Techniques for concept extraction, such as sparse autoencoders and transcoders, aim to extract high-level symbolic concepts from low-level nonsymbolic representations. When these extracted concepts are used for downstream tasks such as model steering and unlearning, it is essential to understand their guarantees, or lack thereof. In this work, we present a unified theoretical framework for unsupervised concept extraction, in which we frame the task of concept extraction as identifying a generative model. We present a general meta-theorem for identifiability, which reduces the problem of establishing identifiability guarantees to the problem of characterizing the intersection of two sets. As we demonstrate on a range of widely-used approaches, this meta-theorem substantially simplifies the task of proving such guarantees, thus paving the way for the development of new, principled approaches for concept extraction.
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.
Grégoire Dhimoïla, Victor Boutin, Agustin Martin Picard +2
Reliable generalization in conditional latent variable models requires understanding both identifiability and extrapolation: how observed variation across attributes determines latent structure, and how that structure determines distributions at unseen attributes. However, existing identifiability and extrapolation guarantees are largely model-specific, with separate analyses in nonlinear ICA, causal representation learning, perturbation modeling, and related conditional latent variable models. We introduce concept modulation models (CMMs), an attribute-indexed class of conditional generative models with structure A→Λ→C→X, where attributes select modulators, modulators induce latent concept laws, and concepts generate observed features. CMMs lift transition-based identifiability to conditional settings by showing that feature agreement on observed attributes induces a latent concept transition constrained by the CMM class. We express these constraints through attribute potentials, log-density ratios between attribute-conditioned concept laws, separating the generic lifting step from model-specific rigidity arguments. The same potentials control extrapolation: agreement at unseen attributes holds exactly when the transported attribute-potential identities extend to those attributes. This yields algebraic extrapolation criteria, identifies the common potential-based proof objects behind several existing identifiability and extrapolation results, and, when combined with the model-specific rigidity arguments in those works, recovers their stated conclusions.
Understanding how neural models represent human-interpretable concepts is challenging. Prior work has explored linear concept subspaces from diverse perspectives, such as probing and concept erasure. We introduce a unified framework to study these subspaces along two axes: \textit{containment}, which tests if a concept is fully represented in a subspace but not outside it, and \textit{disentanglement}, which tests for isolation from other concepts. In experiments on both text and speech models, we first highlight that concept subspaces may not be uniquely determined, and discuss the implications for concept subspace analysis. Then, we compare properties of concept subspaces estimated using five estimators, proposed in different communities. We find that (1) the choice of estimator impacts the containment and disentanglement properties; (2) the state-of-the-art concept erasure method, LEACE, performs well on both testing axes, but still struggles to generalize to unseen data; and (3) in HuBERT speech representations, phone information is both contained and disentangled from speaker information, while speaker information is hard to contain in a compact subspace, despite being disentangled from phones.