用质数平方根构建可调向量表示，实现高正交性与隐私保护。
Primal: A Unified Deterministic Framework for Quasi-Orthogonal Hashing and Manifold Learning
- 基于质数平方根生成确定性特征映射，避免随机投影。
- 低频时线性化非凸结构，高频时产生最大熵哈希，性能优于高斯基线。
- 适合需要正交性或隐私保护的机器学习任务，如压缩感知与分布式学习。
我们提出 Primal，一种利用质数平方根数论独立性的确定性特征映射框架，构建鲁棒且可调的向量表示。不同于标准随机投影（如随机傅里叶特征），该方法利用 Besicovitch 性质生成无理频率调制，保证无限不重复相位轨迹。我们定义两种算法变体：(1) StaticPrime，生成时间位置编码，实验逼近理论 Welch 界的准正交性；(2) DynamicPrime，输入依赖的可调投影层。动态框架的核心创新在于通过单一缩放参数 σ 统一两类数学功能：低频下作为保距核映射，有效线性化非凸几何（如螺旋形），实现高保真信号重建与压缩感知；高频下引发混沌相位缠绕，使投影变为最大熵单向哈希，适用于超维度计算与隐私保护的分裂学习。实证表明，该框架在正交性保持与分布紧致性上优于归一化高斯基线，是计算高效、数学严谨的随机矩阵投影替代方案。代码已开源：https://github.com/VladimerKhasia/primal。
原文摘要 · Abstract (English)
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
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