Laplace-HDC: Understanding the geometry of binary hyperdimensional computing (2404.10759v2)

Abstract: This paper studies the geometry of binary hyperdimensional computing (HDC), a computational scheme in which data are encoded using high-dimensional binary vectors. We establish a result about the similarity structure induced by the HDC binding operator and show that the Laplace kernel naturally arises in this setting, motivating our new encoding method Laplace-HDC, which improves upon previous methods. We describe how our results indicate limitations of binary HDC in encoding spatial information from images and discuss potential solutions, including using Haar convolutional features and the definition of a translation-equivariant HDC encoding. Several numerical experiments highlighting the improved accuracy of Laplace-HDC in contrast to alternative methods are presented. We also numerically study other aspects of the proposed framework such as robustness and the underlying translation-equivariant encoding.

• The paper introduces Laplace-HDC, a new binary HDC method that embeds the Laplace kernel to overcome spatial encoding deficiencies in image data.
• It proposes translation-equivariant mechanisms and Haar convolutional features to maintain spatial integrity in hyperdimensional representations.
• Rigorous numerical experiments validate that the enhanced Laplace-HDC method outperforms conventional binary HDC in both robustness and accuracy.

Exploring Hyperdimensional Computing through Laplace-HDC: A New Binary Encoding Paradigm

Introduction

In the field of hyperdimensional computing (HDC), which mimics the brain's high-dimensional operational framework, binary HDC has been a commonly adopted model due to its computational simplicity and alignment with hardware implementations. This paper extends the understanding of binary HDC by introducing an encoding method named Laplace-HDC, which leverages principles grounded in the Laplace kernel within the construction of hypervectors. The paper's central thesis revolves around addressing the shortcomings in capturing spatial data inherent in typical binary HDC frameworks, proposing robust solutions that also encapsulate translation-equivariant encoding.

HDC Encoding Geometry and Spatial Encoding Challenges

Analysis of Binary HDC Binding Operation

The paper begins by dissecting the geometry influenced by the HDC binding operation, revealing innate deficiencies when employed in image data encoding. Traditional binary HDC efficiently handles generic data types but falters in maintaining the spatial integrity necessary for tasks like image processing.

Laplace Kernel in HDC

A notable contribution of this paper is establishing the natural emergence of the Laplace kernel within the binary HDC context. By structuring hypervector construction to adhere to this kernel, there is improved maintenance in the geometric relationships amongst data points, significantly enhancing encoding efficacy. Critically, the authors meticulously articulate the mathematical framework supporting this approach, providing a robust theoretical foundation that guides practical implementation.

Spatial Information Retention Strategies

Confronting the identified challenges head-on, the paper proposes innovative strategies to incorporate spatial information directly into the HDC schema. These include employing Haar convolutional features and developing translation-equivariant HDC mechanisms. Importantly, the introduction of translation-equivariance in encoding offers a novel perspective and approach, respecting data locality and continuity which are crucial for image and video processing tasks.

Empirical Validation and Theoretical Analysis

Numerical Experiments

The Laplace-HDC model's effectiveness is corroborated through rigorous numerical experiments, showing superior performance compared to conventional methods in both robustness and accuracy. Spatial and temporal data sets, particularly image data, see marked improvements with the application of the proposed modifications.

Theoretical Implications

From a theoretical perspective, the introduction and validation of the Laplace kernel in binary HDC settings enrich the computational framework, allowing for a deeper understanding of the operational dynamics and limitations of traditional models. By addressing the root causes of spatial information loss, the paper sets a precedent for future research to explore more sophisticated representations and operations within the high-dimensional computing arena.

Future Directions and Speculations

Expanding Encoding Techniques

The exploration into translation-equivariant encoding opens potential pathways for more advanced manipulations and transformations within HDC. This could lead to developments in invariant feature recognition across various transformations, beneficial for complex recognition tasks in computer vision and beyond.

Integration with Neural Network Features

Given the success of neural networks in handling raw image data, integrating learned features from these models into the HDC framework could further enhance performance. This integration could leverage the strengths of both paradigms, potentially leading to breakthroughs in efficiency and scalability for machine learning models.

Conclusion

The exploration into the Laplace-HDC presents a significant shift in tackling the inherent limitations of binary hyperdimensional computing, especially regarding spatial data representation. By grounding the enhancements in rigorous theoretical foundations and demonstrating practical efficacy through empirical studies, this paper contributes profoundly to both the theory and application of HDC. Future explorations following this research are poised to unlock even more robust and versatile computational models that closer mimic the human brain's capabilities.