Necessary and Sufficient Conditions for Compositional Representations

Humans naturally use compositional representations for flexible recognition and expression, but current machine learning lacks such ability. Despite many efforts in specific cases, there is still absence of theories and tools to study it systematically. In this paper, we leverage group theory to mathematically prove necessary and sufficient conditions for two fundamental questions of compositional representations. (1) What are the properties for a set of components to be expressed compositionally. (2) What are the properties for mappings between compositional and original representations. We provide examples to better understand the borders of the conditions. We also use the theory to provide a new explanation of why attention mechanism helps compositionality. We hope this work will help to advance understanding of compositionality and improvement of artificial intelligence towards human level.