persistent homology for multivariate data …...motivation understandingthe‘shape’ofdata...
TRANSCRIPT
![Page 1: Persistent homology for multivariate data …...Motivation Understandingthe‘shape’ofdata Instances Attributes Unstructured dataScatterplot matrix Parallel coordinates u.velocity](https://reader034.vdocuments.mx/reader034/viewer/2022050422/5f9151a12f022a146754d1b8/html5/thumbnails/1.jpg)
Persistent homology for multivariate datavisualization
Bastian Rieck
Interdisciplinary Center for Scientific ComputingHeidelberg University
![Page 2: Persistent homology for multivariate data …...Motivation Understandingthe‘shape’ofdata Instances Attributes Unstructured dataScatterplot matrix Parallel coordinates u.velocity](https://reader034.vdocuments.mx/reader034/viewer/2022050422/5f9151a12f022a146754d1b8/html5/thumbnails/2.jpg)
MotivationUnderstanding the ‘shape’ of data
Attributes
Inst
ance
s
Scatterplot matrix Parallel coordinatesUnstructured data
u.ve
loci
tyv.
velo
city
air.t
empe
ratu
rem
ean.
sea.
leve
l.pre
ssur
esu
rface
.tem
pera
ture
tota
l.pre
cipi
tatio
n
u.velocity v.velocity air.temperature mean.sea.level.pressure surface.temperature total.precipitation
−10
0
10
20
Corr:0.134
Corr:−0.096
Corr:−0.104
Corr:−0.0927
Corr:0.0791
−10
0
10
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●● Corr:−0.0525
Corr:0.00273
Corr:−0.0626
Corr:−0.0639
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0e+00 2e−04 4e−04 6e−04
0.00
0.25
0.50
0.75
1.00
u.velocity v.velocity air.temperature mean.sea.level.pressure surface.temperature total.precipitationvariable
valu
e
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Agenda
1 Theory: Algebraic topology
2 Theory: Persistent homology
3 Applications
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Part I
Theory: Algebraic topology
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Algebraic topology
Algebraic topology is the branch of mathematics that usestools from abstract algebra to studymanifolds. The basicgoal is to find algebraic invariants that classify topologicalspaces up to homeomorphism.
Adapted from https://en.wikipedia.org/wiki/Algebraic_topology.
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Manifolds
A d-dimensional Riemannian manifoldM in someRn, with d � n,is a space where every point p ∈M has a neighbourhood that‘locally looks’ likeRd.
A 2-dimensional manifold
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Homeomorphisms
A homeomorphism between two spaces X and Y is a continuousfunction f : X → Y whose inverse f−1 : Y → X exists and iscontinuous as well.
Intuitively, we may stretch, bend—but not tear and glue the twospaces.
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Algebraic invariants
An invariant is a property of an object that remains unchangedupon transformations such as scaling or rotations.
Example
Dimension is a simple invariant: R2 6= R3 because 2 6= 3.
In general
LetM be the family of manifolds. An invariant permits us to definea function f : M×M → {0, 1} that tells us whether two manifoldsare different or ‘equal’ (with respect to that invariant).
No invariant is perfect—there will be objects that have the sameinvariant even though they are different.
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Betti numbersA useful topological invariant
Informally, they count the number of holes in different dimensionsthat occur in a data set.
β0 Connected componentsβ1 Tunnelsβ2 Voids...
...
Space β0 β1 β2
Point 1 0 0Circle 1 1 0Sphere 1 0 1Torus 1 2 1
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Signature property
If βXi 6= βY
i , we know that X 6∼= Y. The converse is not true,unfortunately:
Space β0 β1
X 1 1Y 1 1
We have β0 = 1 and β1 = 1 for X and Y, but still X 6∼= Y.
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Simplicial complexes
0-simplex 1-simplex 2-simplex 3-simplex
Valid Invalid
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Simplicial complexesExample
The simplicial complex representation is compact and permits thecalculation of the Betti numbers using an efficient matrix reductionscheme.
