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Statistical Inference for Geometric Data

Jisu KIM

Carnege Mellon University

Sep 19, 2018

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Introduction

Minimax Rates for Geometric Parameters of a ManifoldMinimax Rates for Estimating the Dimension of a ManifoldThe Origin of the Reach: Better Understanding Regularity ThroughMinimax Estimation Theory

Statistical Inference For Homological FeaturesStatistical Inference for Cluster Trees

Statistical Inference and Computation for Persistent HomologyPersistent homology of KDE filtration on rips complexR Package TDA: Statistical Tools for Topological Data Analysis

Conclusion

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High dimensional data suffers from the curse ofdimensionality.

1[Hastie et al., 2009, Ch2, Figure 2.6]3 / 62

The curse of dimensionality is mitigated when there is a lowdimensional geometric structure.

2http://www.skybluetrades.net/blog/posts/2011/10/30/machine-learning/4 / 62

Geometric structures in the data provide information.

3http://www.mpa-garching.mpg.de/galform/virgo/millennium/poster_half.jpg5 / 62

Statistic Inference for Geometric Data is explored.

I Minimax Rates for Geometric Parameters of a ManifoldI Minimax Rates for Estimating the Dimension of a Manifold (Kim,

Rinaldo, Wasserman, 2016)I The Origin of the Reach: Better Understanding Regularity Through

Minimax Estimation Theory (Aamari, Kim, Chazal, Michel, Rinaldo,Wasserman, 2017)

I Statistical Inference For Homological FeaturesI Statistical Inference for Cluster Trees (Kim, Chen, Balakrishnan,

Rinaldo, Wasserman, 2016)I Statistical Inference and Computation for Persistent Homology

I Statistical inference on persistent homology of KDE filtration on ripscomplex (Shin, Kim, Rinaldo, Wasserman, 2018?)

I R Package TDA: Statistical Tools for Topological Data Analysis(Fasy, Kim, Lecci, Maria, Milman, Rouvreau, 2014a)

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Introduction

Conclusion

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A manifold is a low dimensional geometric structure thatlocally resembles Euclidean space.

4http://www.skybluetrades.net/blog/posts/2011/10/30/machine-learning/8 / 62

The maximum risk of an estimator is its worst expectederror.

I the maximum risk of an estimator θn is the worst expected error thatthe estimator θn can make.

supP∈P

[`(θn(X ), θ(P)

I X = (X1, · · · ,Xn) is drawn from a fixed distribution P, where P iscontained in set of distributions P.

I estimator θn is any function of data X .I The loss function `(·, ·) measures the error of the estimator θn.

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The minimax rate describes the statistical difficulty ofestimating a parameter.

I The minimax rate Rn is the risk of an estimator that performs bestin the worst case, as a function of sample size.

Rn = infθn

supP∈P

[`(θn(X ), θ(P)

I estimator θn is any function of data X .I The loss function `(·, ·) measures the error of the estimator θn.

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We measure the statistical difficulty of estimating geometricparameters of a manifold by their minimax rate.

I Minimax Rates for Estimating the Dimension of a Manifold (Kim,Rinaldo, Wasserman, 2016)

I The Origin of the Reach: Better Understanding Regularity ThroughMinimax Estimation Theory (Aamari, Kim, Chazal, Michel, Rinaldo,Wasserman, 2017)

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Introduction

Conclusion

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Manifold learning finds an underlying manifold to reducedimension.

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The intrinsic dimension of a manifold needs to be estimated.

I Most manifold learning algorithms require the intrinsic dimension ofthe manifold as input.

I The intrinsic dimension is rarely known in advance and therefore hasto be estimated.

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Minimax rate for estimating the dimension

Rn = infˆdimn

supP∈P

ˆdimn(X ) 6= dim(P))]

I estimator ˆdimn is any function of data X .I 0− 1 loss function is considered, so for all x , y ∈ R,`(x , y) = 1(x 6= y).

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Minimax rate for estimating the dimension

Theorem(Proposition 28 and 29)

n−2n . infˆdimn

supP∈P

ˆdimn 6= dim(P))]

. n−1

m−1 n.

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Introduction

Conclusion

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The reach is the maximum radius of a ball that can roll overthe manifold.

DefinitionThe reach of M, denoted by τ(M), can be defined as

τ(M) = infa 6=b∈M

‖b − a‖222d(b − a, TaM)

where TaM is the tangent space of M at a.

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The reach is a regularity parameter in many geometricalinference problem.

I The reach is a key paramter in:I Dimension estimationI Homology inferenceI Volume estimationI Manifold clusteringI Diffusion maps

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Minimax rate for estimating the reach

Rn = infτn

supP∈P

[∣∣∣∣1

τ(P)− 1τn(X )

∣∣∣∣q]

I estimator τn is any function of data X .I inverse lq loss function is considered, so for all x , y ∈ R,`(x , y) =

∣∣∣ 1x− 1

∣∣∣q.

