It has been discovered that latent-Euclidean variational autoencoders (VAEs) admit, in various capacities, Riemannian structure. We adapt these arguments but for complex VAEs with a complex latent stage. We show that complex VAEs reveal to some level Kähler geometric structure. Our methods will be tailored for decoder geometry. We derive the Fisher information metric in the complex case under a latent complex Gaussian with trivial relation matrix. It is well known from statistical information theory that the Fisher information coincides with the Hessian of the Kullback-Leibler (KL) divergence. Thus, the metric Kähler potential relation is exactly achieved under relative entropy. We propose a Kähler potential derivative of complex Gaussian mixtures that acts as a rough proxy to the Fisher information metric while still being faithful to the underlying Kähler geometry. Computation of the metric via this potential is efficient, and through our potential, valid as a plurisubharmonic (PSH) function, large scale computational burden of automatic differentiation is displaced to small scale. Our methods leverage the law of total covariance to bridge behavior between our potential and the Fisher metric. We show that we can regularize the latent space with decoder geometry, and that we can sample in accordance with a weighted complex volume element. We demonstrate these strategies, at the exchange of sample variation, yield consistently smoother representations and fewer semantic outliers.

翻译：已有研究发现，潜在欧几里得变分自编码器（VAE）在不同程度上承认黎曼结构。我们调整了这些论证，但针对具有复数潜在空间的复变分自编码器。我们证明复变分自编码器在某种程度上揭示了凯勒几何结构。我们的方法将针对解码器几何进行定制。我们在潜在复高斯分布且关系矩阵平凡的情况下推导了复数情形的费希尔信息度量。从统计信息论可知，费希尔信息与Kullback-Leibler（KL）散度的海森矩阵一致。因此，在相对熵下精确实现了度量凯勒势关系。我们提出了复高斯混合的凯勒势导数，它作为费希尔信息度量的粗略近似，同时仍忠实于底层的凯勒几何。通过该势函数计算度量是高效的，并且通过我们作为多重次调和（PSH）函数的势，自动微分的大规模计算负担被转移至小规模计算。我们的方法利用全协方差定律来桥接势函数与费希尔度量之间的行为。我们证明可以通过解码器几何对潜在空间进行正则化，并且能够依据加权的复体积元进行采样。我们证明这些策略以样本变异为代价，能持续产生更平滑的表征和更少的语义异常值。