Barycentric Subspaces and

Affine Spans in Manifolds

GSI 30-10-2015

Xavier Pennec

Asclepios team, INRIA Sophia-Antipolis –

Mediterranée, France

and

Côte d’Azur University (UCA)

Statistical Analysis of Geometric Features

Computational Anatomy deals with noisy

Geometric Measures

Tensors, covariance matrices

Curves, tracts

Surfaces, shapes

Images

Deformations

Data live on non-Euclidean manifolds

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Manifold dimension reduction

When embedding structure is already manifold (e.g. Riemannian):

Not manifold learning (LLE, Isomap,…) but submanifold learning

Low dimensional subspace approximation?

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Manifold of cerebral ventricles

Etyngier, Keriven, Segonne 2007.

Manifold of brain images

S. Gerber et al, Medical Image analysis, 2009.

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Barycentric Subspaces

and Affine Spans in Manifolds

PCA in manifolds: tPCA / PGA / GPCA / HCA

Affine span and barycentric subspaces

Conclusion

5

Bases of Algorithms in Riemannian Manifolds

Reformulate algorithms with Expx and Logx

Vector -> Bi-point (no more equivalence classes)

Exponential map (Normal coordinate system):

Expx = geodesic shooting parameterized by the initial tangent

Logx = development of the manifold in the tangent space along geodesics

Geodesics = straight lines with Euclidean distance

Local global domain: star-shaped, limited by the cut-locus

Covers all the manifold if geodesically complete

6

Statistical tools: Moments

Frechet / Karcher mean minimize the variance

𝜎2(𝑥) = 𝑑𝑖𝑠𝑡2 𝑥, 𝑧 𝑝 𝑧 𝑑𝑀(𝑧)𝑀

Tensor moments of a random point with density p

𝔐1 𝑥 = 𝑥𝑧 𝑝 𝑧 𝑑𝑀(𝑧)𝑀 Tangent mean field

𝔐2(𝑥) = 𝑥𝑧 ⊗ 𝑥𝑧 𝑝 𝑧 𝑑𝑀(𝑧)𝑀 Covariance field

Exponential barycenters are critical pts of variance (P(C) =0)

𝔐1 𝑥 = 𝑥 𝑧 𝑝 𝑧 𝑑𝑀(𝑧)𝑀

= 0 (implicit definiton of 𝑥 )

Covariance [and higher order moments]

𝔐2(𝑥 ) = 𝑥 𝑧 ⊗ 𝑥 𝑧 𝑝 𝑧 𝑑𝑀(𝑧)𝑀

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[Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec, JMIV06, NSIP’99 ]

Tangent PCA

Maximize the squared distance to the mean (explained variance)

Algorithm

Find the Karcher mean 𝑥 minimizing 𝜎2 𝑥 = 𝑑𝑖𝑠𝑡2(𝑥, 𝑥𝑖)𝑖

Unfold data on tangent space at the mean

Diagonalize covariance Σ 𝑥 ∝ 𝑥 𝑥𝑖𝑖 𝑥 𝑥𝑖𝑡

Generative model:

Gaussian (large variance) in the horizontal subspace

Gaussian (small variance) in the vertical space

Find the subspace of 𝑇𝑥𝑀 that best explains the variance

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Principal Geodesic / Geodesic Principal Component Analysis

Minimize the squared Riemannian distance to a low

dimensional subspace (unexplained variance)

PGA (Fletcher et al., 2004, Sommer 2014):

space generated by geodesics rays originating from Karcher mean:

𝐺𝑆 𝑥,𝑤1, …𝑤𝑘 = exp𝑥 𝛼𝑖𝑤𝑖𝑖 𝑓𝑜𝑟 𝛼 ∈ 𝑅𝑘

Geodesic PCA (GPCA, Huckeman et al., 2010):

space generated by principle geodesics that cross at one point

(principle mean, may be different from Karcher mean)

Generative model:

Unknown (uniform ?) distribution within the subspace

Gaussian distribution in the vertical space

All different models in curved spaces (no Pythagore thm)

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Problems of tPCA / PGA

Analysis is done relative to on point

What if this point is a poor description of the data?

Multimodal distributions

Uniform distribution on subspaces

Large variance w.r.t curvature

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Courtesy of S. Sommer

Bimodal distribution on S2

Courtesy of S. Sommer

Patching the Problems of tPCA / PGA

Improve the flexibity of the geodesics

1D regression with higher order splines [Vialard, Singh, Niethammer]

Control of dimensionality for n-D Polynomials on manifolds?

