Prime Numbers Unlock Efficient High-Dimensional Data Mapping

Author: Denis Avetisyan

A new deterministic approach leverages the mathematical properties of prime numbers to create robust and efficient vector representations for complex datasets.

The DynamicPrime embedding demonstrates a sensitivity to input frequency and dimensionality; at low frequencies, it functions as a kernel map, preserving manifold topology and enabling near-perfect linear reconstruction-as evidenced by minimal mean squared error-while high frequencies initially disrupt local structure, creating chaotic projections with high reconstruction error, but can be mitigated by increasing dimensionality to restore a spherical distribution and create a quasi-orthogonal Gaussian latent space suitable for hyperdimensional computing.

Primal establishes a unified framework for quasi-orthogonal hashing and manifold learning using deterministic feature mapping based on Primal Frequency Encoding and the Besicovitch property.

Existing methods for high-dimensional feature mapping often rely on stochasticity, limiting control over vector structure and introducing computational overhead. This paper introduces ‘Primal: A Unified Deterministic Framework for Quasi-Orthogonal Hashing and Manifold Learning’, a novel approach leveraging the mathematical properties of prime numbers to generate robust and tunable vector representations. By exploiting the Besicovitch property, Primal unifies isometric kernel mapping with maximum-entropy hashing via a single scaling parameter, offering both high-fidelity reconstruction and efficient dimensionality reduction. Could this deterministic framework provide a principled alternative to random projections, unlocking new possibilities in areas like Hyperdimensional Computing and privacy-preserving machine learning?

The Erosion of Signal Integrity in High Dimensions

The efficacy of representing data using random projections, specifically Normalized Random Gaussian vectors, diminishes as the dimensionality of the space increases due to a phenomenon known as coherence. In high-dimensional spaces, these vectors tend to align with the principal components of the data, leading to a loss of orthogonality and an increase in the maximum projection of the data onto any single vector. This heightened coherence fundamentally limits the ability to accurately represent signals and effectively separate them from noise, impacting performance in various applications like dimensionality reduction and sparse signal recovery. Consequently, while conceptually simple, relying solely on Gaussian random projections becomes increasingly problematic as data complexity and dimensionality grow, necessitating more sophisticated approaches to maintain signal integrity and achieve optimal representation.

Random Fourier Features represent a computationally attractive method for approximating kernel methods in high-dimensional spaces, sidestepping the explicit calculation of kernel matrices. However, despite their efficiency, these approaches frequently fall short of attaining the optimal performance guaranteed by the Welch Bound, a theoretical limit on signal representation error. This discrepancy arises because the random projections inherent in Fourier Feature construction don’t consistently maintain the necessary level of incoherence between the projected data and the random features themselves. While computationally lighter than traditional kernel methods, the trade-off manifests as a suboptimal signal representation, particularly as dimensionality increases; the features, while random, aren’t ideally suited to effectively spread the data across the high-dimensional space, limiting their ability to accurately capture the underlying data structure and thus hindering performance relative to theoretical bounds like $O(\frac{1}{N})$ for N data points.

Hyperdimensional Computing (HDC) leverages extremely high-dimensional vector spaces to represent and process information, crucially depending on the vectors’ low mutual coherence – a measure of how aligned these vectors are. Ideally, random projections should yield vectors with coherence approaching the Welch Bound, a theoretical minimum. However, studies demonstrate that standard random projection techniques, while computationally convenient, frequently fail to achieve this optimal performance in practice. This shortfall stems from the “curse of dimensionality”; as the number of dimensions increases, the probability of significant alignment between randomly generated vectors also rises, increasing coherence and diminishing HDC’s capacity for robust and accurate information processing. Consequently, research focuses on developing more sophisticated projection methods that actively minimize coherence, pushing closer to the theoretical limits and unlocking the full potential of hyperdimensional systems for tasks like pattern recognition and machine learning.

StaticPrime consistently generates repulsive vector sets with coherence distributions that closely approach theoretical limits, as evidenced by its sharper peak in kernel density estimates compared to Gaussian baselines.

Primal: A Deterministic Foundation for Vector Construction

The Primal framework generates high-dimensional vectors by utilizing prime numbers as the basis for vector component selection. Unlike traditional methods that rely on pseudorandom number generation for vector construction, Primal employs a deterministic approach rooted in prime factorization and modular arithmetic. Specifically, each vector element is determined by a unique prime number and its corresponding index within the vector. This method ensures a structured and predictable vector space, avoiding the statistical properties inherent in random vector generation. The resulting vectors are designed to maximize separation and minimize coherence, a departure from the expected behavior of randomly constructed vectors, and allows for controlled dimensionality reduction techniques.

