Derandomizing Dense Binary Hypervector Codebooks for Quantized Scalars

2026-09-29

AI Summary

This study addresses the deviation of scalar similarities from prescribed patterns in finite-dimensional stochastic binary codebooks caused by sampling noise. To mitigate this, we propose a transition-based derandomization framework that innovatively decouples derandomization constraints by separating target patterns, variants, and generators. By precisely constraining initial Hamming weights, update count fluctuations, and balance properties, the method controls initialization and update mechanisms to match exponential or linear decay profiles. We derive exact analytical expressions for finite-dimensional errors and, through simulations grounded in hyperdimensional computing theory, demonstrate that each constraint effectively eliminates specific similarity mismatches. This work provides practical guidance for codebook generation in hardware-constrained scenarios.

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.

Problem

• Hyperdimensional computing
• Dense binary codebooks
• Quantized scalars
• Derandomization
• Similarity mismatch

Innovation

• Hyperdimensional Computing
• Dense Binary Codebooks
• Derandomization Framework
• Finite-Dimensional Analysis
• Target Similarity Matching