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

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 σ. 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

• The paper introduces a deterministic embedding framework that harnesses prime square roots to construct non-repeating, quasi-orthogonal bases with a hierarchical spectral bias.
• It proposes two variants—StaticPrime for efficient sequence generation and DynamicPrime for tunable projection between isometric manifold learning and high-entropy hashing—validated via improved orthogonality and reduced RMS error.
• Empirical evaluations show that Primal closely approaches the Welch bound and enhances classification capacity, demonstrating potential for privacy-preserving machine learning and hyperdimensional computing.

Primal: Deterministic Quasi-Orthogonal Hashing and Manifold Learning

Introduction and Motivation

The paper "Primal: A Unified Deterministic Framework for Quasi-Orthogonal Hashing and Manifold Learning" (2511.20839) establishes a deterministic alternative to stochastic embedding methods, addressing the inherent limitations of Random Fourier Features (RFF) and conventional Gaussian projections in high-dimensional representation spaces. Central to Primal is the exploitation of the Besicovitch property—the linear independence over $\mathbb{Q}$ of prime square roots—to construct frequency bases that guarantee non-repeating phase trajectories. The framework introduces two algorithmic variants: StaticPrime, for sequence generation and quasi-orthogonal position encodings; and DynamicPrime, which provides a tunable projection capable of traversing the space between manifold-preserving linearization and maximum-entropy hashing.

Methodology and Theoretical Contributions

Primal constructs its embedding basis deterministically via ordered prime square roots. For a given input dimension $d$ and output dimension $D=2k$, the prime-based weight matrix $W \in \mathbb{R}^{k \times d}$ is indexed from a precomputed prime table. The resulting basis is inherently multi-scale, where lower frequencies capture global structure and higher frequencies encode fine-grained detail. This deterministic ordering differentiates Primal from random matrix baselines by introducing a hierarchical spectral bias.

DynamicPrime defines its feature transformation as $z = [\cos(\mathbf{v}), \sin(\mathbf{v})]$ with $\mathbf{v} = 2\pi\sigma(Wx)$. The model parameter $\sigma$ governs the operating regime: low $\sigma$ yields isometric embeddings facilitating exact manifold learning and reconstruction, while high $\sigma$ produces non-invertible, high-entropy projections for cryptographic applications and hyperdimensional computing.

Injectivity guarantees are strictly characterized: exact inversion via the Moore-Penrose pseudoinverse is only feasible when $D \geq 2d$ and no phase wrapping occurs, thus defining the boundaries of regime transition.

StaticPrime specializes for temporal or positional encoding, using only $k$ primes for lightweight deterministic sequence generation. This variant is computationally efficient relative to Transformer-style learnable position embeddings, requiring only $\mathcal{O}(D)$ parameters instead of dense random matrices.

Empirical Evaluation

Distributional Tightness and Orthogonality

StaticPrime exhibits sharp improvements over normalized random Gaussian codes in hyperspherical packing efficiency. The cosine similarity landscape of the generated codebook demonstrates a pronounced ridge at zero similarity, quantifying $\sim 0.5\%$ improvement in RMS reduction over Gaussian baselines, while the RMS error profile remains $\sim 1.8\%$ lower globally as dimensionality and sequence length scale.

Figure 1: StaticPrime yields superior orthogonality and distributional tightness compared to the Gaussian baseline, with a distinct log-density ridge at zero and lower RMS error surfaces.

Welch Bound Optimality

Aggregate analysis of optimality ratios reveals that StaticPrime narrowly approaches the Welch lower bound, outperforming Gaussian vectors that regularly exhibit coherence levels up to $5\times$ the theoretical minimum. The coherence distribution for StaticPrime displays a sharply concentrated peak nearer to the ideal, underscoring the spectral repulsion induced by prime modulation.

Figure 2: Distribution of optimality ratios and excess coherence highlights StaticPrime’s closeness to the Welch bound, relative to heavy-tailed Gaussian baselines.

