This work considers the fundamental problem of learning an unknown object from training data using a given model class. We introduce a unified framework that allows for objects in arbitrary Hilbert spaces, general types of (random) linear measurements as training data and general types of nonlinear model classes. We establish a series of learning guarantees for this framework. These guarantees provide explicit relations between the amount of training data and properties of the model class to ensure near-best generalization bounds. In doing so, we also introduce and develop the key notion of the variation of a model class with respect to a distribution of sampling operators. To exhibit the versatility of this framework, we show that it can accommodate many different types of well-known problems of interest. We present examples such as matrix sketching by random sampling, compressed sensing with isotropic vectors, active learning in regression and compressed sensing with generative models. In all cases, we show how known results become straightforward corollaries of our general learning guarantees. For compressed sensing with generative models, we also present a number of generalizations and improvements of recent results. In summary, our work not only introduces a unified way to study learning unknown objects from general types of data, but also establishes a series of general theoretical guarantees which consolidate and improve various known results.

翻译：本文研究了利用给定模型类从训练数据中学习未知对象的基本问题。我们提出了一个统一框架，该框架适用于任意希尔伯特空间中的对象、一般类型的（随机）线性测量作为训练数据以及一般类型的非线性模型类。我们为该框架建立了一系列学习保障，这些保障明确给出了训练数据量与模型类属性之间的关系，以确保达到近乎最优的泛化边界。在此过程中，我们还引入并发展了模型类关于采样算子分布的变化这一关键概念。为展示该框架的通用性，我们证明了它可以容纳许多不同类型的经典问题。我们给出了诸如随机采样的矩阵素描、各向同性向量的压缩感知、回归中的主动学习以及生成模型的压缩感知等示例。在所有案例中，我们展示了已知结果如何成为我们一般学习保障的直接推论。对于生成模型的压缩感知，我们还给出了近期结果的一系列推广和改进。总之，我们的工作不仅引入了从一般数据类型中学习未知对象的统一研究方法，还建立了一系列通用的理论保障，巩固并改进了各种已知结果。