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Basic ideaCalculating boundaries
The boundary of the triangle is:
∂2{a, b, c} = {b, c}+ {a, c}+ {a, b}
The set of edges does not have boundary:
∂1 ({b, c}+ {a, c}+ {a, b})
= {c}+ {b}+ {c}+ {a}+ {b}+ {a}
= 0
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Fundamental lemma
For all p, we have ∂p−1 ◦ ∂p = 0: Boundaries do not have a boundarythemselves.
This permits us to calculate Betti numbers of simplicial complexesby reducing a boundarymatrix to its Smith Normal Form usingGaussian elimination.
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Summary
We want to differentiate between different objects.
This endeavour requires algebraic invariants.
One invariant, the Betti numbers, measures intuitive aspects ofour data.
Their calculation requires a simplicial complex and a boundaryoperator.
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Summary
We want to differentiate between different objects.
This endeavour requires algebraic invariants.
One invariant, the Betti numbers, measures intuitive aspects ofour data.
Their calculation requires a simplicial complex and a boundaryoperator.
Bastian Rieck Persistent homology for multivariate data visualization 13
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Summary
We want to differentiate between different objects.
This endeavour requires algebraic invariants.
One invariant, the Betti numbers, measures intuitive aspects ofour data.
Their calculation requires a simplicial complex and a boundaryoperator.
Bastian Rieck Persistent homology for multivariate data visualization 13
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Summary
We want to differentiate between different objects.
This endeavour requires algebraic invariants.
One invariant, the Betti numbers, measures intuitive aspects ofour data.
Their calculation requires a simplicial complex and a boundaryoperator.
Bastian Rieck Persistent homology for multivariate data visualization 13
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Summary
We want to differentiate between different objects.
This endeavour requires algebraic invariants.
One invariant, the Betti numbers, measures intuitive aspects ofour data.
Their calculation requires a simplicial complex and a boundaryoperator.
Bastian Rieck Persistent homology for multivariate data visualization 13
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Part II
Theory: Persistent homology
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Real-world multivariate data
Unstructured point clouds
n items withD attributes; n×Dmatrix
Non-random sample fromRD
Manifold hypothesis
There is an unknown d-dimensionalmanifoldM ⊆ RD, with d � D, from whichour data have been sampled.
2-manifold inR3
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Agenda
1 Convert our input data into a simplicial complex K.
2 Calculate the Betti numbers of K.
3 Use the Betti numbers to compare data sets.
(Fair warning: It won’t be so simple)
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Converting unstructured data into a simplicial complex
Require: Distance measure (e.g. Euclidean distance), maximumscale threshold ε.
Construct the Vietoris–Rips complex Vε by adding a k-simplexwhenever all of its (k− 1)-dimensional faces are present.
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Calculating Betti numbers directly from VεUnstable behaviour
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Calculating persistent Betti numbersPersistent homology
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How does this work in practice?An example for β0
Have a function f : Vε → R on the vertices of the Vietoris–Ripscomplex.
Extend it to a function on the whole simplicial complex bysetting f(σ) = max{f(v) | v ∈ σ}.
Analyse the connectivity changes in the sublevel sets of f, i.e.sets of the form
L−α(f) = {v | f(v) 6 α}.
This can be done by traversing the values of f in increasingorder and stopping at ‘critical points’.
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How does this work in practice?An example for β0
Have a function f : Vε → R on the vertices of the Vietoris–Ripscomplex.
Extend it to a function on the whole simplicial complex bysetting f(σ) = max{f(v) | v ∈ σ}.
Analyse the connectivity changes in the sublevel sets of f, i.e.sets of the form
L−α(f) = {v | f(v) 6 α}.
This can be done by traversing the values of f in increasingorder and stopping at ‘critical points’.
Bastian Rieck Persistent homology for multivariate data visualization 19
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How does this work in practice?An example for β0
Have a function f : Vε → R on the vertices of the Vietoris–Ripscomplex.
Extend it to a function on the whole simplicial complex bysetting f(σ) = max{f(v) | v ∈ σ}.
