A Random Walk in 10 Dimensions (2021)
7 days ago
- #high-dimensional physics
- #machine learning
- #random walks
- Physics in high dimensions is increasingly important in modern dynamics, including string theory and complex systems like population dynamics.
- High-dimensional landscapes feature more mountain ridges than peaks, impacting evolution, complex systems, and machine learning.
- Visualizing data beyond four dimensions is challenging, leading to misconceptions about high-dimensional spaces.
- Random walks in high dimensions are common in complex systems, such as molecular evolution, where genetic mutations create vast possibilities.
- Ten dimensions are used as a practical example to study high-dimensional physics due to computational limits and unvisualizability.
- Diffusion in ten dimensions behaves similarly to lower dimensions but with more complexity due to increased degrees of freedom.
- Self-avoiding walks (SAWs) are unnecessary in high dimensions because the probability of path crossing is negligible.
- Random walks in maximally rough landscapes show that percolation theory explains mobility in high-dimensional spaces.
- Mountain ridges are common in high dimensions, while peaks are rare, influencing evolutionary biology and machine learning.
- Neutral networks in high-dimensional fitness landscapes allow species to evolve without crossing low-fitness valleys.
- Deep learning benefits from high-dimensional neutral networks, enabling escape from local minima and effective optimization.
- The geometry of high-dimensional random walks may offer insights into human intelligence and consciousness.