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

Geometric diffusions as a tool for harmonic analysis and structure definition of dataBy R. R. Coifman et al.

The second-round discussion* on

* The first-round discussion was led by Xuejun;

* The third-round discussion is to be led by Nilanjan.

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.

• Markov Random Walk

KDxxKxxK

xxpaA

jji

jiijij

1

),(),(

)]|([][

• Symmetric Kernel

ji

jji

iji

jiij a

xxKxxK

xxKa ~

),(),(

),(~

Diffusion Maps

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

2/12/1~ ADDA

• Relationship

• Diffusion maps

)(

)(

)( 11

00

x

x

x m

m

m

• Spectral Decomposition of A

Diffusion Maps

n

jninnij xxa )()(~ 2

where 01 210

• Spectral Decomposition of Am

n

jninmn

mij xxa )()(~ 2)(

• Diffusion distance of m-step

2

)()()(,

2

)()(

~2~~)(

jmim

mij

mjj

miijim

xx

aaaxxD

• Interpretation

dzzxazxaxxD jm

im

jim

2)2/()2/(,

2 ),(~),(~)(

Diffusion Distance

The diffusion distance measures the rate of connectivity between xi and xj by paths of length m in the data.

Diffusion vs. Geodesic Distance

),(),( .. BCDBAD geodgeod ),(),( BCDBAD mm

Data Embedding

)(,),(),(001100 xxxxx d

md

mm

• By mapping the original data into (often )

• The diffusion distance can be accurately approximated

dRx 0dRx0dd

)(1),( 22 mm eOyxyxD

Example: curves

Umist face database: 36 pictures (92x112 pixels) of the same person being randomly permuted.

Goal: recover the geometry of the data set.

.,0)cos(

on tosimilar very a graph obtains oneordered,- re isnumbers ofset this When image.each to

numbera real assigns ioneigenfunct second The 1

t

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.

Original set: 1275 images (75x81 pixels) of the word “3D”.

Example: surface

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.

1

0 iondecompositwavelet

1

scalecoarsest

00

s

s

sss fAAfAf

• Multi-scale analysis of diffusion

Discretize the semi-group {At:t>0} of the powers of A at a logarithmic scale

which satisfy

Diffusion Wavelet

Diffusion Wavelet

• The detail subspaces

• Downsampling, orthogonalization, and operator compression

- diffusion maps: X is the data set

}:{ , jkjj Xk

}:{0 Xkk

A - diffusion operator, G – Gram-Schmidt ortho-normalization, M - AG

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.

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.

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.

To be discussed a third-round led by Nilanjan

Thanks!