Fermat Distance

Fermat distance is a density-based metric that measures distances along paths influenced by the probability density of data points, offering an alternative to Euclidean distance, particularly in high-dimensional spaces or when data exhibits complex geometric structures. Current research focuses on improving the accuracy and efficiency of Fermat distance computation, particularly through the use of normalizing flows and score matching for density estimation, and exploring its application in diverse fields such as uncertainty quantification in machine learning and even integer factorization. This metric's ability to adapt to data density makes it valuable for tasks like clustering and improving the performance of machine learning models, offering a powerful tool for analyzing complex datasets and enhancing model interpretability.