Artificial intelligence, particularly the subfield of machine learning, has seen a paradigm shift towards data-driven models that learn from and adapt to data. This has resulted in unprecedented advancements in various domains such as natural language processing and computer vision, largely attributed to deep learning, a special class of machine learning models. Deep learning arguably surpasses traditional approaches by learning the relevant features from raw data through a series of computational layers. This thesis explores the theoretical foundations of deep learning by studying the relationship between the architecture of these models and the inherent structures found within the data they process. In particular, we ask What drives the efficacy of deep learning algorithms and allows them to beat the so-called curse of dimensionality-i.e. the difficulty of generally learning functions in high dimensions due to the exponentially increasing need for data points with increased dimensionality? Is it their ability to learn relevant representations of the data by exploiting their structure? How do different architectures exploit different data structures? In order to address these questions, we push forward the idea that the structure of the data can be effectively characterized by its invariances-i.e. aspects that are irrelevant for the task at hand. Our methodology takes an empirical approach to deep learning, combining experimental studies with physics-inspired toy models. These simplified models allow us to investigate and interpret the complex behaviors we observe in deep learning systems, offering insights into their inner workings, with the far-reaching goal of bridging the gap between theory and practice.

通过研究深度学习模型的架构和数据内在结构之间的关系，本文探讨了深度学习的理论基础，并通过实验和物理启发式玩具模型的结合，揭示了它们的内在工作原理，旨在填补理论与实践之间的鸿沟。

通过学习不变表示来打破深度神经网络中的维数诅咒