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An estimator of the reach

The reach of M is

τ(M) = infa 6=b∈M

‖b − a‖222d(b − a, TaM)

DefinitionGiven observation X = (X1, . . . ,Xn), we estimate the reach as

τn(X ) = inf1≤i 6=j≤n

‖Xj − Xi‖222d(Xj − Xi , TXiM)

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Minimax rate for estimating the reach

Theorem(Theorem 45 and Proposition 50)

n−qd . inf

τnsupP∈P

[∣∣∣∣1

τ(P)− 1τn

∣∣∣∣q]

. n−2q

3d−1 .

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Introduction

Conclusion

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Geometric holes in the data provide information.

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The number of holes is used to summarize geometricalfeatures.

I Geometrical objects :I A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W,

X, Y, Z,

I , , , , .5

I The number of holes of different dimensions is considered.

1. β0 =# of connected components

2. β1 =# of loops (holes inside 1-dim sphere)

3. β2 =# of voids (holes inside 2-dim sphere) : if dim ≥ 3

5Professor Valerie Ventura25 / 62

Example : Objects are classified by homologies.

1. β0 =# of connected components

2. β1 =# of loops

β0 \ β1 0 1 2

C, G, I, J, L, M,

1 N, S, U, V, W, Z, A, R, D, O, , P, Q BE, F, T, Y, H, K, X

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Statistical inference for homological features.

I Statistical Inference for Cluster Trees (Kim, Chen, Balakrishnan,Rinaldo, Wasserman, 2016)

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Introduction

Conclusion

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The cluster tree is the hierarchy of the high density clusters.

DefinitionFor a density function p, its cluster tree Tp is a function where Tp(λ) isthe set of connected components of the upper level set x : p(x) ≥ λ.

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A confidence set helps denoising the empirical tree.I An asymptotic 1− α confidence set Cα is a collection of trees with

the property that

P(Tp ∈ Cα) = 1− α+ o(1).

Ring data, alpha = 0.05la

0.0 0.2 0.4 0.6 0.8 1.0

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We use the bootstrap to compute 1− α confidence set Cα.

I We let Tph be the cluster tree from the kernel density estimator ph,where

ph(x) =1

(x − Xi

and the confidence set as the ball centered at Tph and radius tα, i.e.

Cα = T : d∞(T ,Tph) ≤ tα .

Theorem(Theorem 57) Above confidence set Cα satisfies

P(Th ∈ Cα

)= 1− α+ O

(log7 n

)1/6 .

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The pruned trees according to the confidence set recover theactual cluster trees.

Ring data, alpha = 0.05

0.0 0.2 0.4 0.6 0.8 1.0

Mickey mouse data, alpha = 0.05

0.0 0.2 0.4 0.6 0.8 1.0

Yingyang data, alpha = 0.05

0.0 0.2 0.4 0.6 0.8 1.00

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Introduction

Conclusion

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Homology of finite sample is different from homology ofunderlying manifold, hence it cannot be directly used for theinference.

I When analyzing data, we prefer robust features where features of theunderlying manifold can be inferred from features of finite samples.

I Homology is not robust:

Underlying circle: β0 = 1, β1 = 1

100 samples: β0 = 100, β1 = 0

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Persistent homology computes homologies on collection ofsets, and tracks when topological features are born andwhen they die.

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Persistent homology of the underlying manifold can beinferred from persistent homology of finite samples.

Circle

0.0 0.5 1.0 1.5

200 samples

0.0 0.5 1.0 1.50.

5Birth

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Confidence band for persistent homology separateshomological signal from homological noise.

Circle

0.0 0.5 1.0 1.5

200 samples

0.0 0.5 1.0 1.50.

5Birth

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Statistical inference for persistent homology.

I Persistent homology of KDE filtration on rips complex (Shin, Kim,Rinaldo, Wasserman, 2018?)

I R Package TDA: Statistical Tools for Topological Data Analysis(Fasy, Kim, Lecci, Maria, Milman, Rouvreau, 2014a)

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Introduction

Conclusion

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Computing a confidence band for the persistent homologyincurs computing on a grid of points, which is infeasible inhigh dimensional space.

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Computing the persistent homology of density function ondata points reduces computational complexity.

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How can we compute a confidence band for the persistenthomology with computation on data points?

I (Shin, Kim, Rinaldo, Wasserman, 2018?) : extending work from Fasyet al. [2014b], Bobrowski et al. [2014], Chazal et al. [2011].

Circle

0.0 0.5 1.0 1.5

200 samples

0.0 0.5 1.0 1.50.

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We rely on the kernel density estimator to extracttopological information of the underlying distribution.

I The kernel density estimator is

ph(x) =1

(x − Xi

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We are considering the upper level set of the average kerneldensity estimator on the support.

