Recursive Binding on a Budget: Subspace Carving in Order-p Tensor Memories

2026-06-09

AI Summary

This work addresses the dimensionality explosion in Tensor Product Representations (TPRs) during deep recursive symbolic reasoning and the compromised capacity and structural fidelity of existing vector-symbolic architectures that rely on superposition within fixed-dimensional spaces. To overcome these limitations, the paper introduces Orthogonal Subspace Carving (OSC), a mechanism that projects filler vectors into the null space of role bases and aggregates them into a fixed-order tensor, thereby decoupling recursive binding from both tensor order and structural depth. OSC enables deep recursion under constant memory overhead, achieving substantially higher memory efficiency in high-superposition regimes—where component vector dimensions are orders of magnitude smaller than the memory tensor—while preserving structural fidelity and scalability. Furthermore, the study reveals TPR as a special case of binding in Clifford algebra and provides a Clifford-algebraic formulation of OSC.

Abstract

Tensor Product Representations provide the structural fidelity required for symbolic reasoning in models but suffer from exponential dimensionality growth when encoding deep recursive structures. Conversely, Vector Symbolic Architectures maintain constant dimensionality but sacrifice capacity and fidelity due to noisy compression via superposition. In this work, we propose Orthogonal Subspace Carving (OSC), a memory architecture that binds fillers to roles by projecting onto the null space of the role basis before aggregating into a fixed order-p tensor. OSC uses projections to enforce geometric orthogonality between bound structures within a static memory trace. We show that this mechanism decouples the tensor order from the structural depth, enabling deep recursive binding within a constant memory footprint. By performing retrieval via recognition, this construction allows for component vectors that are orders of magnitude smaller than the memory tensor, giving superior memory efficiency in settings involving high superposition. We also show that TPR is a special case of binding in Clifford algebra, and give a Clifford formulation of OSC.

Problem

• Tensor Product Representations
• Vector Symbolic Architectures
• recursive binding
• memory efficiency
• structural depth

Innovation

• Orthogonal Subspace Carving
• Tensor Product Representations
• Vector Symbolic Architectures
• Clifford algebra
• recursive binding