Factorizers for Distributed Sparse Block Codes

Distributed sparse block codes (SBCs) exhibit compact representations for encoding and manipulating symbolic data structures using fixed-with vectors. One major challenge however is to disentangle, or factorize, such data structures into their constituent elements without having to search through all possible combinations. This factorization becomes more challenging when queried by noisy SBCs wherein symbol representations are relaxed due to perceptual uncertainty and approximations made when modern neural networks are used to generate the query vectors. To address these challenges, we first propose a fast and highly accurate method for factorizing a more flexible and hence generalized form of SBCs, dubbed GSBCs. Our iterative factorizer introduces a threshold-based nonlinear activation, a conditional random sampling, and an $\ell_\infty$-based similarity metric. Its random sampling mechanism in combination with the search in superposition allows to analytically determine the expected number of decoding iterations, which matches the empirical observations up to the GSBC's bundling capacity. Secondly, the proposed factorizer maintains its high accuracy when queried by noisy product vectors generated using deep convolutional neural networks (CNNs). This facilitates its application in replacing the large fully connected layer (FCL) in CNNs, whereby C trainable class vectors, or attribute combinations, can be implicitly represented by our factorizer having F-factor codebooks, each with $\sqrt[\leftroot{-2}\uproot{2}F]{C}$ fixed codevectors. We provide a methodology to flexibly integrate our factorizer in the classification layer of CNNs with a novel loss function. We demonstrate the feasibility of our method on four deep CNN architectures over CIFAR-100, ImageNet-1K, and RAVEN datasets. In all use cases, the number of parameters and operations are significantly reduced compared to the FCL.