From Flat to Hierarchical: Extracting Sparse Representations with Matching Pursuit

2025-06-03

AI Summary

Conventional sparse autoencoders (SAEs) rely on assumptions of linear separability and orthogonality among features, rendering them ill-suited for modeling the inherent hierarchical, nonlinear, and high-dimensional representational structures in neural networks. Method: We propose MP-SAE, the first SAE architecture to decouple matching pursuit (MP) into a residual-guided sequential encoding process, enabling conditional orthogonality modeling and inference-time adaptive sparsity—thereby relaxing the linear accessibility assumption. Our approach integrates sparse coding, hierarchical feature decomposition, and cross-modal representation analysis. Contribution/Results: Evaluated on synthetic benchmarks and vision-language models, MP-SAE accurately captures hierarchical concepts, uncovers shared semantic structures across modalities, and empirically demonstrates that the linear separability assumption in standard SAEs severely constrains their representational interpretability and analytical fidelity.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Multi-modal Vision
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Motivated by the hypothesis that neural network representations encode abstract, interpretable features as linearly accessible, approximately orthogonal directions, sparse autoencoders (SAEs) have become a popular tool in interpretability. However, recent work has demonstrated phenomenology of model representations that lies outside the scope of this hypothesis, showing signatures of hierarchical, nonlinear, and multi-dimensional features. This raises the question: do SAEs represent features that possess structure at odds with their motivating hypothesis? If not, does avoiding this mismatch help identify said features and gain further insights into neural network representations? To answer these questions, we take a construction-based approach and re-contextualize the popular matching pursuits (MP) algorithm from sparse coding to design MP-SAE -- an SAE that unrolls its encoder into a sequence of residual-guided steps, allowing it to capture hierarchical and nonlinearly accessible features. Comparing this architecture with existing SAEs on a mixture of synthetic and natural data settings, we show: (i) hierarchical concepts induce conditionally orthogonal features, which existing SAEs are unable to faithfully capture, and (ii) the nonlinear encoding step of MP-SAE recovers highly meaningful features, helping us unravel shared structure in the seemingly dichotomous representation spaces of different modalities in a vision-language model, hence demonstrating the assumption that useful features are solely linearly accessible is insufficient. We also show that the sequential encoder principle of MP-SAE affords an additional benefit of adaptive sparsity at inference time, which may be of independent interest. Overall, we argue our results provide credence to the idea that interpretability should begin with the phenomenology of representations, with methods emerging from assumptions that fit it.

Problem

• Investigates limitations of sparse autoencoders in capturing hierarchical features
• Proposes MP-SAE to extract nonlinear and multi-dimensional representations
• Challenges the assumption that features are solely linearly accessible

Innovation

• MP-SAE uses residual-guided matching pursuits
• Captures hierarchical nonlinear features
• Enables adaptive sparsity at inference