论文标题
在线性反问题中感知无监督学习的定理
Sensing Theorems for Unsupervised Learning in Linear Inverse Problems
论文作者
论文摘要
解决不适合的线性反问题需要了解基础信号模型。在许多应用程序中，此模型是先验未知的，必须从数据中学到。但是，不可能使用通过单个不完整的测量操作员获得的观察结果来学习模型，因为在操作员的零空间中没有有关信号模型的信息，从而导致鸡和蛋的问题：要学习需要重建的信号，而是要重建信号，我们需要知道该模型。克服此限制的两种方法是使用多个测量运算符，或者假设信号模型对某些组动作不变。在本文中，我们提供了仅从测量数据中学习信号模型的必要和足够的感应条件，这仅取决于模型的维度以及该模型不变的集团操作的运算符数量或属性。由于我们的结果是学习算法的不可知论，因此它们阐明了从不完整数据中学习的基本局限性，并在各种实用算法集中具有含义，例如字典学习，矩阵完成和深层神经网络。
Solving an ill-posed linear inverse problem requires knowledge about the underlying signal model. In many applications, this model is a priori unknown and has to be learned from data. However, it is impossible to learn the model using observations obtained via a single incomplete measurement operator, as there is no information about the signal model in the nullspace of the operator, resulting in a chicken-and-egg problem: to learn the model we need reconstructed signals, but to reconstruct the signals we need to know the model. Two ways to overcome this limitation are using multiple measurement operators or assuming that the signal model is invariant to a certain group action. In this paper, we present necessary and sufficient sensing conditions for learning the signal model from measurement data alone which only depend on the dimension of the model and the number of operators or properties of the group action that the model is invariant to. As our results are agnostic of the learning algorithm, they shed light into the fundamental limitations of learning from incomplete data and have implications in a wide range set of practical algorithms, such as dictionary learning, matrix completion and deep neural networks.
