Derandomizing binary codebooks improves scalar level similarity matching
Derandomizing Dense Binary Hypervector Codebooks for Quantized Scalars
Information TheoryArtificial IntelligenceNeural and Evolutionary Computing
Summary
When computers use very long strings of bits to represent numbers, they want similar numbers to have similar strings. But randomness in how these strings are made can cause errors, especially when the strings aren't infinitely long. The authors created a way to carefully choose these bit strings to reduce those errors and better match how similar numbers should be represented. They showed mathematically and through simulations how their method controls different types of errors, helping machines represent numbers more accurately with limited bit lengths.
What this means in practice
- •For hardware engineers: Design binary scalar representations that minimize similarity errors under hardware constraints for efficient computing devices.
- •For embedded system developers: Build more precise quantized scalar coding schemes to improve resource-limited machine learning and signal processing systems.
Authors
Dmitri Rachkovskij, Evgeny Osipov, Olexander Volkov, Denis Kleyko, Vaclav Snasel
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.