I Let X1, . . . ,Xn ∼ P, then the average kernel density estimator is

ph(x) = E [ph(x)] =1hd

(x − X

I We are considering the upper level sets of the average kernel densityestimator

DLL>0 , where DL := x ∈ supp(P) : ph(x) ≥ L .

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We are considering the upper level set of the average kerneldensity estimator on the support.

I We are considering the upper level sets of the average KDE

DLL>0 , where DL := x ∈ supp(P) : ph(x) ≥ L .

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We are targeting the persistent homology of the upper levelset of the average kernel density estimator on the support.

I We are considering the upper level sets of the average KDE

DLL>0 , where DL := x ∈ supp(P) : ph(x) ≥ L ,

and targeting its persistent homology PHsupp(P)∗ (ph).

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We use the Rips complex to estimate the target persistenthomology.

I For X ⊂ Rm and r > 0, the Rips complex R(X , r) is defined as

R(X , r) =[Xi1 , . . . ,Xik ] : d(Xij ,Xil ) < 2r , 1 ≤ ∀j 6= l ≤ k, k = 1, . . . , n

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We estimate the target level set by considering the Ripscomplex generated from the level set of the KDE.

I For X ⊂ Rm and r > 0, the Rips complex R(X , r) is defined as

R(X , r) =[Xi1 , . . . ,Xik ] : d(Xij ,Xil ) < 2r , 1 ≤ ∀j 6= l ≤ k, k = 1, . . . , n

I The KDE (kernel density estimator) is

ph(x) =1

(x − Xi

I For Xn = X1, . . . ,Xn, we consider the Rips complex generatedfrom the level set of the KDE

R(X ph

n,L, r)

L>0, where X ph

n,L := Xi ∈ Xn : ph(Xi ) ≥ L .

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We estimate the target level set by considering the Ripscomplex generated from the level set of the KDE.

I For Xn = X1, . . . ,Xn, we estimate the target level set by the levelsets of the KDE on Rips complexes,

R(X ph

n,L, r)

L>0, where X ph

n,L := Xi ∈ Xn : ph(Xi ) ≥ L .

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We estimate the target persistent homology by thepersistent homology of the KDE filtration on Ripscomplexes.

I We estimate the target persistent homology by the persistenthomology of the level sets of the KDE on Rips complexes,

R(X ph

n,L, r)

L>0, where X ph

n,L = Xi ∈ Xn : ph(Xi ) ≥ L .

and denote the persistent homology as PHR∗ (ph, r).

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We estimate the target level set by Rips complexes from theKDE level sets.

I We approximate the target level set

DLL>0 , where DL := x ∈ supp(P) : ph(x) ≥ L ,by the level sets of the KDE on Rips complexes,

R(X ph

n,L, r)

L>0, where X ph

n,L = Xi ∈ Xn : ph(Xi ) ≥ L .

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We estimate the target persistent homology by thepersistent homology of the KDE filtration on Ripscomplexes.

I We estimate the target persistent homology

PHsupp(P)∗ (ph),

by the persistent homology of the KDE filtration on Rips complexes,

PHR∗ (ph, r).

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The persistent homology of the KDE filtration on Ripscomplexes is consistent.

Theorem(Theorem 74)

dB(PHR∗ (phn , rn),PH

supp(P)∗ (phn)

(√log(1/hn)

nhdn+ ‖rn‖∞

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Confidence set

I An asymptotic 1− α confidence set Cα is a random set of persistenthomologies satisfying

P(PHsupp(P)∗ (phn) ∈ Cα) ≥ 1− α+ o(1).

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Confidence set for the persistent homology of the KDEfiltration.

I We let the confidence set as the ball centered at PHR∗ (phn , rn) and

radius bα, i.e.

Cα =P :, dB

(P,PHR

∗ (phn , rn))≤ bα

This is a valid confidence set by the following theorem.

Theorem(Theorem 78)

supp(P)∗ (phn) ∈ Cα

)≥ 1− α+ o(1).

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Introduction

Conclusion

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R Package TDA provides an R interface for C++ librariesfor Topological Data Analysis.

I website:https://cran.r-project.org/web/packages/TDA/index.html

I Author: Brittany Terese Fasy, Jisu Kim, Fabrizio Lecci, ClémentMaria, David Milman, and Vincent Rouvreau.

I R is a programming language for statistical computing and graphics.I R has short development time, while C/C++ has short execution

time.I R package TDA provides an R interface for C++ library

GUDHI/Dionysus/PHAT, which are for Topological Data Analysis.

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R Package TDA provides an R interface for C++ librariesfor Topological Data Analysis.

