Theoretical origin of the logarithmic finite-size correction

Determine the theoretical origin of the empirical logarithmic finite-size correction required to improve the finite-size scaling collapse of the binary cross-entropy reconstruction error in the heteroscedastic variational autoencoder analysis of the contact process.

Background

The paper analyzes reconstruction errors from a heteroscedastic variational autoencoder trained on contact-process configurations near the active-absorbing phase transition. The mean-squared error and binary cross-entropy are examined using finite-size scaling forms based on directed-percolation critical exponents.

For the binary cross-entropy, the authors find that the numerical data collapse improves substantially when an empirical logarithmic correction factor involving L/ln L is included. However, they do not derive this correction from the underlying critical theory and explicitly leave its theoretical origin unresolved. Establishing whether the logarithmic factor reflects a genuine universal correction, a model-specific finite-size effect, or an artifact of the reconstruction observable remains an open problem.