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