Deep-learned variational auto-encoders (VAE) have shown remarkable capabilities for lossy image compression. These neural networks typically employ non-linear convolutional layers for finding a compressible representation of the input image. Advanced techniques such as vector quantization, context-adaptive arithmetic coding and variable-rate compression have been implemented in these auto-encoders. Notably, these networks rely on an end-to-end approach, which fundamentally differs from hybrid, block-based video coding systems. Therefore, signal-dependent encoder optimizations have not been thoroughly investigated for VAEs yet. However, rate-distortion optimized encoding heavily determines the compression performance of state-of-the-art video codecs. Designing such optimizations for non-linear, multi-layered networks requires to understand the relationship between the quantization, the bit allocation of the features and the distortion. Therefore, this paper examines the rate-distortion performance of a variable-rate VAE. In particular, one demonstrates that the trained encoder network typically finds features with a near-optimal bit allocation across the channels. Furthermore, one approximates the relationship between distortion and quantization by a higher-order polynomial, whose coefficients can be robustly estimated. Based on these considerations, the authors investigate an encoding algorithm for the Lagrange optimization, which significantly improves the coding efficiency.