I # of downloads (2014-08-18 - 2018-09-19) : 21820

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Introduction

Conclusion

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Presented papers

I Jaehyeok Shin, Jisu Kim, Alessandro Rinaldo, Larry Wasserman,Persistent homology of KDE filtration on rips complex

I Eddie Aamari, Jisu Kim, Frédéric Chazal, Bertrand Michel,Alessandro Rinaldo, Larry Wasserman, Estimating the Reach of aManifold [2017] [ arXiv | HAL | BibTeX ]

I Jisu Kim, Yen-Chi Chen, Sivaraman Balakrishnan, AlessandroRinaldo, Larry Wasserman, Statistical Inference for Cluster Trees[2017] [ arXiv | NIPS | BibTeX ]

I Jisu Kim, Alessandro Rinaldo, Larry Wasserman, Minimax Rates forEstimating the Dimension of a Manifold [2016] [ arXiv | BibTeX ]

I Brittany T. Fasy, Jisu Kim, Fabrizio Lecci, Clément Maria, David L.Millman, Vincent Rouvreau, Introduction to the R package TDA.[2014] [ arXiv | BibTeX ]

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Other papers

I Kwangho Kim, Jisu Kim, Barnabás Póczos, Distribution Regressionin Semi-supervised Learning

I Kwangho Kim, Jisu Kim, Edward H. Kennedy, Causal effects basedon distributional distances [2018] [ arXiv | BibTeX ]

I Kijung Shin, Bryan Hooi, Jisu Kim, Christos Faloutsos, Think beforeYou Discard: Accurate Triangle Counting in Graph Streams withDeletions [2018] [ paper | appendix | BibTeX ]

I Kijung Shin, Bryan Hooi, Jisu Kim, Christos Faloutsos, DenseAlert:Incremental Dense-Subtensor Detection in Tensor Streams [2017] [paper | appendix | BibTeX ]

I Kijung Shin, Bryan Hooi, Jisu Kim, Christos Faloutsos, D-Cube:Dense-Block Detection in Terabyte-Scale Tensors [2017] [ paper |appendix | BibTeX ]

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Thank you!

Eddie Aamari SivaramanBalakrishnan

Frédéric Chazal

Yen-Chi Chen Jessi Cisewski Christos Faloutsos

Brittany T. Fasy

Stephan Günnemann

Giseon Heo Bryan Hooi Edward Kennedy

Kwangho Kim Jay-Yoon Lee Clément Maria

Bertrand Michel

David L. Millman

Jaeseong Oh BarnabásPóczos

Alessandro Rinaldo

Vincent Rouvreau

Jaehyeok Shin Kijung Shin Daren Wang Larry Wasserman

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Stability and Statistical Inference for Persistent Homology

Minimax Rates for Estimating the Dimension of a ManifoldRegularity conditionsUpper BoundLower BoundUpper Bound and Lower Bound for General Case

The Origin of the Reach: Better Understanding Regularity ThroughMinimax Estimation Theory

Reach and its GeometryReach estimator and its analysisMinimax Estimates

Statistical Inference for Cluster Trees

Reference

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Bottleneck distance gives a metric on the space of persistenthomology.

DefinitionLet D1, D2 be multiset of points. Bottleneck distance is defined as

dB(D1, D2) = infγ

supx∈D1

‖x − γ(x)‖∞,

where γ ranges over all bijections from D1 to D2.

0.0 0.5 1.0 1.5

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Bottleneck distance gives a metric on the space of persistenthomology.

DefinitionLet D1, D2 be multiset of points. Bottleneck distance is defined as

dB(D1, D2) = infγ

supx∈D1

‖x − γ(x)‖∞,

where γ ranges over all bijections from D1 to D2.

0.0 0.5 1.0 1.5

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Confidence band for the persistent homology is a randomquantity containing the persistent homology with highprobability.

Let M be a compact manifold, and X = X1, · · · ,Xn be n samples. LetfM and fX be corresponding functions whose persistent homology is ofinterest. Given the significance level α ∈ (0, 1), (1− α) confidence bandcn = cn(X ) is a random variable satisfying

P (dB(Dgm(fM), Dgm(fX )) ≤ cn) ≥ 1− α.

Circle

0.0 0.5 1.0 1.5

200 samples

0.0 0.5 1.0 1.5

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Confidence band for the persistent homology is a randomquantity containing the persistent homology with highprobability.

Let M be a compact manifold, and X = X1, · · · ,Xn be n samples. LetfM and fX be corresponding functions whose persistent homology is ofinterest. Given the significance level α ∈ (0, 1), (1− α) confidence bandcn = cn(X ) is a random variable satisfying

P (dB(Dgm(fM), Dgm(fX )) ≤ cn) ≥ 1− α.

Circle

0.0 0.5 1.0 1.5

200 samples

0.0 0.5 1.0 1.5

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Confidence band for the persistent homology can becomputed using the bootstrap algorithm.

