A first-round discussion* on Geometric diffusions as a tool for harmonic analysis and structure definition of data By R. R. Coifman et al. *The second-round discussion is to be led by Shihao

Spectral clustering Markov random walk – the diffusion operator Spectral decomposition Diffusion maps

Spectral decomposition of m-step random walk The diffusion map embeds the data into a Euclidean space in which the Euclidean distance is equal to the diffusion distance. Diffusion distance of m-step random walk The diffusion distance measures the rate of connectivity between xi and xj.

Multi-scale analysis of diffusion Interpret A as a dilation operator Discretize the semi-group {At:t>0} of the powers of A at a logarithmic scale which satisfy

Downsampling, orthogonalization, and operator compression The detail subspaces Downsampling, orthogonalization, and operator compression A - diffusion operator, G – Gram-Schmidt ortho-normalization, M - AG - diffusion maps: X is the data set

Diffusion multi-resolution analysis on the circle Diffusion multi-resolution analysis on the circle. Consider 256 points on the unit circle, starting with 0,k=k and with the standard diffusion. Plot several scaling functions in each approximation space Vj.

To be discussed a second-round led by Shihao Thanks!