencoder ensemble (NE_ESAE) (Zhou et al., 2022) leverage module diversity, hierarchical sample structure, and flexible classification fusion to optimize feature extraction and decision-making.
• Single Architecture Ensemble (SAE) frameworks employ automatic search over early-exit and multi-input multi-output configurations to achieve ensemble-like behavior within a unified neural architecture (Ferianc et al., 9 Feb 2024).
• Sequential anchored ensembles approximate Bayesian posteriors efficiently by chaining sequential training phases with high-auto-correlation anchors, reducing computational expense compared to standard ensembles (Delaunoy et al., 2021).

A plausible implication is that future directions in MP-SAE research will emphasize modular, adaptive, and residual-guided architectures integrated within scalable, computationally efficient ensemble schemes, supporting interpretability, robustness, and application to real-world multimodal datasets and tasks.

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Ensembled MP-SAE approaches represent an intersection of iterative sparse representation techniques and ensemble learning, providing a principled and empirically validated framework for extracting hierarchical, diverse, and interpretable latent structures from complex neural activations. Their theoretical guarantees, empirical performance, and extensibility to modular and multimodal architectures make them a robust foundation for future work in interpretable machine learning and scalable feature ensemble systems.