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Compositional Generalization Requires Linear, Orthogonal Representations in Vision Embedding Models

Arnas Uselis, Andrea Dittadi, Seong Joon Oh

TL;DR

This work formalizes three desiderata for compositional generalization under standard training and shows they impose necessary geometric constraints: representations must decompose linearly into per-concept components, and these components must be orthogonal across concepts.

Abstract

Compositional generalization, the ability to recognize familiar parts in novel contexts, is a defining property of intelligent systems. Although modern models are trained on massive datasets, they still cover only a tiny fraction of the combinatorial space of possible inputs, raising the question of what structure representations must have to support generalization to unseen combinations. We formalize three desiderata for compositional generalization under standard training (divisibility, transferability, stability) and show they impose necessary geometric constraints: representations must decompose linearly into per-concept components, and these components must be orthogonal across concepts. This provides theoretical grounding for the Linear Representation Hypothesis: the linear structure widely observed in neural representations is a necessary consequence of compositional generalization. We further derive dimension bounds linking the number of composable concepts to the embedding geometry. Empirically, we evaluate these predictions across modern vision models (CLIP, SigLIP, DINO) and find that representations exhibit partial linear factorization with low-rank, near-orthogonal per-concept factors, and that the degree of this structure correlates with compositional generalization on unseen combinations. As models continue to scale, these conditions predict the representational geometry they may converge to. Code is available at https://github.com/oshapio/necessary-compositionality.

Compositional Generalization Requires Linear, Orthogonal Representations in Vision Embedding Models

TL;DR

This work formalizes three desiderata for compositional generalization under standard training and shows they impose necessary geometric constraints: representations must decompose linearly into per-concept components, and these components must be orthogonal across concepts.

Abstract

Compositional generalization, the ability to recognize familiar parts in novel contexts, is a defining property of intelligent systems. Although modern models are trained on massive datasets, they still cover only a tiny fraction of the combinatorial space of possible inputs, raising the question of what structure representations must have to support generalization to unseen combinations. We formalize three desiderata for compositional generalization under standard training (divisibility, transferability, stability) and show they impose necessary geometric constraints: representations must decompose linearly into per-concept components, and these components must be orthogonal across concepts. This provides theoretical grounding for the Linear Representation Hypothesis: the linear structure widely observed in neural representations is a necessary consequence of compositional generalization. We further derive dimension bounds linking the number of composable concepts to the embedding geometry. Empirically, we evaluate these predictions across modern vision models (CLIP, SigLIP, DINO) and find that representations exhibit partial linear factorization with low-rank, near-orthogonal per-concept factors, and that the degree of this structure correlates with compositional generalization on unseen combinations. As models continue to scale, these conditions predict the representational geometry they may converge to. Code is available at https://github.com/oshapio/necessary-compositionality.
Paper Structure (47 sections, 16 theorems, 117 equations, 41 figures, 4 tables)

This paper contains 47 sections, 16 theorems, 117 equations, 41 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Setup: A Framework for compositionality
  4. Setup: concept spaces and data collection process
  5. Compositional representations and models
  6. Compositional generalization and desiderata
  7. Instantiating the framework with CLIP
  8. Implications of compositionality on representations
  9. Geometry of $f$ under common training settings
  10. Sufficiency of linear and cross-concept orthogonal $f$
  11. Packing and minimum dimension
  12. Surveying necessary conditions in pretrained models
  13. Linear factorization in pre-trained models
  14. Compositional generalization and linear factorization
  15. Orthogonality of factors
  16. ...and 32 more sections

Key Result

Proposition 1

Let $\Pi=(f,\mathcal{H},A,\mathcal{T})$ be the tuple instantiated in def:scg, with linear heads $\mathcal{H}$ and $A$ given by GD+CE. Suppose that the training sets follow random sampling with validity rule $R(T) = 1 \text{ if } |T| = 2^{k-1} + 1$. Assume des:axiom_of_divisibilitydes:axiom_of_transf

Figures (41)

  • Figure 1: What enables compositional generalization in vision-language embedding models? Training data contain common configurations (left: a cat on a person) but lack rare ones (right: a person on a cat). Yet the same text-based queries, e.g. "A photo of a person", must work on both, even when the latter was never seen during training. We investigate what properties encoder $f$ must satisfy for such transfer to succeed.
  • Figure 2: Interpreting previous works' sampling designs $T$ and validity rules $R$. Each panel shows a concept space as a grid (2D or 3D), with blue cells denoting training combinations and red cells denoting held-out test combinations.
  • Figure 3: Relationship between (generalizing) compositional models. Divisibility (orange) and transferability (blue) requirements.
  • Figure 4: Instantiating the framework with CLIP-like embedding models for analysis.
  • Figure 5: Stable and unstable examples of feature representations. The top panel shows an unstable configuration, where depending on the sample, the readout either does not transfer or unstably. Bottom panel shows a stable configuration.
  • ...and 36 more figures

Theorems & Definitions (39)

  • Definition 1: Concept space
  • Definition 2: Training support, validity class, and training dataset
  • Definition 3: (Linearly) compositional model
  • Definition 4: Compositional generalization
  • Proposition 1: Compositional generalization implies linear factorization
  • Proposition 2: Sufficiency summary (informal)
  • Proposition 3: Minimum dimension for linear probes
  • Definition 5: Intervention on a concept value
  • Definition 6: Cross dataset at $\bm{c}$
  • Definition 7: Dataset marginal counts
  • ...and 29 more