Johnson-Lindenstrauss Compression with Neuroscience-Based Constraints

Understand

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 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.