Nested “algebraic” subspaces

Principle nested spheres [Jung, Dryden, Marron 2012]

Quotient of Lie group action [Huckemann, Hotz, Munk, 2010]

No general semi-direct product space structure in general

Riemannian manifolds

Iterated Frame Bundle Development [HCA, Sommer GSI 2013]

Iterated construction of subspaces

Parallel transport in frame bundle

Intrinsic asymmetry between components

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Courtesy of S. Sommer

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Barycentric Subspaces

and Affine Spans in Manifolds

PCA in manifolds: tPCA / PGA / GPCA / HCA

Affine span and barycentric subspaces

Conclusion

Affine subspaces in Euclidean spaces

Affine subspaces in a Euclidean space

Aff x0, v1, … vk = {𝑥 = 𝑥0 + 𝜆1v1 + 𝜆2v2 +⋯𝜆𝑘vk}

Affine span of (k+1) points (𝒙𝒊 = 𝒙 + 𝒗𝒊)

Aff x0, x1, … xk = {x = 𝜆𝑖𝑖 𝑥𝑖 𝑤𝑖𝑡ℎ 𝜆𝑖𝑘𝑖=1 = 1}

= x ∈ 𝑅𝑛 𝑠. 𝑡 𝜆𝑖𝑖 (𝑥𝑖−𝑥 = 0, 𝜆 ∈ 𝑃𝑘∗}

Weighted means / barycenters with homogeneous coordinates

𝑃𝑘∗ = 𝜆0 ∶ 𝜆1: … ∶ 𝜆𝑘 ∈ R

𝑘+1 𝑠. 𝑡. 𝜆1 ≠ 0

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Affine span in Riemannian manifolds

Key ideas:

Look at data points from the mean

(mean has to be unique)

Look at several reference points from

any point of the manifold subspace

Barycentric coordinates

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A. Manesson-Mallet. La géométrie Pratique, 1702

Notations for Riemannien manifolds

(k+1)-pointed Riemannian manifold

𝑀∗ 𝑥0, … 𝑥𝑘 = 𝑀 / ∪ 𝐶 𝑥𝑖

𝑑𝑖𝑠𝑡 𝑥, 𝑥𝑖 and log𝑥(𝑥𝑖) are smooth on 𝑀∗

Tensor moments of the (k+1) reference points

𝔐0 𝜆 = 𝜆𝑖𝑖 Density

𝔐1 𝑥, 𝜆 = 𝜆𝑖𝑖 𝑥𝑥𝑖 Tangent mean at x

𝔐2(𝑥, 𝜆) = 𝜆𝑖𝑖 𝑥𝑥𝑖⊗𝑥𝑥𝑖 Covariance at x

𝔐0 𝜆 ,𝔐𝑘(𝑥, 𝜆) are smooth tensor fields on 𝑀∗ 𝑥0, … 𝑥𝑘

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Barycentric subspaces and Affine span

Fréchet/Karcher barycentric subspaces (KBS / FBS) Normalized weighted variance: σ2(x,λ) = λ𝑖𝑑𝑖𝑠𝑡

2 𝑥, 𝑥𝑖 / λ𝑖

Set of absolute / local minima of the weighted variance

Works in stratified spaces (may go accross different strata)

Exponential barycentric subspace* Weighted exponential barycenters: 𝔐1 𝑥, 𝜆 = 𝜆𝑖𝑖 𝑥𝑥𝑖 = 0

EBS 𝑥0, … 𝑥𝑘 = 𝑥 ∈ 𝑀∗ 𝑥0, … 𝑥𝑘 𝔐1 𝑥, 𝜆 = 0}

Affine span* = closure of EBS in M

𝐴𝑓𝑓 𝑥0, … 𝑥𝑘 = 𝐸𝐵𝑆 𝑥0, … 𝑥𝑘

* Beware: the definitions have been changed w.r.t. the paper

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Barycentric subspaces and Affine span

Global minima are subset of local ones: 𝑭𝑩𝑺 ⊂ 𝑲𝑩𝑺

Exp. barycenters are critical points of w-variance on M*

𝛻σ2(x,λ)= −2𝔐1 x, λ = 0 𝑲𝑩𝑺 ∩ 𝑴∗ ⊂ 𝑬𝑩𝑺

Caractérisation of local minima: Hessian (if non degenerate)