Analyse the connectivity changes in the sublevel sets of f, i.e.sets of the form
L−α(f) = {v | f(v) 6 α}.
This can be done by traversing the values of f in increasingorder and stopping at ‘critical points’.
Bastian Rieck Persistent homology for multivariate data visualization 19
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How does this work in practice?An example for β0
Have a function f : Vε → R on the vertices of the Vietoris–Ripscomplex.
Extend it to a function on the whole simplicial complex bysetting f(σ) = max{f(v) | v ∈ σ}.
Analyse the connectivity changes in the sublevel sets of f, i.e.sets of the form
L−α(f) = {v | f(v) 6 α}.
This can be done by traversing the values of f in increasingorder and stopping at ‘critical points’.
Bastian Rieck Persistent homology for multivariate data visualization 19
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Persistent homology & persistence diagramsOne-dimensional example
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Uses for persistence diagrams
A persistence diagram is a multi-scale summary of topologicalactivity in a data set. But the diagrams go well and beyond a simplecomparison of Betti numbers!
L’algèbre est généreuse, elle donne souvent plus qu’on luidemande.
—D’Alembert
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Distance calculations
W2(X, Y) =
√inf
η : X→Y
∑x∈X
‖x− η(x)‖2∞
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Distance calculations
W2(X, Y) =
√inf
η : X→Y
∑x∈X
‖x− η(x)‖2∞
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Distance calculations
W2(X, Y) =
√inf
η : X→Y
∑x∈X
‖x− η(x)‖2∞
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Sensitivity of distancesWasserstein versus function space distances
Only the Wasserstein distance does not distort the ‘shape’ of noisein the data.
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Sensitivity of distancesWasserstein versus function space distances
Only the Wasserstein distance does not distort the ‘shape’ of noisein the data.
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Part III
Applications
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Published projects
Persistence rings Simplicial chain graphs
bagged trees
regression trees
enet
enet tuned
knn
lm tuned
lm
m5
mars
plspls tuned
random forest
ridge
ridge tuned
rlm pca
rlm
rpart
svm
svm tuned
cubist
Model landscapesRieck, Mara, Leitte: Multivariate data analysis using persistence-based �ltering and topological signatures
Rieck, Leitte: Structural analysis of multivariate point clouds using simplicial chains Rieck, Leitte: Enhancing comparative model analysis using persistent homology
MDS1.65
t-SNE2.52
Isomapk=8, 3.55
Evaluating embeddingsRieck, Leitte: Persistent homology for the evaluation of dimensionality reduction schemes
Agreement analysisRieck, Leitte: Agreement analysis of quality measures for dimensionality reduction
Data descriptor landscapesRieck, Leitte: Comparing dimensionality reduction methods using data descriptor landscapes
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Published projects
Persistence rings Simplicial chain graphs
bagged trees
regression trees
enet
enet tuned
knn
lm tuned
lm
m5
mars
plspls tuned
random forest
ridge
ridge tuned
rlm pca
rlm
rpart
svm
svm tuned
cubist
Model landscapesRieck, Mara, Leitte: Multivariate data analysis using persistence-based �ltering and topological signatures
Rieck, Leitte: Structural analysis of multivariate point clouds using simplicial chains Rieck, Leitte: Enhancing comparative model analysis using persistent homology
MDS1.65
t-SNE2.52
Isomapk=8, 3.55
Evaluating embeddingsRieck, Leitte: Persistent homology for the evaluation of dimensionality reduction schemes
Agreement analysisRieck, Leitte: Agreement analysis of quality measures for dimensionality reduction
Data descriptor landscapesRieck, Leitte: Comparing dimensionality reduction methods using data descriptor landscapes
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Qualitative visualizations: Persistence rings
0.5 1
0.5
1
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Qualitative visualizations: Persistence rings
0.5 1
0.5
1
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Qualitative visualizations: Persistence rings
0.5 1
0.5
1
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Qualitative visualizations: Simplicial chain graphs
MotivationAnalyse the connectivity of topological features for ensembles ofdata sets: Different runs of an experiment, different times at whichmeasurements are being taken…
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Central idea
Obtain geometrical descriptions of topological features (‘holes’)while calculating persistent homology.
