Diffusion Geometries, and multiscale

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Diffusion Geometries, and multiscale Harmonic
Analysis on graphs and complex data sets.
Multiscale diffusion geometries,
Ontologies and knowledge building
Ronald Coifman
Applied Mathematics Yale university.

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Conventional nearest neighbor search , compared
with a diffusion search. The data is a pathology
slide ,each pixel is a digital document (spectrum
below for each class )
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One of our goals is to report on mathematical
tools used in machine learning, document and web
browsing, bio informatics, and many other data
mining activities. The remarkable observation
is that basic Geometric Harmonic Analysis of
empirical Markov processes provides a unified
mathematical structure which encapsulates most
successful methods in these areas. These
methods enable global descriptions of objects
verifying microscopic relations (like
calculus). We relate these ideas to methods of
classical Harmonic analysis , like Calderon
Zygmund theory in which Fourier analysis and
multiscale geometry merge.
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• This simple point is illustrated below

Each puzzle piece is linked to its neighbors ( in
feature space ) the network of links forms a
sphere. A parametrization of the sphere can be
obtained from the eigenvectors of the inference
relation (diffusion operator)
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A simple empirical diffusion matrix A can be
constructed as follows Let represent
normalized data ,we soft truncate the
covariance matrix as
A is a renormalized Markov version of
this matrix The eigenvectors of this matrix
provide a local non linear principal component
analysis of the data . Whose entries are the
diffusion coordinates These are also the
eigenfunctions of the discrete Graph Laplace
Operator.
This map is a diffusion (at time t) embedding
into Euclidean space
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The First two eigenfunctions organize the small
images which were provided in random order, in
fact assembling the 3D puzzle.
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A two dimensional map created by the Diffusion
Map algorithm for 400 MMPI-2 examinees. The
distance between two people was measured as the
difference between their responses. The color
corresponds to the score each examinee received
on the depression scale. New subjects need to be
placed in this tabulation of responders.
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The following image indicates that graphs may
have clusters at different scales.
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A very simple way to build a hierarchical
multiscale structure is as follows. We define
the diffusion distance between two subsets E and
F as
Start by considering small disjoint clusters of
nearest neighbors . Form a graph of these
clusters where the distance is defined with t1 .
Repeat on the graph of these clusters doubling
the time , etc
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4 Gaussian Clouds
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A simple application of signal processing on
data ,or data filters is Feature based diffusion
algorithms . Given an image, associate with each
pixel p a vector v(p) of features . For example
a spectrum, or the 5x5 subimage centered at the
pixel ,or any combination of features . Define a
Markov filter as
The various powers of A or polynomials in A
provide filters which account for feature
similarity between pixels .
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Feature diffusion filtering of the noisy Lenna
image is achieved by associating with each pixel
a feature vector (say the 5x5 subimage centerd at
the pixel) this defines a Markov diffusion matrix
which is used to filter the image ,as was done in
for the spiral in the preceding slide
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The data is given as a random cloud , the filter
organizes the data. The colors are
not part of the data