Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

翻译：通过生成模型表示超高维数据的流形已在实践中展现出计算效率。然而，这要求数据流形具有全局参数化。为表示任意拓扑的流形，我们提出学习变分自编码器的混合模型。其中，每个编码器-解码器对代表流形的一个坐标卡。我们针对模型权重的最大似然估计提出损失函数，并选择能提供坐标卡及其逆解析表达式的架构。一旦流形被学习，我们将其用于解决逆问题：通过最小化限制在已学习流形上的数据保真项。为解决该最小化问题，我们提出在已学习流形上的黎曼梯度下降算法。我们在低维玩具示例以及特定图像流形的去模糊和电阻抗层析成像任务中验证了方法的性能。