دانلود کتاب هندسه هوش – مبانی شبکه‌های تبدیل‌کننده در یادگیری عمیق

The study of transformers, particularly in the context of natural language processing and machine learning, has revolutionized the way we understand and process data. The central theme of this revolution is the concept of representation: how data, whether it be words in a sentence, pixels in an image, or nodes in a graph, is transformed into a mathematical structure that a machine can manipulate. Understanding this process from a mathematical perspective requires us to delve into the theory of vector spaces and linear algebra, which form the backbone of representation theory. In transformer models, data is represented as vectors in high-dimensional spaces. These vectors capture not only the intrinsic properties of the data but also the relationships and interactions between different data points. The operations performed on these vectors, such as attention mechanisms and linear transformations, rely heavily on the principles of vector spaces. Therefore, a deep understanding of vector spaces, subspaces, and bases is essential for comprehending how transformers encode and manipulate information. Moreover, transformers leverage the concept of symmetry— a principle deeply rooted in group theory and representation theory. Symmetry allows us to understand how certain transformations, such as rotations or translations, affect the data representations. By exploring these symmetries within the framework of vector spaces, we can gain insights into the invariances and equivariances that make transformer models so powerful. This chapter serves as a foundation for the mathematical framework that underpins transformers. We begin by introducing the fundamental concepts of vector spaces and linear algebra. These concepts will not only provide the necessary tools to analyze and understand transformer models but will also reveal the deep connections between geometry, symmetry, and intelligence. As we progress, we will see how the abstract mathematical notions introduced here manifest in the practical operations of transformers, setting the stage for more advanced topics in later chapters. Through mathematical exploration, we aim to build an intuition for how transformers operate, grounded in the precise language of vector spaces and their transformations. This will enable us to appreciate the elegance and power of transformers from a purely mathematical standpoint, where the focus is not on implementation or code, but on the underlying mathematical structures that drive these models.