An Input Skip Can Reparameterize a VAE Without Improving the Generator

An input skip can reduce reconstruction error while leaving a model's distribution of new samples unchanged. For a Gaussian variational autoencoder (VAE) with constant input skip $0\leqα<1$, a model without a skip can be written in skip form by multiplying its decoder output by $b=1-α$ and its noise variance by $b^2$. The distribution of observations and latent variables and its exact generative scores stay the same, while reconstruction mean squared error (MSE) is multiplied by $b^2$. Separate training can produce different generators because optimization depends on the parameter coordinates. In a Gaussian experiment with known density, changing skip and observation variance together reduces MSE but worsens energy score and the estimate of negative log density from importance sampling. On seven annual Deep-ocean Assessment and Reporting of Tsunamis (DART) records, energy score differences at the same observation variance have both signs on four original records and favor the skip on three records from 2018. Probabilistic principal component analysis (PPCA) has lower mean energy scores than every neural DART fit. With the encoder and decoder kept the same, estimating observation variance from training reconstruction error can also worsen energy score. Assessing generation requires more than reconstruction error.