Exact constant in the compressed sensing RIP threshold for Gaussian projections

Determine the exact constant C in the measurement bound m ≥ C·k·ln(n/k) that guarantees the restricted isometry property for Gaussian random projection matrices, thereby specifying the precise latent-recovery threshold used to scale system dimension in the simulations.

Background

To contextualize simulation results, the authors normalize system dimension using the compressed sensing scaling m≈k·ln(n/k), which reflects the order at which Gaussian matrices satisfy the restricted isometry property with high probability. However, the exact constant factor multiplying k·ln(n/k) is not established.

Pinning down this constant would sharpen the boundary between regimes where latent features are provably recoverable versus not, improving interpretability of empirical results and calibrating alignment measurements under superposition.

References

The exact constant is not known in general, so we use m_cs = k ln(n/k) as a natural unit for the system dimension, distinguishing the regime in which latent features are in principle recoverable (CS; m ≥ m_cs) from the regime in which they are not (No CS; m < m_cs).

— Measuring the Representational Alignment of Neural Systems in Superposition  (2604.00208 - Liu et al., 31 Mar 2026) in Section: Simulating the Impact of Superposition on Alignment, Simulation Setup