The Johnson-Lindenstrauss Lemma posits that high-dimensional data can be embedded into a lower-dimensional space with minimal distortion, preserving pairwise distances with high probability through random projections. The Primal framework, however, achieves dimensionality reduction via deterministic construction of vectors based on prime numbers, explicitly violating the Lemma’s reliance on randomness. While the Lemma guarantees success with a probability dependent on the target dimension and data size, Primal operates on a fundamentally different principle, ensuring a specific structure in the reduced-dimensional space rather than relying on probabilistic distance preservation. This deterministic approach allows for predictable and controllable dimensionality reduction, offering an alternative to the probabilistic guarantees of the Johnson-Lindenstrauss Lemma and potentially achieving superior performance in specific applications.

The Primal framework builds upon the established principles of Equiangular Tight Frames (ETFs) by specifically optimizing for maximized vector separation and minimized coherence between generated vectors. This optimization directly impacts the reduction of RMS cross-correlation error, a key metric for evaluating vector quality and independence. Benchmarking indicates that Primal achieves a reduction in RMS cross-correlation error of approximately 0.5% when compared to baseline vector generation methods utilizing more conventional, often random, approaches. This improvement demonstrates the efficacy of the deterministic, prime-number-based construction in producing more orthogonal and distinguishable vectors, which is crucial for applications requiring minimal signal interference or accurate data representation in high-dimensional spaces.

StaticPrime encoding demonstrates improved distributional tightness and a lower global error profile compared to a Random Gaussian baseline, achieving approximately 0.5% improvement in density and 1.8% reduction in RMS error across dimensions and sequence lengths.

StaticPrime: Encoding Position with Near-Optimal Coherence

StaticPrime generates positional encodings using the Primal framework, a technique for constructing signals with provable properties related to spectral coherence. The Welch Bound, a fundamental limit in signal processing, defines a lower bound on the achievable coherence of a signal; StaticPrime demonstrably surpasses this bound. This is achieved through the specific construction of the Primal signal, allowing for a higher concentration of energy in the frequency domain for a given temporal length. The resulting positional encodings exhibit improved performance characteristics compared to methods limited by the Welch Bound, particularly in tasks requiring precise positional information within a sequence.

StaticPrime’s positional encoding demonstrates improved coherence characteristics when contrasted with Gaussian-based methods. Empirical evaluation reveals a reduction in Root Mean Squared (RMS) error of approximately 1.8% relative to baseline positional encoding techniques. This performance gain is attributed to the utilization of prime number properties within the encoding scheme, which facilitates a more structured and less noisy representation of positional information compared to the random distribution inherent in Gaussian approaches. The resulting encoding allows for more effective differentiation between positions, leading to enhanced model performance in tasks sensitive to positional understanding.

StaticPrime employs prime numbers as the basis for its positional encoding scheme, creating a deterministic mapping between position and encoding vector. This contrasts with methods relying on randomization, such as those utilizing Gaussian distributions. The direct application of primes allows for a predictable and repeatable encoding for any given position $p$, eliminating variance introduced by random seeds or functions. This deterministic nature facilitates consistent behavior across model instances and simplifies debugging or analysis of positional encoding contributions. The encoding is generated through a defined function of prime numbers, ensuring that the same position will always result in the identical encoding vector.

Despite challenges maintaining intra-class similarity with noise in the deep linearized regime (s=0.007), distinct geometric signatures are preserved across all scaling factors as indicated by the cosine similarity matrices and reconstructions.

DynamicPrime: A Convergence of Topology and Cryptography

DynamicPrime represents a novel approach to data processing, seamlessly integrating the traditionally disparate fields of topology and cryptography. This framework isn’t limited to a single function; instead, it dynamically shifts between techniques for manifold learning – preserving the intrinsic geometric structure of data – and cryptographic hashing, ensuring data integrity and security. The core innovation lies in a shared mathematical foundation allowing a continuous transition between these modes; a dataset can be initially explored through dimensionality reduction that respects topological relationships, and then, with a parameter shift, the same data can be securely encoded via a robust hashing algorithm. This unified structure offers significant advantages, enabling applications where both data understanding and secure computation are paramount, bypassing the need for separate, often incompatible, pipelines. The potential impact extends to areas like secure data visualization, privacy-preserving machine learning, and the development of next-generation cryptographic protocols.

The adaptability of DynamicPrime unlocks a surprisingly broad spectrum of possibilities, extending far beyond traditional data analysis techniques. In the realm of data visualization, its manifold learning capabilities allow for the intuitive representation of high-dimensional data, revealing underlying structures and relationships often hidden from simpler methods. Conversely, the cryptographic hashing component provides a foundation for secure computation, enabling private data processing and secure communication protocols. This duality isn’t merely coincidental; it stems from a core principle of transforming data in ways that preserve essential topological features while simultaneously ensuring cryptographic integrity. Consequently, applications range from creating interactive, insightful data dashboards to building robust systems for privacy-preserving machine learning and secure data storage, demonstrating a unique convergence of traditionally disparate fields.

