Topological characterization of an image dataset with Betti numbers and a generative model

TOPOLOGICAL CHARACTERIZATION OF AN

IMAGE DATASET WITH BETTI NUMBERS AND A

GENERATIVE MODEL.

Maxime MAILLOT (Exalead)

Michaël AUPETIT (CEA LIST)

Gérard GOVAERT (UTC-CNRS)

DataSense | 08-07-2014


• Multivariate data exploration• Signals , images, ….

• Classical ML techniques • Clustering: K-Means; Gaussian Mixture Models -> convex clusters

• Dimension reduction : Self-Organizing Maps, MDS, PCA -> Dime Reduct artefacts imposed by the

representation space

•Topological information (from underlying structure) :•Number of connected components•Intrinsic dimension•Topological invariants (Betti numbers)

Context

Cognition and topology

Neuronal encoding of topological information survived Darwinian natural selection showing the importance of this information in our cognitive processes

WHY TOPOLOGICAL INFORMATION?


Retinotopic map of a mouse [Hübener 2003]

Topology and visual perception

Gestalt psychological theory [1920]The whole is more than summing the partsLaw of continuity, proximity, similarity



Topological view

Underlying structure

Statistical view

Underlying density

Geometrical view

Points location or underlying shapes

Predictive model: Our visual system instantly provides a topological model of the population

Descriptive model: sample is enough, no hypothesis about the population underlying the data

Mental map and topology

Topological invariants as an objective representation



Objective map Mof a building B Subjective

map M1

of B

Subjective map M2

of B

Whatever radically different the perception process and experience of each person are, a topological invariant still exists common to both persons’ mental models and the real building’s map:

They share the same connectedness


Patterns reliability and topology

A large family of transformations

Reliability- The processing pipeline from data to decision is more likely to be a homotopy - So topological information is more likely to survive to the distortions of the pipeline- Hence topological information is a more reliable basis for decision facing

uncertainty


U

Isometries Similarities Homeomorphisms Homotopies

U U

Initial space

Betti numbersIntrinsic dimension

Probability density functionsGeometry

SOME HINTS ABOUT TOPOLOGY

Topology in a nutshell

What is the difference between a mug and a doughnut?



Topology in a nutshell

What is the difference between a mug and a doughnut?


Taste is significantly different!


Topological invariants

Two spaces have the same topology iff they are homeomorphic to each other, i.e. they are linked through a continuous function H whose inverse H-1 is also continuous.

Topology classifies spaces based on their topological invariants like the Betti numbers


1-cycle which can contract to a point

Blue and brown 1-cycles cannot collapse to each otherThey form a homology group, the rank of which is 2 (b1=2)

1-cycles which cannot contract to a point

(b0,b1,b2)= (1,2,1)# of connected components# of independent 1-cycles (tunnels)# of independent 2-cycles (cavities)

Measures

Topological

inference

Sensor spaceSample of a robot’s trajectory Image of walls 1 and 2

In the robot-to-sensors distance space

Wall 1

Wall 2

Sensor 3

Sensor 1

Sensor 2

FROM SETS OF POINTS TO BETTI NUMBERS

Simplex family

Simplex assembly

SIMPLICIAL COMPLEX


0-simplex 1-simplex 2-simplex 3-simplex


For any manifold V it exists a simplicial complex C which is homeomorphic to V (C(V) is a triangulation of V)

Two triangulations may have the same Betti numbers while their manifolds are not homeomorphic.


Betti numbers

Computational topology

Simplicial complex



Vietori-Rips complex and Betti numbers

a c

b

d

R=11

10

88

8

a c

b

d8

88

R=9

[Ch

aza

l]

(b0,b1,b2)(N,0,0) (37,6,0) (1,2,0) (1,0,0)

R

Topological persistence and multiscale analytics = persistence of topological structure through scale


RESTRICTED DELAUNAY COMPLEX

[Edelsbrunner, Shah 1997]

M1

M2

From manifold to triangulation



[Edelsbrunner, Shah 1997]

M1

M2

. From manifold to triangulation



Manifold = union of spheres Centered on the atoms’ core (alpha sets the spheres radius

Molecules topology [Edelsbrunner1994]

Alpha-shapes


TOPOLOGY REPRESENTING NETWORKS

Topology Representing Network [Martinetz, Schulten 1994]

Connect 1st and 2nd Nearest Neighbor prototype of each data : Competitive Hebbian Learning (CHL)




1er2nd





1er2nd





1er2nd

ROI = Order 2 Voronoi cells





1er2nd

ROI = Order 2 Voronoi cells




Order 2 Voronoi cells

No noise

Sample with gaussian noise

When a Statistician meets a Topologist…

What is the probability for a HEAD if you flip a coin cut in a Moebius strip?

A GENERATIVE MODEL APPROACH

Moebius strip





HEAD or TAIL? Moebius strip

P( HEAD ) = ?





HEAD or TAIL? Moebius strip

P( HEADACHE ) = 1


Unknown generative manifolds with

possible different topology, different

labels, and possibly overlapping…

GENERATIVE GRAPH [GAILLARD 2010]

Topological inferencefrom the sample to the population

…from which are drawn samples with unknown probability

density…

…corrupted with unknown noise…

Statistical generative model – Where the data come from?

…leading to the actual data observations.


Unknown generative manifolds with

possible different topology, different

labels, and possibly overlapping…


…from which are drawn samples with unknown probability

density…


Statistical generative model – General hypotheses

…leading to the actual data observations.


