Scalable and Interpretable Representation Alignment with Ordinal Similarity

Neural networks encode inputs as high-dimensional vectors, known as representations, that capture how models process data by encoding task-relevant structure and semantics. Representation alignment refers to the degree to which different models, layers, or training conditions produce similar representations for the same inputs, with important implications for model interpretation, selection, and robustness analysis. Existing approaches to measure alignment primarily rely on geometric properties, such as neighborhood and cluster similarity, offering limited insight into the global organization of representations. In this work, we present TopoAlign, a topology-aware framework for visually comparing model representations from a structural perspective. Leveraging mapper graphs from topological data analysis, TopoAlign jointly analyzes graphs constructed from representations of shared inputs across different models or layers. The framework supports a top-down comparative workflow: it first performs global structure alignment via joint force-directed optimization to produce coordinated graph layouts; it then identifies local correspondences through automated detection of structurally matching regions, visualized with Bubble Sets; and finally it enables fine-grained pattern inspection through motif-based queries and membrane-inspired visualizations. We demonstrate TopoAlign through case studies on language and multimodal models, complemented by expert feedback. Our results show that TopoAlign provides meaningful insights into representation structure and alignment from a topological perspective.

Learned representations across models and modalities often exhibit striking structural similarities, suggesting shared underlying concept decompositions. However, concept alignment remains poorly defined: existing approaches optimize different objectives under the same terminology, obscuring what is actually aligned. We propose a unifying framework that decomposes alignment along two axes: what is aligned (representations vs. concepts) and at what level (instance-wise vs. distributional). This induces four corresponding properties -- instance-wise and distributional variants of translation and concept consistency -- and reveals precisely which of these guarantees existing methods provide. We further introduce \InterVenchA, an intervention-based benchmark that separately measures extraction quality, translation quality, and concept consistency. Through theory and experiments, we show that commonly assumed equivalences between alignment objectives fail in practice: optimizing one property does not reliably recover the others, and purely unsupervised objectives fail to recover meaningful instance-level alignment. We then propose the Coupled Sparse Autoencoder (CoSAE), which jointly enforces complementary alignment objectives. Strong alignment emerges only in this regime. Surprisingly, as little as 0.1% paired data is sufficient to recover instance-level alignment when anchoring distributional objectives. Overall, our results show that concept alignment is fundamentally multi-objective: it must be defined, measured, and optimized as such.

The study of representations is widespread across fields, including neuroscience, psychology, and artificial intelligence. While representations are often studied and compared through similarities between stimuli, current methods provide only limited access to the dimensions that shape these representations and are often limited in interpretability. To overcome these challenges, here we introduce Similarity-Based Representation Factorization (SRF), a general computational method for recovering low-dimensional, non-negative, interpretable embeddings from similarity matrices derived from measured data. Across simulations and many neural, behavioral, and computational datasets, SRF recovers interpretable dimensions from diverse forms of representational data, even for very sparsely sampled, incomplete data. The dimensions derived from these datasets match those obtained by task-specific models, predict independent behavioral properties, improve exploratory analysis, and offer higher power for confirmatory hypothesis testing than comparing similarity matrices. Together, these results establish SRF as a general-purpose method with broad applications for uncovering, understanding, and using the dimensions underlying representations.