1. Given a sample X = x1, . . . , xn, compute the kernel densityestimator ph.

2. Draw X ∗ = x∗1 , . . . , x∗n from X = x1, . . . , xn (with replacement),and compute θ∗ =

√n||p∗h(x)− ph(x)||∞, where p∗h is the density

estimator computed using X ∗.3. Repeat the previous step B times to obtain θ∗1 , . . . , θ

4. Compute qα = infq : 1

∑Bj=1 I (θ

∗j ≥ q) ≤ α

5. The (1− α) confidence band for E[ph] is[ph − qα√

n, ph +

qα√n

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Reference

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Reference

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The supporting manifold M is assumed to be bounded.

M ⊂ I := [−KI ,KI ]m ⊂ Rm with KI ∈ (0,∞)

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The reach is assumed to be lower bounded to avoid anarbitrarily complicated manifold.

I P is a set of distributions P that is supported on a bounded manifoldM, with its reach τ(M) ≥ τg , and with other regularity assumptions.

πM (x)

≤ τg

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The reach is assumed to be lower bounded to avoid anarbitrarily complicated manifold.

I M is of local reach ≥ τ`, if for all points p ∈ M, there exists aneighborhood Up ⊂ M such that Up is of reach ≥ τ`.

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Density is bounded away from ∞ with respect to theuniform measure.

I Distribution P is absolutely continuous to induced Lebesgue measurevolM , and dP

dvolM≤ Kp for fixed Kp.

I This implies that the distribution on the manifold is of essentialdimension d .

I Pdκl ,κg ,Kp

denotes set of distributions P that is supported ond-dimensional manifold of (global) reach ≥ τg , local reach ≥ τ`, anddensity is bounded by Kp.

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Reference

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The Maximum Risk of any chosen Estimator Provides anUpper Bound on the Minimax Rate.

Rn = infˆdimn

supP∈P

ˆdimn(X ), dim(P))]

≤ supP∈P

︸︷︷︸the maximum risk of any chosen estimator

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TSP(Travelling Salesman Problem) Path Finds ShortestPath that Visits Each Points exactly Once.

6http://www.heatonresearch.com/fun/tsp/anneal15 / 65

Our Estimator estimates Dimension to be d2 if d1-squaredLength of TSP Generated by the Data is Long.

I When intrinsic dimesion is higher, length of TSP path is likely to belonger.

ˆdimn(X ) = d1 ⇐⇒

∃σ ∈ Sn s.tn−1∑

‖Xσ(i+1) − Xσ(i)‖d1Rm ≤ C ,

where C is some constant that depends only on KI , d1, and m.

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Our Estimator has Maximum Risk of O(n−(

d2d1−1)n).

I Our estimator makes error with probability at most O(n−(

d2d1−1)n

if intrinsic dimension is d2.I Our estimator is always correct when the intrinsic dimension is d1.

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Our Estimator makes Error with Probability at most

(n−(

d2d1−1)n)

if Intrinsic Dimension is d2.

I Based on the following lemma:

Lemma(Lemma 18) Let X1, · · · ,Xn ∼ P ∈ Pd2

κl ,κg ,Kp, then

[n−1∑

‖Xi+1 − Xi‖d1 ≤ L

]. n−

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Our Estimator is always Correct when the IntrinsicDimension is d1.

I Based on following lemma:

Lemma(Lemma 19) Let M be a d1-dimensional manifold with global reach ≥ τgand local reach ≥ τ`, and X1, · · · ,Xn ∈ M. Then there exists C whichdepends only on m, d1 and KI , and there exists σ ∈ Sn such that

n−1∑

‖Xσ(i+1) − Xσ(i)‖d1Rm ≤ C .

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Our estimator is always correct when the intrinsic dimensionis d1.

n−1∑

‖Xσ(i+1) − Xσ(i)‖d1Rm ≤ C .

I When d1 = 1 so that the manifold is a curve, length of TSP path isbounded by length of curve volM(M).

Xσ(1)

Xσ(2)

Xσ(3)

Xσ(n−1)

Xσ(n)

Yn−1

∑Yi ≤ volM (M)

Xσ(n−2)

Yn−2

I Global reach≥ τg implies volM(M) is bounded.20 / 65

n−1∑

‖Xσ(i+1) − Xσ(i)‖d1Rm ≤ C .

I When d1 > 1, Several conditions implied by regularity conditionscombined with Hölder continuity of d1-dimensional space-fillingcurve is used.

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n−1∑

‖Xσ(i+1) − Xσ(i)‖d1Rm ≤ C .

I When d1 > 1, Several conditions implied by regularity conditionscombined with Hölder continuity of d1-dimensional space-fillingcurve is used.

Lemma(Lemma 85, Space-filling curve) There exists surjective mapψd : R→ Rd which is Hölder continuous of order 1/d , i.e.