Unknown generative manifolds …


…from which are drawn samples with unknown

probability density…


Statistical generative model – Simplified hypotheses


Unknown generative manifolds…


…from which are drawn samples with unknown

probability density…


Generative Gaussian Graph (GGG) – Simplified hypotheses


Unknown generative manifolds…

Delaunay graph of some prototypes with class label probability

p1-p

10

Jj

)c,x(p )jc(p)j(p

Gaussian noise with identity covariance

),jx(p

)j(p

Uniform density over each topological component (vertices and edges)

)jc(p

),jx(p

c

j

)jc(p)jc(p

)j(p)j(p

01

01

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Topological summary

Model selection (# vertices): Bayesian Information Criterion


GGG: From data to topological synthesis

)jc(p

)j(p

),jx(p

c

j

)jc(p)jc(p

)j(p)j(p

01

01

Jj

)c,x(p ),jx(p )jc(p)j(pLikelihood

Maximization (EM)


Multivariate data GMM

Delaunay

GENERATIVE SIMPLICIAL COMPLEX [MAILLOT2012]

Generative simplices familly


A

g0

…(Pseudo-Monte Carlo estimation)

DATA SAMPLED FROM A GENERATIVE GAUSSIAN SIMPLEX


d= 0 d= 1 d= 2

σ= 0.1

σ= 0.5

σ= 0.2

GENERATIVE SIMPLICIAL COMPLEX


Expectation-Maximization

π1 < π2 < π3 < ………< πi < …… < πn

BIC max

FROM DATA TO GENERATIVE SIMPLICIAL COMPLEX



Protoypes location initialized with GMM



Delaunay complex built on top of the prototypes

First the edges…




Delaunay complex built on top of the prototypes

First the edges…Then the surfaces…


Likelihood maximization for dimension 1 components

The p proportion of each edge is estimated with EM

Edges with too low proportion do not contribute significantly to the model (wrt Bayesian Information Criterion), they are pruned from the model



Likelihood maximization for dimension 2 components

Proportions of both surfaces and remaining edges are estimated with EM, then pruned wrt BIC



Topological cleaning

If a simplex survived, all its facets are pruned.


RESULTS (1/3)


SPHERE (1,0,1,0…) TORE (1,2,1,0…)

KLEIN BOTTLE (1,1,0…)

RESULTS (2/3)

Images data COIL-100 :


• 100 objects in rotation each represented by 72 images (5°) with 64x64 pixels (projected by PCA on the 71 first principal components)

• O 2D simplices

• Delaunay complex only computed for 1D then 2D elements in the 71D space

• We recover a cycle structure

RESULTS (3/3)

Images data COIL-100 :


• Expected Betti numbers (1,1,0 …)

• (1,2,0 …) correspond to an 8 shape

• The (1,n,0 …) shows that many faces of the objects look similar

• (1,0,0,…) shows a rotatioal invariant object

Example for (1,2,0,…) (like an 8)

CONCLUSIONS

• GSC: first generative model to extract Betti numbers from a data set

• No meta-parameter to tune (EM + BIC)


PERSPECTIVES

• Topological analysis for each connected component separately

• Algorithmic improvements (pseudo-monte-carlo, pruning…)

• Link BIC optimal and Betti numbers

• Deep Networks : how topological invariants could be explicitely encoded within each layer?


THANK YOU FOR YOUR ATTENTION

- MA, Learning Topology with the Generative Gaussian Graph and the EM algorithm. NIPS 2005 Conference proceeding, pp.83-90, 2006.

- Gaillard Pierre, MA, Gérard Govaert. Learning topology of a labeled data set with the supervised generative Gaussian graph. Neurocomputing, 71(7-9): 1283-1299, Elsevier March 2008

- Maillot Maxime, MA, Gérard Govaert. Extraction of Betti numbers based on a generative model. ESANN 2012

- Maillot Maxime, MA, Gérard Govaert. The Generative Simplicial Complex to extract Betti numbers from unlabeled data. Workshop at NIPS 2012

Questions?


QUESTIONS

•Pourquoi un modèle de bruit isovarié? -> pour la complexité du modèle soit attrapée

par le complexe simplicial et les nombres de Betti

•Pourquoi les nombres de Betti? La connexioté semble suffire pour les applications ?

Forme prise par les états d’un système dynamique (épilepsie / cas normal-alerte-

catastrophe… ) pas de cas réel mais mise au point d’un modèle/système de mesure.


•Suggestions:

•- comparaison topologie ND vs topologie 2D pour évaluation distorsions de projections

•- système dynamique changeant de forme et dont la forme indique l’état (bon, alerte,

mauvais)

•- analyse/caractérisation topologique de données

•- contrôle de passage dans zone d’alerte (système dynamique dont on observe l’état

bruité) on veut vérifier que l’on ne peut pas passer directement d’un état bon à un état

mauvais sans passer par l’état d’alerte: extension du SGGG au cas des CS: trous

dans la structure = fuite possible A CLARIFIER

•- Cas de l’analyse de locuteurs sur les lettres (triangle NSI2000): utiliser un locuteur

comme sommet du GSC et positionner les autres par rapport à lui, détecter la forme

des lettres prononcées NON NE MARCHE PAS la forme est similaire à une

homothétie près


Documents

Topological characterization of an image dataset with Betti numbers and a generative model