This is known as the ‘localization problem’ in persistent homology.
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Solving the localization problem
1 Define a geodesic ball in a simplicial complex.
2 Solve all-pairs-shortest-paths problem to find possible sites.
3 Branch-and-bound strategy to improve performance.
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Simplicial chain graphBasic example
Advantage: The space in which we localize the features usually hasa high dimensions, but the graph will always be drawn inR2.
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ResultsData: Tropical Atmosphere Ocean Array
1995 199719961994
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Lessons learned
1 We can obtain features via persistent homology that permit acomparative analysis.
2 Visualizing these features becomes abstract very quickly.
3 Need more ‘quantitative’ topological visualizations.
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Quantitative visualizations: Model landscapes
Example: Solubility analysis
19 different mathematical models
1267 chemical compounds, described by228-dimensional feature vectors
Measured ground truth (solubility values)
Each model is a function f : D→ R. How to evaluate similarities &differences between the models?
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State-of-the-art
Existingmeasures (RMSE or R2) only focus on values of amodel.
The structure/shape is not being used!
Shortcomings: Sensitivity to noise, ‘masking’ the influence ofoutliers…
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Our approach
1 Calculate Vietoris–Rips complex Vε on the molecular descriptors.
2 Use model values & ground truth to obtain a set of functions on Vε.
3 Calculate persistence diagrams for each function.
4 Compare diagrams using the Wasserstein distance.
5 Absolute comparison with ground truth diagram.
6 Relative comparison with all diagrams.
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Our approach
1 Calculate Vietoris–Rips complex Vε on the molecular descriptors.
2 Use model values & ground truth to obtain a set of functions on Vε.
3 Calculate persistence diagrams for each function.
4 Compare diagrams using the Wasserstein distance.
5 Absolute comparison with ground truth diagram.
6 Relative comparison with all diagrams.
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Our approach
1 Calculate Vietoris–Rips complex Vε on the molecular descriptors.
2 Use model values & ground truth to obtain a set of functions on Vε.
3 Calculate persistence diagrams for each function.
4 Compare diagrams using the Wasserstein distance.
5 Absolute comparison with ground truth diagram.
6 Relative comparison with all diagrams.
Bastian Rieck Persistent homology for multivariate data visualization 34
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Our approach
1 Calculate Vietoris–Rips complex Vε on the molecular descriptors.
2 Use model values & ground truth to obtain a set of functions on Vε.
3 Calculate persistence diagrams for each function.
4 Compare diagrams using the Wasserstein distance.
5 Absolute comparison with ground truth diagram.
6 Relative comparison with all diagrams.
Bastian Rieck Persistent homology for multivariate data visualization 34
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Our approach
1 Calculate Vietoris–Rips complex Vε on the molecular descriptors.
2 Use model values & ground truth to obtain a set of functions on Vε.
3 Calculate persistence diagrams for each function.
4 Compare diagrams using the Wasserstein distance.
5 Absolute comparison with ground truth diagram.
6 Relative comparison with all diagrams.
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Our approach
1 Calculate Vietoris–Rips complex Vε on the molecular descriptors.
2 Use model values & ground truth to obtain a set of functions on Vε.
3 Calculate persistence diagrams for each function.
4 Compare diagrams using the Wasserstein distance.
5 Absolute comparison with ground truth diagram.
6 Relative comparison with all diagrams.
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Visualization of relative model differences
bagged trees
regression trees
enet
enet tuned
knn
lm tuned
lm
m5
mars
plspls tuned
random forest
ridge
ridge tuned
rlm pca
rlm
rpart
svm
svm tuned
cubist
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Why is m5 rated differently?Comparison with cubist
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Summary
Topological methods yield quantitative and qualitative informationabout data sets—often, this goes well and beyond the scope ofregular geometric approaches.