0 ≤ ∀s, t ≤ 1, ‖ψd(s)− ψd(t)‖Rd ≤ 2√d + 3|s − t|1/d .

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Mimimax rate is upper bounded by O

(n−(

d2d1−1)n).

Proposition(Proposition 21) Let 1 ≤ d1 < d2 ≤ m. Then

infˆdimn

supP∈Pd1∪Pd2

ˆdimn, dim(P))]

. n−(d2d1−1)n.

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Reference

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A subset T ⊂ [−KI ,KI ]n and set of distributions Pd1

1 , Pd22

are found so that, whenever X = (X1, · · · ,Xn) ∈ T , wecannot distinguish two models.

I The lower bound measures how hard it is to tell whether the datacome from a d1 or d2 -dimensional manifold.

I T , Pd11 and Pd2

2 are linked to the lower bound by using Le Cam’slemma.

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Le Cam’s Lemma provides lower bounds based on theminimum of two densities q1 ∧ q2, where q1, q2 are inconvex hull of Pd1

1 and convex hull of Pd22 , respectively.

Lemma(Lemma 22, Le Cam’s Lemma) Let P be a set of probability measures,and Pd1 ,Pd2 ⊂ P be such that for all P ∈ Pdi , θ(P) = θi for i = 1, 2.For any Qi ∈ co(Pi ), let qi be density of Qi with respect to measure ν.Then

infθsupP∈P

EP [d(θ, θ(P))] ≥d(θ1, θ2)

Qi∈co(Pdi )

∫[q1(x) ∧ q2(x)]dν(x).

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T is constructed so that for any x = (x1, · · · , xn) ∈ T ,there exists a d1-dimensional manifold that satisfiesregularity conditions and passes through x1, · · · , xn.

I Ti ’s are cylinder sets in [−KI ,KI ]d2 , and then T is constructed as

T = Snn∏

i=1Ti , where the permutation group Sn acts on

n∏i=1

Ti as a

coordinate change.

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T is constructed so that for any x = (x1, · · · , xn) ∈ T ,there exists a d1-dimensional manifold that satisfiesregularity conditions and passes through x1, · · · , xn.

I Given x1, · · · , xn ∈ T (blue points), manifold of global reach ≥ τgand local reach ≥ τ` (red line) passes through x1, · · · , xn.

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Pd11 is constructed as set of distributions that are supported

on manifolds that passes through x1, · · · , xn forx = (x1, · · · , xn) ∈ T , and Pd2

2 is a singleton set consistingof the uniform distirbution on [−KI ,KI ]

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If X ∈ T , it is hard to determine whether X is sampledfrom distribution P in either Pd1

1 or Pd22 .

I There exists Q1 ∈ co(Pd11 ) and Q2 ∈ co(Pd2

2 ) such thatq1(x) ≥ Cq2(x) for every x ∈ T with C < 1.

I Then q1(x) ∧ q2(x) ≥ Cq2(x) if x ∈ T , so C∫Tq2(x)dx can serve

as lower bound of minimax rate.I Based on following claim:

Claim(Claim 25) Let T = Sn

n∏i=1

Ti . Then for all x ∈ intT , there exists C > 0

that depends only on κl , KI , and rx > 0 such that for all r < rx ,

Q1 (B(xi , r)) ≥ CQ2 (B(xi , r)) .

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Mimimax rate is lower bounded by Ω(n−2(d2−d1)n

I Lower bound below is now combination of Le Cam’s lemma,constructions of T , Pd1

1 , Pd22 , and claim.

Proposition(Proposition 26)

infˆdim

supP∈Pd1∪Pd2

EP(n) [l( ˆdimn, dim(P))] & n−2(d2−d1)n.

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Reference

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Multinary Classification and 0− 1 Loss are Considered.

Rn = infˆdimn

supP∈P

I Now the manifolds are of any dimensions between 1 and m, so

considered distribution set is P =m⋃

d=1Pd .

I 0− 1 loss function is considered, so for all x , y ∈ R,`(x , y) = I (x = y).

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Mimimax Rate is Upper Bounded by O(n−

1m−1n

), and

Lower Bounded by Ω(n−2n

Proposition(Proposition 28 and 29)

n−2n . infˆdimn

supP∈P

ˆdimn, dim(P))]

. n−1

m−1 n.

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Reference

35 / 65

Reference

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The medial axis of a set M is the set of points that have atleast two nearest neighbors on the set M .

Med(M) = z ∈ Rm : there exists p 6= q ∈ M with‖p − z‖ = ‖q − z‖ = d(z ,M).

Med(M)

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The reach of M , denoted by τM , is the minimum distancefrom Med(M) to M .

τM = infx∈Med(M),y∈M

‖x − y‖ .

Med(M)

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The reach τM gives the maximum offset size of M on whichthe projection is well defined.

τM = infx∈Med(M),y∈M

‖x − y‖ .

