深度学习的前提——假设空间的弯曲和平坦

This paper explores a core assumption in deep learning—the geometric structure of space—specifically, the impact of the curved space and flat space assumptions on the optimization process of deep learning models. Traditional deep learning models are mostly based on the flat space assumption, assuming that input data and model parameters follow a Euclidean geometry during training. However, real-world data often exhibits nonlinear characteristics and complex geometric structures, requiring models to adapt to the characteristics of curved spaces. This paper introduces the curved space assumption to explore how different assumptions about spatial geometry affect the training process, gradient calculation, and optimization path of deep learning models. In particular, it discusses how the introduction of dynamic gradient matrices and spatial curvature enables models to adapt to different spatial structures, improving their performance in handling nonlinear data and extrapolation tasks. Furthermore, this study experimentally validates the application of the curved space assumption in deep learning, especially comparing its effectiveness in complex function approximation and nonlinear regression problems. By comparing with traditional flat space assumption models (such as MLP), the results show that models based on curved space exhibit stronger advantages in both fitting and extrapolation capabilities, especially when dealing with data with strong nonlinear characteristics, achieving more accurate optimization. This research provides a new perspective for understanding spatial geometry assumptions in deep learning models and proposes a more flexible optimization strategy that can better handle nonlinear structures and spatial curvature in practical problems.