Geometric diffusions as a tool for harmonic analysis and structure definition of data

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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
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• Spectral clustering

• Markov random walk the diffusion operator

• Spectral decomposition

• Diffusion maps

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• 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.
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• Multi-scale analysis of diffusion

Interpret A as a dilation operator
Discretize the semi-group Attgt0 of the powers
of A at a logarithmic scale
which satisfy
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• 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

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