We propose a novel randomized algorithm for constructing binary neural networks with tunable accuracy. This approach is motivated by hyperdimensional computing (HDC), which is a brain-inspired paradigm that leverages high-dimensional vector representations, offering efficient hardware implementation and robustness to model corruptions. Unlike traditional low-precision methods that use quantization, we consider binary embeddings of data as points in the hypercube equipped with the Hamming distance. We propose a novel family of floating-point neural networks, G-Nets, which are general enough to mimic standard network layers. Each floating-point G-Net has a randomized binary embedding, an embedded hyperdimensional (EHD) G-Net, that retains the accuracy of its floating-point counterparts, with theoretical guarantees, due to the concentration of measure. Empirically, our binary models match convolutional neural network accuracies and outperform prior HDC models by large margins, for example, we achieve almost 30\% higher accuracy on CIFAR-10 compared to prior HDC models. G-Nets are a theoretically justified bridge between neural networks and randomized binary neural networks, opening a new direction for constructing robust binary/quantized deep learning models. Our implementation is available at \url{https://github.com/GNet2025/GNet}.

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.

We introduce a framework for recovering an image from its rotationally and translationally invariant features based on autocorrelation analysis. This work is an instance of the multi-target detection statistical model, which is mainly used to study the mathematical and computational properties of single-particle reconstruction using cryo-electron microscopy (cryo-EM) at low signal-to-noise ratios. We demonstrate with synthetic numerical experiments that an image can be reconstructed from rotational and translational invariants and show that the reconstruction is robust to noise. These results constitute an important step towards the goal of structure determination of small biomolecules using cryo-EM.