Primal: A Unified Deterministic Framework for Quasi-Orthogonal Hashing and Manifold Learning

TL;DR

Primal introduces a deterministic, prime-based feature mapping framework that unifies manifold learning and hashing, offering improved orthogonality and distribution properties over traditional stochastic methods.

Contribution

It presents a novel deterministic approach leveraging number theory for feature mappings, unifying manifold learning and hashing within a single scalable framework.

Findings

Achieves near-optimal quasi-orthogonality compared to Gaussian baselines.

Effectively linearizes complex geometries like spirals.

Provides a versatile projection method for signal reconstruction and hashing.

Abstract

We present Primal, a deterministic feature mapping framework that harnesses the number-theoretic independence of prime square roots to construct robust, tunable vector representations. Diverging from standard stochastic projections (e.g., Random Fourier Features), our method exploits the Besicovitch property to create irrational frequency modulations that guarantee infinite non-repeating phase trajectories. We formalize two distinct algorithmic variants: (1) StaticPrime, a sequence generation method that produces temporal position encodings empirically approaching the theoretical Welch bound for quasi-orthogonality; and (2) DynamicPrime, a tunable projection layer for input-dependent feature mapping. A central novelty of the dynamic framework is its ability to unify two disparate mathematical utility classes through a single scaling parameter {\sigma}. In the low-frequency regime, the method acts as an isometric kernel map, effectively linearizing non-convex geometries (e.g., spirals) to enable high-fidelity signal reconstruction and compressive sensing. Conversely, the high-frequency regime induces chaotic phase wrapping, transforming the projection into a maximum-entropy one-way hash suitable for Hyperdimensional Computing and privacy-preserving Split Learning. Empirical evaluations demonstrate that our framework yields superior orthogonality retention and distribution tightness compared to normalized Gaussian baselines, establishing it as a computationally efficient, mathematically rigorous alternative to random matrix projections. The code is available at https://github.com/VladimerKhasia/primal