𝐻(x,λ) = −2 𝜆𝑖𝐷𝑥 log𝑥 𝑥𝑖𝑖

= 𝐈𝐝 −𝟏

𝟑𝐑𝐢𝐜 𝕸𝟐 𝐱, 𝝀 + ⋯

Regular and positive pts (non-degenerated critical points)

𝑬𝑩𝑺𝑹𝒆𝒈 𝒙𝟎, … 𝒙𝒌 = 𝒙 ∈ 𝑨𝒇𝒇 𝒙𝟎, …𝒙𝒌 , 𝒔. 𝒕. 𝑯 𝒙, 𝝀∗(𝒙) ≠ 𝟎

𝑬𝑩𝑺+ 𝒙𝟎, … 𝒙𝒌 = { 𝒙 ∈ 𝑨𝒇𝒇 𝒙𝟎, … 𝒙𝒌 , 𝒔. 𝒕. 𝑯 𝒙, 𝝀∗(𝒙) 𝑷𝒐𝒔. 𝒅𝒆𝒇. }

Theorem: 𝑲𝑩𝑺 = 𝑬𝑩𝑺+ plus potentially some degenerate points of the

affine span and some points of the cut locus of the reference points.

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Characterization of the EBS

SVD characterization of EBS

Rewrite 𝔐1 x, λ = 𝑍 𝑥 𝜆 = 0 with 𝑍 𝑥 = [ 𝑥𝑥0, … 𝑥𝑥𝑘]

𝜆∗ 𝑥 ∈ 𝐾𝑒𝑟(𝑍 𝑥 )

SVD: 𝑍 𝑥 = 𝑈 𝑥 𝑆 𝑥 𝑉𝑡(𝑥)

𝐴𝑓𝑓 𝑥0, … 𝑥𝑘 = { 𝑥 ∈ 𝑀∗ 𝑥0, … 𝑥𝑘 , 𝑠. 𝑡. 𝑠𝑘 𝑥 = 0 }

Local parameterization around 𝑥, 𝜆 : 𝛿𝑥 = 𝐻 𝑥, 𝜆 −1 𝑍 𝑥 𝛿𝜆

𝑬𝑩𝑺𝑹𝒆𝒈 𝒙𝟎, … 𝒙𝒌 is a stratified space

(k-m+1) vanishing singular values on the m-strata

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PCA / spectral / POD - like characterizations

Small (k+1)(k+1) matrix

Ω 𝑥 = 𝑍 𝑥 𝑡𝐺 𝑥 𝑍(𝑥) ( Ωij 𝑥 =< 𝑥𝑥𝑖 , 𝑥𝑥𝑗 >𝑥)

Vanishing smallest eigenvalue

𝑥 ∈ 𝐴𝑓𝑓 𝑥0, … 𝑥𝑘 ⟺ det Ω 𝑥 = 0 ⟺ 𝜎𝑘 = 0

Large n.n covariance matrix

Σ 𝑥 ∝ 𝑍 𝑥 𝑍 𝑥 𝑡 = 𝔐2 x, 1

Vanishing (n-k) smallest eigenvalues 𝑥 ∈ 𝐴𝑓𝑓 𝑥0, … 𝑥𝑘 ⟺ 𝜎1(𝑥) ≥ ⋯𝜎𝑘(𝑥) ≥ 𝜎𝑘+1 𝑥 = 𝜎𝑛(𝑥)

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Affine span of a Sphere

(k+1)-pointed Sphere

𝑋 = 𝑥0, 𝑥1, … , 𝑥𝑘 ∈ 𝑆𝑛𝑘

Exclude antipodal points: 𝑆𝑛∗ = 𝑆𝑛/ −𝑋

Exponential barycentric subspace: almost great subspheres

EBS 𝑥0, … 𝑥𝑘 = 𝑆𝑝𝑎𝑛 𝑋 𝑆𝑛∗

Affine span = great subsphere

𝐴𝑓𝑓 𝑥0, … 𝑥𝑘 = 𝐸𝐵𝑆(𝑥0, … 𝑥𝑘) = 𝑆𝑝𝑎𝑛 𝑋 𝑆𝑛

Fréchet/Karcher barycentric subspaces (KBS / FBS)

In practice positive & negative eigenvalue of Hessians: Cf Buss & Fillmore ACM TG 2001

KBS/FBS is an incomplete subset of EBS: manifold with

boundaries (less interesting than affine span)

