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Geometric%20diffusions%20as%20a%20tool%20for%20harmonic%20analysis%20and%20structure%20definition%20of%20data

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Title: Geometric%20diffusions%20as%20a%20tool%20for%20harmonic%20analysis%20and%20structure%20definition%20of%20data


1
The second-round discussion on
Geometric diffusions as a tool for harmonic
analysis and structure definition of data
By R. R. Coifman et al.
The first-round discussion was led by Xuejun
The third-round discussion is to be led by
Nilanjan.
2
Diffusion Maps
  • Purpose
  • - finding meaningful structures and
    geometric descriptions of a
  • data set X.
  • - dimensionality reduction
  • Why?

The high dimensional data is often subject to a
large quantity of constraints (e.g. physical
laws) that reduce the number of degrees of
freedom.
3
Diffusion Maps
  • Markov Random Walk

Many works propose to use first few eigenvectors
of A as a low representation of data (without
rigorous justification).
  • Symmetric Kernel
  • Relationship

4
Diffusion Maps
  • Spectral Decomposition of A

where
  • Spectral Decomposition of Am
  • Diffusion maps

5
Diffusion Distance
  • Diffusion distance of m-step
  • Interpretation

The diffusion distance measures the rate of
connectivity between xi and xj by paths of length
m in the data.
6
Diffusion vs. Geodesic Distance
7
Data Embedding
  • By mapping the original data into
    (often )
  • The diffusion distance can be accurately
    approximated

8
Example curves
Umist face database 36 pictures (92x112 pixels)
of the same person being randomly permuted. Goal
recover the geometry of the data set.
9
Original ordering
Re-ordering
The natural parameter (angle of the head) is
recovered, the data points are re-organized and
the structure is identified as a curve with 2
endpoints.
10
Example surface
Original set 1275 images (75x81 pixels) of the
word 3D.
11
Diffusion Wavelet
  • A function f defined on the data admits a
    multiscale representation of the form
  • Need a method compute and efficiently represent
    the powers Am.

12
Diffusion Wavelet
  • Multi-scale analysis of diffusion

Discretize the semi-group Attgt0 of the powers
of A at a logarithmic scale
which satisfy
13
Diffusion Wavelet
14
  • 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

15
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.
16
Diffusion multi-resolution analysis on the
circle. We plot the compressed matrices
representing powers of the diffusion operator.
Notice the shrinking of the size of the matrices
which are being compressed at the different
scales.
17
Multiscale Analysis of MDPs
1 S. Mahadevan, Proto-value Functions
Developmental Reinforcement Learning,
ICML05. 2 S. Mahadevan, M. Maggioni, Value
Function Approximation with Diffusion Wavelets
and Laplacian Eigenfunctions, NIPS05. 3
M. Maggioni, S. Mahadevan, Fast Direct Policy
Evaluation using Multiscale Analysis of
Markov Diffusion Processes, ICML06.
18
To be discussed a third-round led by Nilanjan
Thanks!
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