Random Walks in 10 Dimensions: Defying Intuition in High-Dimensional Spaces
High-dimensional physics is the norm in modern dynamics, from string theory's ten dimensions to complex systems. However, high dimensions present the 'curse of dimensionality': visualization is impossible, overfitting is rampant, and intuition fails. This article uses a 10-dimensional random walk to illustrate high-dimensional space characteristics. In high dimensions, mountain ridges are far more common than peaks, profoundly impacting evolution, complex systems, and machine learning. Random walks efficiently explore high-dimensional spaces, even maximally rough landscapes, potentially traversing the entire space. This helps understand the evolution of complex structures in life and how to avoid local minima in deep learning.
Read more