Johnson-Lindenstrauss Compression with Neuroscience-Based Constraints

TL;DR

This paper constructs and proves the optimality of sparse, sign-consistent Johnson-Lindenstrauss matrices, supporting the idea that neural tissue can perform efficient, correlation-preserving data compression under biological constraints.

Contribution

It introduces a new construction of sparse, sign-consistent JL matrices and proves their near-optimality, bridging neural constraints with mathematical data compression techniques.

Findings

Sparse, sign-consistent JL matrices can be constructed and are nearly optimal.

Neural inhibition plays a crucial role in efficient data compression in the brain.

Supports the hypothesis that neural pathways implement JL-like compression mechanisms.

Abstract

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. 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.