Exact exponent for KL-approximation of a Gaussian by finite mixtures

Determine the exact exponential decay rate, as a function of the signal variance σ, of the best Kullback–Leibler approximation error (m, N(0, σ^2), KL) by m-atomic Gaussian location mixtures. Equivalently, compute g(σ) := lim_{m→∞} (−1/m) log (m, N(0, σ^2), KL) to characterize the precise σ-dependence of the exponent in the capacity gap C − C_m for the Gaussian channel with an input cardinality constraint m.

Background

The paper studies approximation of Gaussian location mixtures by finite mixtures and connects this to the finite-constellation capacity gap in Gaussian channels. For the Gaussian mixing distribution P = N(0, σ^2), the authors correct a previously flawed lower bound and show the approximation error decays exponentially in m, with the optimal exponent scaling on the order of 1/σ for fixed σ. However, while tight exponential decay is established, the precise function governing the exponent as a function of σ is not identified.

This open problem asks for a sharp characterization of the exponent governing the asymptotic rate of convergence in KL divergence (and hence the capacity gap), refining the current Θ(1/σ) understanding to an exact expression.