Laplace-HDC: Improving Binary Hyperdimensional Computing through Geometric Insights

Core Concepts

The core message of this paper is that the Laplace kernel naturally arises in the similarity structure of binary hyperdimensional computing (HDC) encodings, motivating a new encoding method called Laplace-HDC that improves upon previous HDC approaches. The authors also discuss limitations of binary HDC in encoding spatial information and propose solutions, including using Haar convolutional features and defining a translation-equivariant HDC encoding.

Abstract

The paper studies the geometry of binary hyperdimensional computing (HDC), a computational scheme that encodes data using high-dimensional binary vectors. The authors 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 their new encoding method Laplace-HDC.
The key highlights and insights are:

Laplace-HDC: The authors define a family of admissible kernels Kα and show that when α = 1, the resulting expected similarity kernel is approximately the Laplace kernel. This motivates their Laplace-HDC encoding method, which outperforms previous HDC approaches.

Limitations of binary HDC: The authors discuss how the similarity structure induced by binary HDC encodings is invariant to global permutations of the data elements, which can lead to a loss of spatial information for data like images.

Spatial encoding: To address the limitation above, the authors propose using Haar convolutional features and defining a translation-equivariant HDC encoding scheme that better preserves spatial information.

Robustness: The authors demonstrate the robustness of Laplace-HDC to noise, showing that the classification accuracy remains above 80% even with up to 25% bit error rate in the encoded vectors.

Experiments: Numerical experiments are presented that highlight the improved accuracy of Laplace-HDC compared to alternative HDC methods on MNIST and FashionMNIST datasets. The authors also study the translation-equivariance property of their proposed encoding scheme.

Stats

The average accuracy of standard HDC models on the MNIST handwritten digit classification task is 82%.
Several variants of deep neural networks can achieve accuracies above 99.5% on MNIST.

Quotes

"HDC aims to mimic the brain's operation by encoding data with high-dimensional vectors, called hypervectors, while using simple operations, such as the XOR operation."
"A significant challenge associated with HDC models is their relatively low accuracy."

Key Insights Distilled From

Laplace-HDC: Understanding the geometry of binary hyperdimensional computing

by Saeid Pourma... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10759.pdf

Deeper Inquiries

How can the translation-equivariant HDC encoding be further extended to capture more complex spatial relationships in the data, such as rotation and scale invariance

To extend the translation-equivariant HDC encoding to capture more complex spatial relationships in the data, such as rotation and scale invariance, additional transformations and features can be incorporated into the encoding process. One approach is to introduce rotational equivariance by utilizing rotation matrices in the encoding scheme. By applying rotation matrices to the input data before encoding, the HDC model can learn features that are invariant to rotations of the input images. This can be particularly useful in tasks where the orientation of objects in images is important, such as object recognition in computer vision.
Furthermore, to achieve scale invariance, the HDC encoding can be augmented with scale transformation operations. By resizing the input images to different scales before encoding, the model can learn features that are robust to changes in the size of objects in the images. This can be beneficial in applications where the size of objects varies significantly, such as in medical imaging or satellite image analysis.
By combining rotational and scale transformations with the translation-equivariant HDC encoding, the model can learn a richer set of features that capture complex spatial relationships in the data. This enhanced encoding scheme can improve the model's performance on tasks that require robustness to various spatial transformations.

What other kernel functions, beyond the Laplace kernel, could be used to construct admissible HDC encodings, and how would they affect the performance and properties of the resulting models

Beyond the Laplace kernel, several other kernel functions can be used to construct admissible HDC encodings, each with its own impact on the performance and properties of the resulting models. Some alternative kernel functions that could be explored include:

Gaussian Kernel: The Gaussian kernel is a popular choice in kernel methods due to its smoothness and flexibility in capturing complex relationships in the data. By using the Gaussian kernel in HDC encodings, the model can learn non-linear patterns and dependencies in the data, potentially improving accuracy on tasks with intricate spatial structures.

Polynomial Kernel: The polynomial kernel introduces non-linearity by computing the inner product raised to a certain power. By incorporating the polynomial kernel into HDC encodings, the model can capture higher-order interactions between features, allowing for more complex decision boundaries and improved performance on tasks with non-linear relationships.

Sigmoid Kernel: The sigmoid kernel is another non-linear kernel function that can be used in HDC encodings. By leveraging the sigmoid function, the model can learn non-linear transformations of the input data, enabling it to capture more intricate patterns and dependencies in the data.

Each of these kernel functions brings unique characteristics to the HDC encoding process, influencing the model's ability to capture different types of spatial relationships and patterns in the data. Experimenting with various kernel functions can help optimize the HDC model for specific tasks and datasets.

Given the simplicity and robustness of binary HDC, how could these techniques be leveraged in specialized hardware or edge computing applications to enable efficient and reliable machine learning models

The simplicity and robustness of binary HDC techniques make them well-suited for specialized hardware or edge computing applications where efficiency and reliability are paramount. Here are some ways these techniques could be leveraged in such scenarios:

Hardware Acceleration: Binary HDC models can be optimized for hardware acceleration using specialized processors like GPUs or FPGAs. By designing custom hardware architectures tailored to the binary operations used in HDC, significant speedups can be achieved, making real-time inference feasible for edge devices.

Low-Power Computing: The energy-efficient nature of binary HDC makes it ideal for low-power computing environments such as IoT devices. By minimizing the computational complexity and memory requirements of the models, binary HDC can enable efficient machine learning inference on resource-constrained devices.

Edge Computing: Deploying binary HDC models at the edge allows for data processing and inference to occur locally on the device, reducing latency and dependence on cloud services. This is particularly beneficial for applications where real-time decision-making is critical, such as autonomous vehicles or industrial IoT systems.

Robustness to Noise: The robustness of binary HDC models to noise makes them suitable for edge computing environments where data quality may be compromised. By maintaining accuracy in the presence of noisy data, binary HDC models can provide reliable insights and predictions in challenging conditions.

By leveraging the simplicity, efficiency, and robustness of binary HDC techniques, specialized hardware and edge computing applications can benefit from fast, reliable, and energy-efficient machine learning solutions.