提出一种低维内存下实现深度递归绑定的新方法
Recursive Binding on a Budget: Subspace Carving in Order-p Tensor Memories
- 通过投影到角色基的零空间实现结构正交性
- 在固定张量阶数下支持深层递归绑定，内存不变
- 适合高叠加场景，组件向量比内存张量小多个数量级
张量积表示（TPR）能保持符号推理的结构保真度，但在编码深层递归结构时面临指数级维度增长。向量符号架构（VSA）虽维持常数维度，但因超叠加压缩导致噪声和容量损失。本文提出正交子空间切割（OSC），通过将填充物投影至角色基的零空间，再聚合为固定阶p张量来绑定填充物与角色。该机制在静态记忆轨迹中强制绑定结构间的几何正交性，使张量阶与结构深度解耦，从而在恒定内存开销下实现深层递归绑定。通过识别式检索，组件向量可比内存张量小多个数量级，在高超叠加场景中表现卓越。我们还证明TPR是克利福德代数中的特殊绑定形式，并给出了OSC的克利福德表述。
原文摘要 · Abstract (English)
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.
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