Beyond Minimum Distance: The Optimal Leading Coefficient in the High-SNR Error-Probability Expansion for AWGN Spherical Codes

What the math gives to ML

The paper distinguishes maximizing the minimum pairwise distance from minimizing the full high-SNR error asymptotic: once the dominant exponent is fixed, the number and arrangement of nearly closest pairs determine the leading coefficient. Its constructive insight is that allowing a vanishing sacrifice in a few pair distances can create geometric freedom to move many other pairs farther away, reducing their aggregate error contribution. This suggests an SNR- or temperature-annealed hyperspherical embedding objective that tracks soft pairwise error mass rather than only the single worst pair. The most credible transfer is to normalized class prototypes or metric-learning embeddings, where the objective can be tested against ordinary max-margin and contrastive losses.