DynamicPrime Scaling and Regime Transition

Manipulation of the scaling factor $\sigma$ demonstrates a smooth transition between manifold learning and chaotic hashing. In low-frequency settings, DynamicPrime preserves and linearizes complex topologies, enabling near-perfect reconstruction. As $\sigma$ increases or $D$ decreases, phase wrapping disrupts locality, yielding chaotic projections that maximize entropy but preclude inversion.

Figure 3: DynamicPrime transitions from topology-preserving isometric mapping (low $\sigma$) to high-entropy cryptographic hashing (high $\sigma$), with latent spaces evolving from linear manifolds to quasi-orthogonal Gaussian blobs.

The embedding geometry aligns with theoretical Clifford Torus ergodic flows—prime-based frequencies ensure dense, non-intersecting windings across the latent manifold, visible in PCA projections as winding mesh-like structures.

Classification Capacity Analysis

StaticPrime and DynamicPrime retain strong discriminative capacity even in severely compressed (<10D) regimes. At $D=4$, classification remains effective, though noise degrades intra-class similarity in deep linearized settings. At $D=128$, class separation is maximized, with linearized regimes exhibiting near-perfect abstraction and denoising, outperforming full reconstruction which preserves noise.

Figure 4: At output dimension 4, deep linearized regimes suffer under noise but maintain distinct geometric signatures.

Figure 5: At output dimension 128, linearized outputs yield superior class separation with high intra-class cosine similarities, abstracting away incidental noise.

Implications

Machine Learning and Hyperdimensional Computing

The deterministic quasi-orthogonal property allows for compressed, interference-minimized embeddings crucial for vector symbolic architectures, associative memory models, Hyperdimensional Computing, and neuro-symbolic reasoning. Primal approaches the Welch bound, increasing retrievability and memory capacity, enabling dense Key-Value storage and rapid lookup with minimal cross-talk noise.

For Implicit Neural Representations (e.g., SIRENs), the non-repeating, frequency-rich basis mitigates spectral bias, facilitating learning of high-frequency signal components, and potentially accelerating convergence in PINNs and scientific machine learning (PDE solvers, imaging).

Privacy, Edge AI, Telecom, and Biometric Security

The high-frequency hashing regime provides cryptographically secure transformations for Privacy-Preserving Machine Learning and cancelable biometrics. Deterministic nature supports reproducibility for Split Learning and edge AI, where randomness is a liability. In telecommunications, efficient codebook generation for CDMA or Massive mMTC is feasible without iterative optimization, adapting dynamically to user loads.

Continual and Sparse Representation Learning

Primal’s quasi-orthogonal embeddings minimize gradient and representational interference, supporting continual learning and online adaptation. Dimension can be grown seamlessly by prime extensions, with exact backward compatibility. Deterministic dictionary generation benefits sparse coding and compressive sensing—enabling RIP-satisfying, overcomplete dictionaries without learning overhead.

Limitations

Despite its deterministic merits, Primal’s monotonic prime scaling introduces large dynamic ranges. High $D$ may induce numerical aliasing and low-frequency vanishing, challenging floating-point arithmetic and feature utilization. Adaptive scaling or prime subsampling may be necessary in ultra-high-dimension deployments.

Future Directions

Potential lines of inquiry include rigorous analytic bounding of coherence, domain-specific adaptation (natural sciences, edge deployment), deployment optimization, and hardware realization leveraging the analog properties of irrational prime frequencies for optical computing and ultra-fast deterministic hashing.

Conclusion

Primal offers a mathematically rigorous, computationally efficient framework for high-dimensional feature mapping. It supplants stochastic embeddings with deterministic, spectral independence derived from prime modulation, approaching theoretical Welch bound limits in orthogonality and interference minimization. The unified DynamicPrime mechanism allows smooth transition between manifold learning and cryptographic hashing regimes, with broad implications for machine learning, privacy, and scientific computing. Primal stands as a foundational deterministic substitute and structural prior for high-dimensional embeddings, advancing the reliability and efficiency of neuro-symbolic and privacy-preserving systems.