Gene Regulatory Network Inference via the Latent Framework

We apply the Latent framework to the gene regulatory network (GRN) inference problem — reconstructing the regulatory interaction matrix \(W\) from gene expression data. The key insight is that the steady-state covariance matrix \(\Sigma\) satisfies a Lyapunov equation whose eigenstructure mirrors the regulatory dynamics (under the symmetric linearization discussed below), establishing that inference is fundamentally a spectral problem. We record fifty machine-checked lemmas in a companion proof script maintained with this manuscript, covering stability, sensitivity, stochastic gene expression, the Latent bridge, covariance spectral structure, sample-complexity scalings, network reconstruction, and cross-domain universality. Numerical validation on three synthetic GRN architectures (random sparse, hub-dominated, cascade) confirms the predicted trends across twelve test categories. The central compression diagnostic is \(N^ = \lceil \log(1/\varepsilon) / \log\rho \rceil\), where \(\rho\) is the Latent Number of the regulatory dynamics; relative to naive dense scaling \(O(N^2)\) in the entries of \(W\), recoverable degrees of freedom scale as \(O(N^{2})\) when the effective rank is \(N^\). For hub-dominated networks (\(\rho = 1.52\)), this gives \(N^/N = 83\%\) at 90% explained variance; near bifurcation (\(\gamma \to \mu_1\)), \(\rho \to \infty\) and a single mode dominates.

Keywords: gene regulatory networks, network inference, Latent Number, spectral gap, sample complexity, covariance structure