Vector-Space Projection and Unified Complex-Valued Acceleration for Scaling Matrix-Based Brain-Inspired Representations

Abstract

Symbolic reasoning at scale requires representations that are both expressive and hardware-efficient. Hyperdimensional Computing (HDC) provides a promising brain-inspired framework, encoding concepts as high-dimensional vectors composed through efficient algebraic operations. However, the widely used Fourier Holographic Reduced Representation (FHRR) relies on commutative binding, which cannot encode order or directionality. Prior FPGA/ASIC accelerators recover this through costly permutation logic, incurring irregular memory access and high control overhead. Generalized Holographic Reduced Representations (GHRR) instead redefine binding as non-commutative matrix multiplication, naturally encoding sequences, graphs, and hierarchies without permutations—but at the cost of turning every element into a matrix, introducing compute and storage overhead that limits scalability. In this work, we propose a flattening methodology that preserves GHRR's matrix-based encoding while executing training and inference directly in vector space, functionally equivalent to FHRR inference yet free of permutation logic. We further design a custom 28 nm ASIC that fuses binding and similarity into a unified complex-valued datapath, with dual-DMA streaming and runtime normalization for accurate inference. Against a PyTorch baseline on an NVIDIA RTX 4090 GPU, the prototype delivers 1.36×-1.56× higher throughput and 16.2×-18.6× better energy efficiency.