Simple Mechanisms for Representing, Indexing and Manipulating Concepts

TL;DR

The paper tackles the lack of a formal framework for defining and manipulating concepts learned by deep models. It introduces a mathematical construct where primitive concepts are zero sets of polynomials and their signatures are derived from the null space of the second-moment (and polynomial-feature) representations, enabling membership tests and hierarchical composition. A transformer-inspired architecture is proposed that separates concept discovery (via attention) from concept storage (a dictionary of signatures) and can build higher-level concepts through structured operations like intersections and unions. The approach is validated on synthetic datasets, showing that concept signatures cluster inputs by concept, support generalization to unseen lower-level concepts, and scale to more complex hierarchical structures. Overall, the work provides a principled route to represent, index, and manipulate abstract concepts within neural architectures, with potential implications for interpretability and retrieval-based reasoning.

Abstract

Supervised and unsupervised learning using deep neural networks typically aims to exploit the underlying structure in the training data; this structure is often explained using a latent generative process that produces the data, and the generative process is often hierarchical, involving latent concepts. Despite the significant work on understanding the learning of the latent structure and underlying concepts using theory and experiments, a framework that mathematically captures the definition of a concept and provides ways to operate on concepts is missing. In this work, we propose to characterize a simple primitive concept by the zero set of a collection of polynomials and use moment statistics of the data to uniquely represent the concepts; we show how this view can be used to obtain a signature of the concept. These signatures can be used to discover a common structure across the set of concepts and could recursively produce the signature of higher-level concepts from the signatures of lower-level concepts. To utilize such desired properties, we propose a method by keeping a dictionary of concepts and show that the proposed method can learn different types of hierarchical structures of the data.

Figures (4)

• Figure 1: Overview of our learning architecture.
• Figure 2: Synthetic dataset of type $n$-circles+$n$-parabola and experimental findings on it.
• Figure 3: Examples and properties of the signatures.
• Figure 4: Connection of our learning architecture to the original transformers. A combination of feedforward network and attention module will generate $M(X)$ and a subsequent feedforward layer can generate the nullspace signature $T(X)$ by taking higher power of $I-M(X)$. See \ref{['thm:transformer-connection']} for more details.

Theorems & Definitions (52)

• Definition 2.1: Linear $k$-dimensional manifold
• Definition 2.2: Null space signature for a linear manifold
• Proposition 2.1
• Definition 2.3: Well-behaved Algebraic Manifold
• Definition 2.4
• Proposition 2.2
• Definition 2.5: Generative representation of $k$-dimensional manifold
• Theorem 2.1: Polynomial generative representation to algebraic manifold
• Theorem 2.2
• Theorem 2.3