1. Intrinsic Data Geometry from a Training Set
Phd Thesis Proposal
Nicola Rebagliati
Supervisor: Alessandro Verri
December 2007

2. Index Introduction The Training Set Intrinsic Geometry
An interesting operator over manifolds
The Laplace-Beltrami operator
The Laplacian matrix
Preliminary work
Image segmentation
Image signatures
Work plan

3. Introduction

4. The Training Set
In the context of Statistical Learning we are given a set of training data representative of the collection of all possible data:
Where, in the classification case,
That is: Vectors in a Euclidean space and Labels

5. The Training Set The labelling function is unknown
We are looking for a function f* that minimize an error over future data
We should stay away from overfitting and underfitting
A solution...
... that overfits

6. Some example of Training Set-1 1
Images: R^10000
Faces
Not Faces
Patient information: R^15
Exceeding quantity of iron
Right quantity of iron
Actions: R^(10000 x k)
Walking
Resting

7. Intrinsic Dimensionality
Without assumptions vectors may be uniformly distributed in their euclidean space
In many practical cases the degrees of freedom of training data are far less than n.
Images of 100X100 pixels representing squares may have three degrees of freedom, for example.

8. Data form a structure
We may have something more than a simple lower dimensionality
Data may live in a topological space where we can compute angles and distances
That is a Riemannian Manifold

9. Example: The Swiss Roll
Difference between euclidean distance and geodesic distance

10. Semisupervised Learning
The geometry of the data is exploited in the semisupervised learning [BelkinNiyogi2004]
In Semisupervised Learning the Training set is made of Labeled Data and Unlabeled Data
Structural penalty w.r.t. known data
Structural penalty w.r.t. all data
Empirical error

11. Geometrical Information
A manifold has some intrinsic geometrical information [McKeanSinger67] :
The Volume
The Volume of the Boundary
The Euler Characteristic
...
The training set is extracted from the manifold
Is it possible to approximate these information with the training set?

12. An interesting operator over manifold

13. The continuos Case: The Laplace-Beltrami operator
Let M denote a manifold
The Laplace Beltrami operator maps functions to functions, both defined on the manifold
In a while we see why we are interested in its eigenvalues and eigenvectors

14. The heat trace
The heat trace of the Laplacian on a Riemannian manifold can be expressed as:
The heat trace has an asymptotic expansion:
C0 is the volume of the manifold
C1 is proportional to the volume of the Boundary
...
[McKeanSinger]

15. The discrete Case: The Laplacian Matrix
The Training Set can be seen as an undirected weighted graph
Choose a comparing function for weighting graph
edges:
Define the degree of a vertex as
Consider the Weight matrix
Consider the diagonal Degree matrix

16. The discrete Case: The Laplacian Matrix
The Laplacian Matrix is simply:
A normalized version [VonLuxburg06] has some better properties:

17. The discrete Case: The Laplacian Matrix
Now consider the spectral decomposition of the normalized laplacian:
The eigenvalues are ordered as
The training set can be embedded in a possibly low dimensional space through the eigenvectors and eigenvalues:

18. The Convergence
We would like the discrete operator to be consistent with the continuos one
Pointwise distance between the two operator is bounded by an error:
The error is divided in Bias + Variance
Here we have some constants that depend on the possibly unknown underlying manifold

19. Image segmentation
PRELIMINARY WORK

20. Image segmentation
A pixel is given by a digital camera as a tuple (x, y, R, G, B)
We may define features that map a pixel, usually considering also its neighborhood, on a vector in
For example (x, y, R, G, B) -> (R,G,B)

21. Image segmentation Clustering methods group together different vectors
I want to compare clustering results with a ‘ground truth’, or ‘human truth’

22. Segmenting with Laplacian
It can be proved that the second eigenvector of the Laplacian matrix give a clustering that:
Maximize intra-group cohesion
Minimize inter-group cohesion
I consider as weighting function the gaussian:
What sigma should be used?

23. Choose sigma in image segmentation
Heuristic: Choosing sigma such that the underlying manifold, if exists, is best approximated
That is
The training set is sampled over the manifold, so it is a random variable
Choose sigma such that we have the minimum Bias-Variance error
Let’s see a syntetic case

24. Choose sigma in image segmentation
3D Sphere
Error Bound

25. Choose sigma and the balancing principle
The error bound is divided in bias and variance
Algorithm for balancing the two terms
We do not know the constants

26. Image Signatures
PRELIMINARY WORK

27. 3D-shape retrieval in database
In an article Reuter et al. [ReuterWolterPeinecke] propose to use the spectrum of the laplace-beltrami operator of a shape as its signature.
This signature has many desiderable properties within that context:
Representation invariant
Scaling invariant
Continuos w.r.t. Shape deformations

28. Image Signature
In an another article Peinecke et al. [PeineckeReuterWolter] have proposed to look at an image as a 2D-manifold and to use its laplacian spectrum as signature.
It is not clear what are the useful geometric invariants in that case:
Illumination?
Scale?

29. Image Signature: Laplacian Definition
Different Kind of Laplacian Definition
The Normalized Laplacian
The Laplace-Kirchoff operator
The continuos operator Computed using the Finite Element Method (FEM)
Different Results

30. Laplace-Kirchoff operator, radius:1

31. Laplace operator, sigma_xy=30, sigma gray=50

32. Work to do

33. Short Term
Investigate an empirical criterion or an algorithm for choosing a suitable sigma
Tackle the problem of extracting geometrical information from a set of eigenvalues of the Laplacian matrix. In the case of 3D-shapes, using FEM, a solution is proposed in [ReuterPeineckeWolter06]

34. Short Term Developing the idea of image signature
Compare different Laplacian definitions
Explore the invariance possibilities
Experimentation of plugging geometrical information in the learning context

35. Long Term
Wider view for plugging geometrical information in the learning contest
Characterize the manifold of images inside the space of matrices
Torralba, Fergus and Freeman – 80 million tiny images: a large dataset for non-parametric object and scene recognition -