Med(M)

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The reach τM gives the maximum radius of a ball that youcan roll over M .

I When M ⊂ Rm is a manifold,

τM = infq2 6=q1∈M

‖q2 − q1‖22d(q2 − q1,Tq1M)

q1 + Tq1M

d (q2 − q1, Tq1M)

‖q2 − q1‖‖q2−q1‖2

2d(q2−q1,Tq1M)

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τM = infq2 6=q1∈M

‖q2 − q1‖22d(q2 − q1,Tq1M)

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The bottleneck is a geometric structure where the manifoldis nearly self-intersecting.

Definition(Definition 34) A pair of points (q1, q2) in M is said to be a bottleneck ofM if there exists z0 ∈ Med(M) such that q1, q2 ∈ B(z0, τM) and‖q1 − q2‖ = 2τM .

Med(M)

B(z0, τM )

‖q1 − q2‖ = 2τM

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The reach is attained either from the bottleneck (globalcase) or the area of high curvature (local case).

Theorem(Theorem 37) At least one of the following two assertions holds:

I (Global Case) M has a bottleneck (q1, q2) ∈ M2.I (Local case) There exists q0 ∈ M and an arc-length parametrized γ0

such that γ0(0) = q0 and ‖γ′′0 (0)‖ = 1τM

Med(M)

B(z0, τM )

‖q1 − q2‖ = 2τM

B(z0, τM )

Med(M)M

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Reference

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τM = infq 6=p∈M

‖q − p‖22d(q − p,TpM)

q1 + Tq1M

d (q2 − q1, Tq1M)

‖q2 − q1‖‖q2−q1‖2

2d(q2−q1,Tq1M)

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We define the reach estimator τ as the maximum radius of aball that you can roll over the point cloud.

I Let X = x1, . . . , xn be a finite point cloud, then the reachestimator τ is a plugin estimator as

τ(X ) = infxi 6=xj∈X

‖xj − xi‖22d(xj − xi ,TxiM)

q1 + Tq1M

d (q2 − q1, Tq1M)

‖q2 − q1‖‖q2−q1‖2

2d(q2−q1,Tq1M)

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The statistical efficiency of the reach estimator τ is analyzedthrough its risk.

I The risk of the estimator τ is the expected loss the estimator.

EP(n) [` (τ(X ), τM)] .

I X = X1, . . . ,Xn is drawn from a fixed distribution P with itssupport M.

I The loss function used is `(τ, τ ′) =∣∣ 1τ− 1

τ ′

∣∣p, p ≥ 1.

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The risk of the reach estimator τ is analyzed.

I The risk of the estimator τ is the expected loss the estimator

[∣∣∣∣1τM− 1τ(X )

∣∣∣∣p].

I X = X1, . . . ,Xn is drawn from a fixed distribution P with itssupport M.

I The loss function used is `(τ, τ ′) =∣∣ 1τ− 1

τ ′

∣∣p, p ≥ 1.

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The reach estimator has the risk of O(n−

2p3d−1

I The reach estimator has the risk of O(n−

)for the global case.

I The reach estimator has the risk of O(n−

2p3d−1

)for the local case.

Med(M)

B(z0, τM )

‖q1 − q2‖ = 2τM

B(z0, τM )

Med(M)M

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The reach estimator has the maximum risk of O(n−

the global case.Proposition(Proposition 40) Assume that the support M has a bottleneck. Then,

[∣∣∣∣1τM− 1τ(X )

∣∣∣∣p]

. n−pd .

Med(M)

B(z0, τM )

‖q1 − q2‖ = 2τM

50 / 65

The reach estimator has the maximum risk of O(n−

2p3d−1

for the local case.Proposition(Proposition 44) Suppose there exists q0 ∈ M and a geodesic γ0 withγ0(0) = q0 and ‖γ′′0 (0)‖ = 1

τM. Then,

[∣∣∣∣1τM− 1τ(X )

∣∣∣∣p]

. n−2p

3d−1 .

B(z0, τM )

Med(M)M

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Reference

52 / 65

The statistical difficulty of the reach estimation problem isanalyzed by the minimax rate.

I Minimax rate is the risk of an estimator that performs best in theworst case, as a function of sample size.

Rn = infτn

supP∈P

EPn [` (τn(X ), τM)] .

I X = X1, . . . ,Xn is drawn from a fixed distribution P with itssupport M, where P is contained in set of distributions P.

I An estimator τn is any function of data X .I The loss function used is `(τ, τ ′) =

∣∣ 1τ− 1

τ ′

∣∣p, p ≥ 1.

53 / 65

The statistical difficulty of the reach estimation problem isanalyzed by the minimax rate.

I Minimax rate is the risk of an estimator that performs best in theworst case, as a function of sample size.

Rn = infτn

supP∈P

[∣∣∣∣1τM− 1τn(X )

∣∣∣∣p].

