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：利用生成模型表示高维数据流形已被证明在实践中具有计算效率。然而，这要求数据流形允许全局参数化。为了表示任意拓扑结构的流形，我们提出学习一种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局部坐标图。我们提出了用于模型权重最大似然估计的损失函数，并选择了一种能提供坐标图及其逆解析表达的架构。一旦流形被学习完成，我们将其用于求解逆问题，方法是在已学习流形上最小化数据保真项。为解决这一最小化问题，我们提出在已学习流形上执行黎曼梯度下降算法。我们通过低维示例实验，以及在特定图像流形上进行去模糊和电阻抗断层成像的应用，验证了所提方法的性能。