1. 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

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

3. 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.

4. 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

5. 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

6. 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.

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