To make sense of millions of raw data and represent them efficiently, practitioners rely on representation learning. Recently, deep connections have been shown between these approaches and the spectral decompositions of some underlying operators. Historically, explicit spectral embeddings were built from graphs constructed on top of the data. In contrast, we propose two new methods to build spectral embeddings: one based on functional analysis principles and kernel methods, which leads to algorithms with theoretical guarantees, and the other based on deep networks trained to optimize principled variational losses, which yield practically efficient algorithms. Furthermore, we provide a new sampling algorithm that leverages learned representations to generate new samples in a single step.

翻译：为了理解海量原始数据并有效表示它们，实践者依赖表示学习。最近，这些方法与某些底层算子的谱分解之间展现出深层联系。传统上，显式谱嵌入是基于数据上构建的图建立的。相比之下，我们提出了两种构建谱嵌入的新方法：一种基于泛函分析原理和核方法，可得到具有理论保证的算法；另一种基于训练优化原则性变分损失的深度网络，能产生实际高效的算法。此外，我们提供了一种新的采样算法，该算法利用学习到的表示，通过单步生成新样本。