Future work
1 Performance improvements: Smaller complexes, otherdistance measures, …
2 Ensemble data & ‘average’ topological structures
3 Connection to geometric features in data
Thank you for your attention!
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Summary
Topological methods yield quantitative and qualitative informationabout data sets—often, this goes well and beyond the scope ofregular geometric approaches.
Future work
1 Performance improvements: Smaller complexes, otherdistance measures, …
2 Ensemble data & ‘average’ topological structures
3 Connection to geometric features in data
Thank you for your attention!
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Summary
Topological methods yield quantitative and qualitative informationabout data sets—often, this goes well and beyond the scope ofregular geometric approaches.
Future work
1 Performance improvements: Smaller complexes, otherdistance measures, …
2 Ensemble data & ‘average’ topological structures
3 Connection to geometric features in data
Thank you for your attention!
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Part IV
Bonus slides
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Quantitative visualizations: Dimensionality reduction
Which of these embeddings are suitable for my data?
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Data descriptorsDefinition & examples
Any function f : D ⊆ Rn → R is an admissible data descriptor. fshould measure ‘salient properties’ of a multivariate data set.
Density Eccentricity Local linearity
Low High
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Data descriptorsDefinition & examples
Any function f : D ⊆ Rn → R is an admissible data descriptor. fshould measure ‘salient properties’ of a multivariate data set.
Density
Eccentricity Local linearity
Low High
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Data descriptorsDefinition & examples
Any function f : D ⊆ Rn → R is an admissible data descriptor. fshould measure ‘salient properties’ of a multivariate data set.
Density Eccentricity
Local linearity
Low High
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Data descriptorsDefinition & examples
Any function f : D ⊆ Rn → R is an admissible data descriptor. fshould measure ‘salient properties’ of a multivariate data set.
Density Eccentricity Local linearity
Low High
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Workflow
Persistenthomology
Datadescriptors
Embeddings
Original data
W2
W2
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Visualizing global quality
QualityEmbedding = W2
(DOriginal, DEmbedding
)HLLE
1.66
t-SNE2.26
Isomap2.26
PCA3.42
SPE10.67
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Correspondence to our intuitionGlobal quality information
HLLE1.66
t-SNE2.26
Isomap2.26
PCA3.42
SPE10.67
t-SNE
Isomap PCA SPE
HLLEOriginal Data
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Local quality information
Quality
GoodMediumBad
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ResultsIsomap faces
Image source: A Global Geometric Framework for Nonlinear Dimensionality Reduction (Joshua B. Tenenbaum, Vin de Silva,John C. Langford), 2000.
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ResultsIsomap faces: Parameter study
MDS1.65
t-SNE2.52
Isomapk=8, 3.55
Isomapk=16, 3.62
Isomapk=10, 5.07
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Quantitative visualizations: Multiple data descriptors
Data descriptors
Dat
a se
tEm
be
dd
ing
s
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Feature vector approach
Embedding Data descriptors
Feature vector
Data W2 W2 W2
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Data descriptor landscapes
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Data descriptor landscapes
density
eccentricity
linearity
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Data descriptor landscapes
density
eccentricity
linea
rity
Isomap (k=8)
Isomap (k=9)
HLLE LTSA
LLE
PCA
Low (better) High (worse)
Norm
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Data descriptor landscapes
density
eccentricity
linea
rity
Isomap (k=8)
Isomap (k=9)
HLLE
LLE
PCA
Low (better) High (worse)
Norm
LTSA
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Data descriptor landscapesGrouping behaviour
SPE (k=20)
PCA
FA (n=1000)
LLE (k=30)density
eccentricity
linea
rity
FA (n=2000)
FA (n=5000)
LLE (k=25)
LLE (k=50)
LLE (k=45)
LLE (k=55)
LLE (k=99)
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Data descriptor landscapesGrouping behaviour
PCA
LLE (k=30)density
eccentricity
linea
rity
FA (n=2000)
FA (n=5000)
LLE (k=25)
LLE (k=50)
LLE (k=45)
LLE (k=99)
FA (n=1000)
LLE (k=55)
SPE (k=20)
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