Although the variational autoencoder (VAE) and its conditional extension
(CVAE) are capable of state-of-the-art results across multiple domains, their
precise behavior is still not fully understood, particularly in the context of
data (like images) that lie on or near a low-dimensional manifold. For example,
while prior work has suggested that the globally optimal VAE solution can learn
the correct manifold dimension, a necessary (but not sufficient) condition for
producing samples from the true data distribution, this has never been
rigorously proven. Moreover, it remains unclear how such considerations would
change when various types of conditioning variables are introduced, or when the
data support is extended to a union of manifolds (e.g., as is likely the case
for MNIST digits and related). In this work, we address these points by first
proving that VAE global minima are indeed capable of recovering the correct
manifold dimension. We then extend this result to more general CVAEs,
demonstrating practical scenarios whereby the conditioning variables allow the
model to adaptively learn manifolds of varying dimension across samples. Our
analyses, which have practical implications for various CVAE design choices,
are also supported by numerical results on both synthetic and real-world
datasets.

研究证明变分自编码器和其条件扩展在多个领域可以达到最新成果，小样本图像中具有低维曼诺尔性质的特殊数据，其复杂行为仍不为人所知晓，本文从数据分布的角度出发，通过实验证明变分自编码器的全局最小值可以学习到正确的曼诺尔维度，在这个前提下讨论了更普适的条件扩展以适应不同数据分布的情况。

使用条件变分自编码器学习流形维度

Learning Manifold Dimensions with Conditional Variational Autoencoders

Domains such as logo synthesis, in which the data has a high degree of
multi-modality, still pose a challenge for generative adversarial networks
(GANs). Recent research shows that progressive training (ProGAN) and mapping
network extensions (StyleGAN) enable both increased training stability for
higher dimensional problems and better feature separation within the embedded
latent space. However, these architectures leave limited control over shaping
the output of the network, which is an undesirable trait in the case of logo
synthesis. This paper explores a conditional extension to the StyleGAN
architecture with the aim of firstly, improving on the low resolution results
of previous research and, secondly, increasing the controllability of the
output through the use of synthetic class-conditions. Furthermore, methods of
extracting such class conditions are explored with a focus on the human
interpretability, where the challenge lies in the fact that, by nature, visual
logo characteristics are hard to define. The introduced conditional style-based
generator architecture is trained on the extracted class-conditions in two
experiments and studied relative to the performance of an unconditional model.
Results show that, whilst the unconditional model more closely matches the
training distribution, high quality conditions enabled the embedding of finer
details onto the latent space, leading to more diverse output.

本文探讨了一种对 StyleGAN 体系结构的条件扩展方法，通过使用合成类别条件提高网络的可控性和结果的分辨率，同时研究了提取这些类别条件的方法以增加人的可解释程度。实验证明，条件模型相对于无条件的模型可以更好的嵌入细节，产生更多样化和高质量的输出结果。