Learning disentangled representations of concepts and re-composing them in unseen ways is crucial for generalizing to out-of-domain situations. However, the underlying properties of concepts that enable such disentanglement and compositional generalization remain poorly understood. In this work, we propose the principle of interaction asymmetry which states: "Parts of the same concept have more complex interactions than parts of different concepts". We formalize this via block diagonality conditions on the $(n+1)$th order derivatives of the generator mapping concepts to observed data, where different orders of "complexity" correspond to different $n$. Using this formalism, we prove that interaction asymmetry enables both disentanglement and compositional generalization. Our results unify recent theoretical results for learning concepts of objects, which we show are recovered as special cases with $n\!=\!0$ or $1$. We provide results for up to $n\!=\!2$, thus extending these prior works to more flexible generator functions, and conjecture that the same proof strategies generalize to larger $n$. Practically, our theory suggests that, to disentangle concepts, an autoencoder should penalize its latent capacity and the interactions between concepts during decoding. We propose an implementation of these criteria using a flexible Transformer-based VAE, with a novel regularizer on the attention weights of the decoder. On synthetic image datasets consisting of objects, we provide evidence that this model can achieve comparable object disentanglement to existing models that use more explicit object-centric priors.

翻译：学习概念的分离表示并以未见方式重新组合它们，对于泛化至域外情境至关重要。然而，能够实现这种分离和组合泛化的概念底层特性仍鲜为人知。在本研究中，我们提出交互不对称性原理，其表述为："同一概念各部分间的相互作用比不同概念各部分间的相互作用更为复杂"。我们通过生成器（将概念映射至观测数据）的$(n+1)$阶导数块对角化条件对此进行形式化，其中不同"复杂度"阶次对应不同的$n$值。利用该形式体系，我们证明交互不对称性能够同时实现分离表示与组合泛化。我们的结果统一了近期关于物体概念学习的理论成果，这些成果可视为$n\!=\!0$或$1$时的特例。我们给出了直至$n\!=\!2$的理论结果，从而将先前工作拓展至更灵活的生成器函数，并推测相同证明策略可推广至更大的$n$值。实践层面，我们的理论表明：为实现概念分离，自编码器应在解码过程中惩罚其潜在容量及概念间的交互作用。我们提出采用基于Transformer的灵活变分自编码器实现这些准则，并在解码器注意力权重上引入新型正则化器。在由物体构成的合成图像数据集上，我们提供的证据表明该模型能达到与现有使用更显式物体中心先验模型相当的物体分离效果。