Year: 2026
Type: preprint
Abstract: Hyperdimensional computing and vector symbolic architectures often represent quantized scalar levels by dense binary codebooks whose level-to-level similarity is intended to follow a prescribed function of scalar separation. At finite dimensionality, randomized scalar codebook constructions deviate from this target because of sampling noise, random-start imbalance, update-count fluctuations, component dependence, and finite-capacity effects. We develop a transition-based derandomization framework for dense binary scalar codebooks across two target-similarity families, with similarity decaying exponentially or linearly with level separation. The framework separates the target similarity law, the derandomization variant, and the concrete generator construction, making explicit how initialization, selection, update, and capacity-handling mechanisms shape the induced similarity profile. We formalize derandomization variants that separately constrain initial Hamming weight, update-count variability, and update balance, thereby controlling distinct sources of finite-dimensional error. For each family and variant, we derive the induced mean similarity, identify realization-wise and mean target-matching regimes, and derive exact finite-dimensional expressions for bias, variance, and root-mean-square error. Simulations across dimensions, quantization ranges, reference scalar levels, and generator constructions validate the theory and show how each constraint removes or reduces a specific source of similarity mismatch. The results provide practical guidance for choosing scalar codebook generators that more closely match a desired similarity law under finite-dimensional and hardware-relevant constraints.
Source: arXiv (Cornell University)
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Vector-symbolic architectures as an algebraic programming language for neuromorphic computers, G3676281905, G4335719035, Harnessing Structure, Recurrence, and Randomness for Efficient Neural Networks, G539454170, Visuell symbolförankring med distribuerade representationer, Beräkning i Superposition, tiny AI 2: Förbättra AI med hyperdimensionella beräkningar, Center for Artificial Intelligence and Quantum Computing in System Brain Research, G8052140988, G8055062168, NAISS
OpenAlex ID: W7215074067