A Unifying Framework for Concept-Based Representational Similarity

Technical Summary: A Unifying Framework for Concept-Based Representational Similarity

Problem Statement

Modern deep learning models often exhibit striking structural similarities in their internal representations, suggesting shared underlying concept decompositions across architectures, initializations, and modalities. However, the notion of "concept alignment" remains poorly defined. Existing literature uses the same terminology to optimize different objectives, obscuring which specific properties are actually being aligned. Furthermore, it is unclear whether optimizing one alignment objective (e.g., translation) reliably recovers others (e.g., concept consistency), or whether purely unsupervised, distributional objectives can recover meaningful instance-level alignment. The paper argues that current methods lack a rigorous framework to distinguish between aligning representations versus aligning concepts, and between instance-wise versus distributional alignment.

Methodology: A Unifying Framework

The authors propose a formal framework that decomposes concept alignment along two axes:

1. What is aligned: Representations (translation) vs. Concepts (concept consistency).
2. Level of alignment: Instance-wise (specific data points) vs. Distributional (statistical properties of the data).

This decomposition yields four fundamental properties:

• Instance-wise Translation ($P_{tr}$): The ability to map a representation from system $i$ to system $j$ ($(\psi^{-1}_j \circ \psi_i)(x_i) = x_j$).
• Instance-wise Concept Consistency ($P_{cont}$): The ability of both systems to recover the same compressed view of a shared concept space ($\psi_i(x_i) = \psi_j(x_j)$).
• Distributional Translation ($P^{dist}_{tr}$): The pushforward measure of the translation matches the target distribution.
• Distributional Concept Consistency ($P^{dist}_{cont}$): The pushforward measures of the concept spaces match.

The framework operationalizes these properties as regularization terms in the training loss of Sparse Autoencoders (SAEs). The authors analyze the theoretical relationships between these properties, noting that while translation and concept consistency are theoretically equivalent under strict injectivity and reconstruction assumptions, these conditions often fail in practice.

To evaluate these properties, the authors introduce InterVenchA, a diagnostic benchmark extending SAEBench. It separates evaluation into:

• Extraction Quality ($M_{extract}$): How well the SAE reconstructs and disentangles features within a single domain.
• Translation Quality ($M_{tr}$): How well concepts learned in one domain transfer to reconstruct another (independent of consistency).
• Concept Consistency ($M_{cont}$): How well the same set of concepts captures task-relevant information across domains (independent of translation).

Proposed Solution: Coupled Sparse Autoencoder (CoSAE)

Based on the finding that no single objective suffices, the authors propose CoSAE, a modular SAE framework that jointly enforces complementary alignment objectives. CoSAE combines:

• Reconstruction loss ($L_{dcy}$).
• Instance-wise translation loss ($L_{tr}$) on a small subset of paired data.
• Instance-wise concept consistency loss ($L_{cont}$) on the same paired data.
• Distributional alignment losses ($L^{dist}_{tr}$ and $L^{dist}_{cont}$) on the full dataset.

Crucially, the method relies on "anchoring" distributional objectives with a minimal amount of paired data (as little as 0.1% of the dataset) to resolve the unidentifiability of the transport problem.

Key Results

Through experiments on synthetic data, vision models (ViT, DinoV2, SigLIP), and multimodal encoders (CLIP, OpenCLIP), the authors demonstrate:

1. Failure of Equivalences: Commonly assumed equivalences between alignment objectives do not hold in practice.

   • Optimizing translation does not reliably recover concept consistency, and vice versa.
   • Cycle consistency (often used as a surrogate for translation in pair-free settings) is not a faithful proxy for alignment; high cycle consistency can coexist with poor alignment.
   • Purely distributional objectives fail to recover instance-level alignment due to the unidentifiability of the transport problem.

2. Necessity of Multi-Objective Optimization: Strong alignment emerges only when translation and concept consistency are explicitly regularized together.

   • In real embeddings, optimizing $L_{tr}$ alone improves translation but not consistency; optimizing $L_{cont}$ alone improves consistency but degrades extraction quality.
   • Only the combination of both yields high scores in both metrics.

3. Effectiveness of Scarce Supervision: The "anchoring" strategy is highly effective. Using just 1 in 1,000 matched pairs to anchor distributional objectives allows CoSAE to recover strong instance-level alignment, matching or exceeding the performance of models trained with full instance-wise supervision.

4. Performance: CoSAE outperforms state-of-the-art methods, including Crosscoders, Universal SAEs (USAEs), and SAE-Aligned (SAE-A), across all alignment metrics ($M_{tr}$ and $M_{cont}$). Notably, Crosscoders, which enforce consistency via architectural constraints (summing codes), fail to maintain strong standalone encoders, whereas CoSAE maintains robust individual encoders.

5. Downstream Utility: The alignment learned by CoSAE supports competitive zero-shot transfer performance (e.g., ImageNet classification) comparable to methods using dense projectors, despite CoSAE relying on sparse autoencoders.

Significance and Claims

The paper claims that concept alignment is fundamentally a multi-objective problem that cannot be captured by a single surrogate objective. The authors argue that:

• Existing methods often conflate distinct properties, leading to ambiguous interpretations of "alignment."
• A rigorous definition distinguishing between representation vs. concept alignment and instance vs. distributional levels is necessary for progress.
• The strongest practical performance arises not from optimizing one objective perfectly, but from combining complementary regularizers with minimal supervision.

The authors position CoSAE not just as a new model, but as a constructive consequence of a clearer theoretical understanding of alignment. They emphasize that meaningful alignment requires respecting the structure of the problem in both evaluation and method design, moving away from single-score metrics toward a coupled family of objectives.

Limitations Acknowledged:

• The framework focuses on shared structure and does not explicitly model idiosyncratic or domain-specific features beyond self-reconstruction.
• Evaluation relies on proxy metrics rather than ground-truth concepts, meaning semantic alignment and functional transfer may not always coincide.
• The combined objective introduces hyperparameter sensitivity.
• Empirical validation is limited to specific architectures and datasets, though the framework is intended to be generalizable.