topological characterization of an image dataset with betti numbers and a generative model
DESCRIPTION
Topological characterization of an image dataset with Betti numbers and a generative model. Context. Multivariate data exploration Signals , images, …. Classical ML techniques Clustering : K- Means ; Gaussian Mixture Models -> convex clusters - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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?
DataSense | 08-07-2014
Retinotopic map of a mouse [Hübener 2003]
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Topology and visual perception
Gestalt psychological theory [1920]The whole is more than summing the partsLaw of continuity, proximity, similarity
WHY TOPOLOGICAL INFORMATION?
DataSense | 08-07-2014
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
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Mental map and topology
Topological invariants as an objective representation
WHY TOPOLOGICAL INFORMATION?
DataSense | 08-07-2014
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
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WHY TOPOLOGICAL INFORMATION?
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
DataSense | 08-07-2014
U
Isometries Similarities Homeomorphisms Homotopies
U U
Initial space
Betti numbersIntrinsic dimension
Probability density functionsGeometry
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SOME HINTS ABOUT TOPOLOGY
Topology in a nutshell
What is the difference between a mug and a doughnut?
DataSense | 08-07-2014
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SOME HINTS ABOUT TOPOLOGY
Topology in a nutshell
What is the difference between a mug and a doughnut?
DataSense | 08-07-2014
Taste is significantly different!
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SOME HINTS ABOUT TOPOLOGY
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
DataSense | 08-07-2014
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
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FROM SETS OF POINTS TO BETTI NUMBERS
Simplex family
Simplex assembly
SIMPLICIAL COMPLEX
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0-simplex 1-simplex 2-simplex 3-simplex
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FROM SETS OF POINTS TO BETTI NUMBERS
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.
DataSense | 08-07-2014
Betti numbers
Computational topology
Simplicial complex
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DataSense | 08-07-2014
FROM SETS OF POINTS TO BETTI NUMBERS
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
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DataSense | 08-07-2014
RESTRICTED DELAUNAY COMPLEX
[Edelsbrunner, Shah 1997]
M1
M2
From manifold to triangulation
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DataSense | 08-07-2014
RESTRICTED DELAUNAY COMPLEX
[Edelsbrunner, Shah 1997]
M1
M2
. From manifold to triangulation
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DataSense | 08-07-2014
RESTRICTED DELAUNAY COMPLEX
Manifold = union of spheres Centered on the atoms’ core (alpha sets the spheres radius
Molecules topology [Edelsbrunner1994]
Alpha-shapes
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DataSense | 08-07-2014
TOPOLOGY REPRESENTING NETWORKS
Topology Representing Network [Martinetz, Schulten 1994]
Connect 1st and 2nd Nearest Neighbor prototype of each data : Competitive Hebbian Learning (CHL)
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DataSense | 08-07-2014
TOPOLOGY REPRESENTING NETWORKS
Topology Representing Network [Martinetz, Schulten 1994]
1er2nd
Connect 1st and 2nd Nearest Neighbor prototype of each data : Competitive Hebbian Learning (CHL)
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DataSense | 08-07-2014
TOPOLOGY REPRESENTING NETWORKS
Topology Representing Network [Martinetz, Schulten 1994]
1er2nd
Connect 1st and 2nd Nearest Neighbor prototype of each data : Competitive Hebbian Learning (CHL)
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DataSense | 08-07-2014
TOPOLOGY REPRESENTING NETWORKS
Topology Representing Network [Martinetz, Schulten 1994]
1er2nd
ROI = Order 2 Voronoi cells
Connect 1st and 2nd Nearest Neighbor prototype of each data : Competitive Hebbian Learning (CHL)
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DataSense | 08-07-2014
TOPOLOGY REPRESENTING NETWORKS
Topology Representing Network [Martinetz, Schulten 1994]
1er2nd
ROI = Order 2 Voronoi cells
Connect 1st and 2nd Nearest Neighbor prototype of each data : Competitive Hebbian Learning (CHL)
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DataSense | 08-07-2014
TOPOLOGY REPRESENTING NETWORKS
Order 2 Voronoi cells
No noise
Sample with gaussian noise
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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
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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
HEAD or TAIL? Moebius strip
P( HEAD ) = ?
DataSense | 08-07-2014
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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
HEAD or TAIL? Moebius strip
P( HEADACHE ) = 1
DataSense | 08-07-2014
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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.
DataSense | 08-07-2014
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Unknown generative manifolds with
possible different topology, different
labels, and possibly overlapping…
GENERATIVE GRAPH [GAILLARD 2010]
…from which are drawn samples with unknown probability
density…
…corrupted with unknown noise…
Statistical generative model – General hypotheses
…leading to the actual data observations.
DataSense | 08-07-2014
Unknown generative manifolds …
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GENERATIVE GRAPH [GAILLARD 2010]
…from which are drawn samples with unknown
probability density…
…corrupted with unknown noise…
Statistical generative model – Simplified hypotheses
DataSense | 08-07-2014
Unknown generative manifolds…
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GENERATIVE GRAPH [GAILLARD 2010]
…from which are drawn samples with unknown
probability density…
…corrupted with unknown noise…
Generative Gaussian Graph (GGG) – Simplified hypotheses
DataSense | 08-07-2014
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
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-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
GENERATIVE GRAPH [GAILLARD 2010]
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)
DataSense | 08-07-2014
Multivariate data GMM
Delaunay
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GENERATIVE SIMPLICIAL COMPLEX [MAILLOT2012]
Generative simplices familly
DataSense | 08-07-2014
A
g0
…(Pseudo-Monte Carlo estimation)
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DATA SAMPLED FROM A GENERATIVE GAUSSIAN SIMPLEX
DataSense | 08-07-2014
d= 0 d= 1 d= 2
σ= 0.1
σ= 0.5
σ= 0.2
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GENERATIVE SIMPLICIAL COMPLEX
DataSense | 08-07-2014
Expectation-Maximization
π1 < π2 < π3 < ………< πi < …… < πn
BIC max
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FROM DATA TO GENERATIVE SIMPLICIAL COMPLEX
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FROM DATA TO GENERATIVE SIMPLICIAL COMPLEX
Protoypes location initialized with GMM
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FROM DATA TO GENERATIVE SIMPLICIAL COMPLEX
Delaunay complex built on top of the prototypes
First the edges…
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FROM DATA TO GENERATIVE SIMPLICIAL COMPLEX
DataSense | 08-07-2014
Delaunay complex built on top of the prototypes
First the edges…Then the surfaces…
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FROM DATA TO GENERATIVE SIMPLICIAL COMPLEX
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
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FROM DATA TO GENERATIVE SIMPLICIAL COMPLEX
Likelihood maximization for dimension 2 components
Proportions of both surfaces and remaining edges are estimated with EM, then pruned wrt BIC
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FROM DATA TO GENERATIVE SIMPLICIAL COMPLEX
Topological cleaning
If a simplex survived, all its facets are pruned.
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RESULTS (1/3)
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SPHERE (1,0,1,0…) TORE (1,2,1,0…)
KLEIN BOTTLE (1,1,0…)
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RESULTS (2/3)
Images data COIL-100 :
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• 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
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RESULTS (3/3)
Images data COIL-100 :
DataSense | 08-07-2014
• 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)
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CONCLUSIONS
• GSC: first generative model to extract Betti numbers from a data set
• No meta-parameter to tune (EM + BIC)
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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?
DataSense | 08-07-2014
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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?
DataSense | 08-07-2014
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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.
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•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
DataSense | 08-07-2014