Jekyll2021-05-31T15:43:00-05:00https://liutianlin0121.github.io/blog/feed.xmlResearch blogUpdates on random findings and thoughts related to my research. By Tianlin Liu.A detailed derivation of the diffusion map2021-05-29T00:00:00-05:002021-05-29T00:00:00-05:00https://liutianlin0121.github.io/blog/2021/05/29/difussion-mapsThe diffusion map is an extensively used dimensionality reduction technique. It represents high-dimensional data points using an underlying graph, which non-linearly encodes the geometrical similarities between data points. Despite its popularity, few tutorials introduce the diffusion map in precision that I find satisfying. This blog delves deep into the derivation.The expected angle between two isotropic random vectors2021-05-01T00:00:00-05:002021-05-01T00:00:00-05:00https://liutianlin0121.github.io/blog/2021/05/01/angles-between-random-vectorsIn Section 3.2 of the book High-dimensional Probability by Roman Vershynin, it was mentioned in the caption to Figure 3.4 thatAre coordinates of an isotropic random vector uncorrelated?2021-04-17T00:00:00-05:002021-04-17T00:00:00-05:00https://liutianlin0121.github.io/blog/2021/04/17/isotropic-rvI was reading Section 3.3 of the nicely written book High-dimensional Probability by Roman Vershynin. At the beginning of that section, it was mentioned that: