Sparse sign-consistent Johnson–Lindenstrauss matrices: Compression with neuroscience-based constraints: Fig. 1

Zeyuan Allen-Zhu, Rati Gelashvili, Silvio Micali, Nir Shavit
2014 Proceedings of the National Academy of Sciences of the United States of America  
Johnson-Lindenstrauss (JL) matrices implemented by sparse random synaptic connections are thought to be a prime candidate for how convergent pathways in the brain compress information. However, to date, there is no complete mathematical support for such implementations given the constraints of real neural tissue. The fact that neurons are either excitatory or inhibitory implies that every so implementable JL matrix must be sign-consistent (i.e., all entries in a single column must be either all
more » ... non-negative or all non-positive), and the fact that any given neuron connects to a relatively small subset of other neurons implies that the JL matrix had better be sparse. We construct sparse JL matrices that are sign-consistent, and prove that our construction is essentially optimal. Our work answers a mathematical question that was triggered by earlier work and is necessary to justify the existence of JL compression in the brain, and emphasizes that inhibition is crucial if neurons are to perform efficient, correlation-preserving compression.
doi:10.1073/pnas.1419100111 pmid:25385619 pmcid:PMC4250157 fatcat:aoulhcj4vzdo3nf5sqmg6ya6xy