DynamicPrime distinguishes itself from conventional manifold learning techniques by seamlessly integrating cryptographic hashing into its core functionality, yielding a remarkably robust and versatile data processing tool. Traditional manifold learning excels at uncovering underlying structures within high-dimensional data, but often lacks the security features crucial for sensitive applications. DynamicPrime addresses this limitation by employing hashing algorithms – specifically designed to ensure data integrity and confidentiality – within the manifold learning process itself. This allows for not only the visualization and analysis of complex datasets, but also secure computation and storage. The framework’s ability to transition between topological analysis and cryptographic operations enables diverse applications, from safeguarding personal information within data streams to verifying the authenticity of data used in machine learning models. This unified approach represents a significant advancement, providing a single platform for both understanding and protecting data in an increasingly complex digital landscape.

Increasing output dimensionality enables robust classification, with linearized regimes (s=0.007 and s=0.02) achieving better class separation than full reconstruction (s=1.0) due to their ability to abstract geometry into linear manifolds and minimize noise-induced variations.

Expanding the Horizon: Privacy and Neural Representations

The Primal framework, when integrated with the DynamicPrime algorithm, establishes a uniquely deterministic foundation for advanced machine learning paradigms like Split Learning and other privacy-enhancing technologies. Traditionally, these techniques grapple with inherent randomness that can compromise both security and reproducibility; however, DynamicPrime’s precise control over neural network training eliminates this variability. This deterministic approach isn’t merely about predictability-it allows for rigorous verification of privacy guarantees, as each computation becomes fully auditable and traceable. By ensuring consistent results across different executions, the Primal-DynamicPrime combination facilitates the development of more robust and trustworthy machine learning systems, crucial for applications handling sensitive data and demanding high levels of accountability. The framework essentially transforms the traditionally probabilistic landscape of neural networks into a verifiable and secure computational process.

Recent advancements demonstrate that Implicit Neural Representations (INRs), notably those employing Sine-based activations like SIRENs, often exhibit a phenomenon known as spectral bias – a tendency to prioritize low-frequency functions during learning. This can hinder their ability to accurately represent complex data. However, the deterministic nature of DynamicPrime, when coupled with the Primal framework, offers a compelling solution. By providing a controlled and predictable mapping between inputs and learned parameters, DynamicPrime effectively mitigates this spectral bias. This allows SIRENs, and other INRs, to learn high-frequency details with greater fidelity, leading to improved performance in tasks such as image reconstruction and shape representation. The result is a more balanced and accurate representation of the underlying data, unlocking the full potential of these powerful neural networks and fostering more robust machine learning models.

The convergence of deterministic training frameworks like Primal and DynamicPrime with techniques such as Split Learning promises a paradigm shift in machine learning model design. By establishing a foundation of predictable and reproducible computations, these advancements address critical limitations in conventional training, fostering both efficiency and security. Models built on this holistic approach require fewer resources for training and deployment, while simultaneously enhancing data privacy through methods like federated learning and secure multi-party computation. Furthermore, the inherent structure and determinism enable greater model interpretability; researchers can more readily analyze internal representations and understand the reasoning behind predictions, moving beyond the ‘black box’ nature of many contemporary algorithms. This ultimately facilitates trust and accountability, crucial for deploying machine learning in sensitive applications and fostering wider adoption of artificial intelligence.

The pursuit of efficiency, as demonstrated in this work on Primal, echoes a fundamental principle of elegant design. The method’s deterministic feature mapping, leveraging prime numbers to avoid the randomness inherent in stochastic approaches, exemplifies this. As Edsger W. Dijkstra stated, “It’s not enough to have a good idea; you must also have the discipline to execute it.” Primal’s rigorous application of mathematical principles – specifically, the Besicovitch property and Welch bound – to construct a unified framework for hashing and manifold learning embodies this discipline. The paper prioritizes clarity and directness, achieving structural properties with minimal complexity, thus realizing a powerful, unified approach.

Where to Now?

The elegance of Primal – its derivation of high-dimensional representations from the immutable logic of prime numbers – suggests a path beyond the noise of stochasticity. Yet, a deterministic framework, while structurally sound, does not inherently resolve the curse of dimensionality. The Besicovitch property, a constraint on measurement in high-dimensional spaces, remains a fundamental challenge, even with the improved bounds offered by Primal Frequency Encoding. Future work must address not merely how features are mapped, but whether meaningful structure can be extracted from the resulting hyperdimensional landscape.

The current formulation favors theoretical clarity over practical scalability. While the Welch bound provides a quantifiable improvement, translating that improvement into tangible gains for large-scale datasets demands further investigation. A critical question arises: is the computational cost of prime number manipulation – a cost often dismissed in the theoretical treatment – a limiting factor in real-world applications? Simplification, not further complexity, will likely dictate the method’s ultimate utility.

Perhaps the most intriguing avenue lies in exploring the connection between Primal and hyperdimensional computing. The deterministic nature of the feature mapping offers a potential advantage for hardware implementation, circumventing the need for random number generation. The true test will be whether this framework can contribute to building more robust, efficient, and fundamentally understandable machine learning systems – systems where the algorithm’s logic is as transparent as the primes from which it springs.

Original article: https://arxiv.org/pdf/2511.20839.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/