I X = X1, . . . ,Xn is drawn from a fixed distribution P with itssupport M, where P is contained in set of distributions P.

I An estimator τn is any function of data X .I The loss function used is `(τ, τ ′) =

∣∣ 1τ− 1

τ ′

∣∣p, p ≥ 1.

54 / 65

The maximum risk of our estimator provides an upperbound on the minimax rate.

Rn = infτn

supP∈P

[∣∣∣∣1τM− 1τn(X )

∣∣∣∣p]

≤ supP∈P

[∣∣∣∣1τM− 1τ(X )

∣∣∣∣p]

︸︷︷︸the maximum risk of our estimator

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Minimax rate is upper bounded by O(n−

2p3d−1

Theorem(Theorem 45)

infτn

supP∈P

[∣∣∣∣1τM− 1τn

∣∣∣∣p]

. n−2p

3d−1 .

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Le Cam’s lemma provides a lower bound based on the reachdifference and the statistical difference of two distributions.

I Total variance distance between two distributions is defined as

TV (P,P ′) = supA∈B(RD )

|P(A)− P ′(A)| .

Lemma(Lemma 46) Let P,P ′ ∈ P with respective supports M and M ′. Then

infτn

supP∈P

[∣∣∣∣1τM− 1τn

∣∣∣∣p]

∣∣∣∣1τM− 1τM′

∣∣∣∣p

(1− TV (P,P ′))2n.

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Two distributions P , P ′ are found so that their reaches differbut they are statistically difficult to distinguish.

infτn

supP∈P

[∣∣∣∣1τM− 1τn

∣∣∣∣p]

∣∣∣∣1τM− 1τM′

∣∣∣∣p

(1− TV (P,P ′))2n.

I The lower bound measures how hard it is to tell whether the data isfrom distributions with different reaches.

I P and P ′ are found so that∣∣∣ 1τM− 1

τM′

∣∣∣p

is large while

(1− TV (P,P ′))2n is small.

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P is a distribution supported on a sphere while P ′ is adistribution supported on a bumped sphere.

59 / 65

Mimimax rate is lower bounded by Ω(n−

Proposition(Proposition 50)

infτn

supP∈P

[∣∣∣∣1τM− 1τn

∣∣∣∣p]

& n−pd .

60 / 65

Reference

61 / 65

We can use `∞ metric to measure a distance between trees.

DefinitionThe l∞ metric between trees are defined as

d∞(Tp,Tq) = sup |p(x)− q(x)| .

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Pruning finds the simpler trees that are in the confidence set.

I We propose two pruning schemes to find trees that are simpler theempirical tree Tph and are in the fconfidence set.

I Pruning only leaves: remove all leaves of length less than 2tα.I Pruning leaves and internal branches: iteratively remove all branches

of cumulative length less than 2tα.

63 / 65

Reference

64 / 65

ReferenceEddie Aamari, Jisu Kim, Frédéric Chazal, Bertrand Michel, AlessandroRinaldo, and Larry Wasserman. Estimating the Reach of a Manifold.ArXiv e-prints, May 2017.

O. Bobrowski, S. Mukherjee, and J. E. Taylor. Topological consistencyvia kernel estimation. ArXiv e-prints, July 2014.

Frédéric Chazal, Leonidas J Guibas, Steve Y Oudot, and Primoz Skraba.Scalar field analysis over point cloud data. Discrete & ComputationalGeometry, 46(4):743–775, 2011.

Brittany T. Fasy, Jisu Kim, Fabrizio Lecci, Clément Maria, David L.Millman, and Vincent Rouvreau. Introduction to the R package TDA.CoRR, abs/1411.1830, 2014a. URLhttp://arxiv.org/abs/1411.1830.

Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, LarryWasserman, Sivaraman Balakrishnan, and Aarti Singh. Confidence setsfor persistence diagrams. Ann. Statist., 42(6):2301–2339, 12 2014b.doi: 10.1214/14-AOS1252. URLhttp://dx.doi.org/10.1214/14-AOS1252.

Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The elements ofstatistical learning: data mining, inference and prediction. Springer, 2edition, 2009. ISBN 978-0-3878-4857-0. URLhttp://statweb.stanford.edu/~tibs/ElemStatLearn/.

Jisu Kim, Yen-Chi Chen, Sivaraman Balakrishnan, Alessandro Rinaldo,and Larry Wasserman. Statistical inference for cluster trees. In D. D.Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors,Advances in Neural Information Processing Systems 29, pages1839–1847. Curran Associates, Inc., 2016. URLhttp://papers.nips.cc/paper/6508-statistical-inference-for-cluster-trees.pdf.

Jisu Kim, Alessandro Rinaldo, and Larry Wasserman. Minimax Rates forEstimating the Dimension of a Manifold. ArXiv e-prints, May 2016.

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