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Limit of affine span for collapsing points

1st order: ESB converges to [restricted] Geodesic Subspace

Wx w1, …wk = {𝑤 = 𝛼𝑖𝑤𝑖𝑖 𝑓𝑜𝑟 𝛼 ∈ 𝑅𝑘}

𝐺𝑆∗ 𝑊𝑥 = {exp𝑥 𝑤 ,𝑤 ∈ 𝑊𝑥 ∩ 𝐷𝑥} is the limit of

EBS 𝑥0, exp𝑥𝑜 𝜖 𝑤1 , … exp𝑥𝑜 𝜖 𝑤𝑘 when 𝜖 → 0.

Sphere: 1st order (k,n)-jet: PGA with great subspheres

2nd order (k,n)-jet include Principle nested spheres [Jung, Dryden,

Marron 2012]

Conjecture

This can be generalized to higher order derivatives

Quadratic, cubic splines [Vialard, Singh, Niethammer]

Quotient of Lie group action [Huckemann, Hotz, Munk, 2010]

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Barycentric Subspaces

and Affine Spans in Manifolds

PCA in manifolds: tPCA / PGA / GPCA / HCA

Affine span and barycentric subspaces

Conclusion

Conclusion

Generalization to a–barycentric subspaces (median, mode)?

σ𝛼(x,λ) = 1

α λ𝑖𝑑𝑖𝑠𝑡

𝛼 𝑥, 𝑥𝑖 / λ𝑖

Well… critical points of σ𝛼(x,λ) are also critical points of

σ2(x,λ′) with 𝜆𝑖′ = 𝜆𝑖 𝑑𝑖𝑠𝑡

𝛼− 2 𝑥, 𝑥𝑖 (i.e. the affine span)

A natural generalization of affine subspaces in Manifolds?

Generalization to affine connection setting?

Generalizes PGA and GPCA and PNS as limit cases Conjecture: splines and “ quotient slices” are also limit cases

Natural flags structure: principle nested relations

[Damon & Marron, JMIV 2014]

Implementation and tests

Alternated Newton / Gauss-Newton optimization

Data should exhibit a large variability w.r.t. curvature

Natural extension to multi-atlas methods

Barycentric subspace analysis?

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Nestedness, Forward and Backward Analysis

k+1 points define a k-barycentric subspace 𝐴𝑓𝑓+ 𝑥0, … 𝑥𝑘

Optimize the point positions minimizing:

𝜎𝑜𝑢𝑡2 𝑥0, … 𝑥𝑘 = 𝑑𝑖𝑠𝑡

2 𝑦𝑗 , 𝑃𝑟𝑜𝑗𝐴𝑓𝑓 𝑥0…𝑥𝑘 (𝑦𝑗)𝑗

Forward analysis:

Iteratively add points 𝑥𝑗 from j=0 to k

𝑥0 = 𝑀𝑒𝑎𝑛 𝑦𝑗 , 𝑥1 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑥 𝜎𝑜𝑢𝑡2 (𝑥0, 𝑥) … PGA-like

Start with 2 points: x0, x1 = argmin(x,y) σout2 (x, y) GPGA-like

Backward analysis:

Iteratively remove one point from (𝑥0, … 𝑥𝑗) from j=0 to k

One optimization only for all points and the discrete ordering

From greedy to global optimization?

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Barycentric Subspace Analysis (BSA)

k ordered points define a Flags of affine spans

𝑥0 ≺ 𝑥1 ≺ ⋯ ≺ 𝑥𝑘 are k +1 n distinct ordered points of M.

𝐹𝐿(𝑥0 ≺ 𝑥1 ≺ ⋯ ≺ 𝑥𝑘) is the sequence of properly nested

subspaces 𝐹𝐿_𝑖 𝑥0 ≺ 𝑥1 ≺ ⋯ ≺ 𝑥𝑘 = 𝐴𝑓𝑓(𝑥0, … 𝑥𝑖)

0 ⊂ 𝐴𝑓𝑓+ 𝑥0 = 𝑥0 ⊂ …𝐴𝑓𝑓

+ 𝑥0, … 𝑥𝑘 … ⊂ 𝐴𝑓𝑓+ 𝑥0, … 𝑥𝑛 = 𝑀

𝜎𝑜𝑢𝑡2 𝑘 = 𝑑𝑖𝑠𝑡2 𝑦𝑗 , 𝑃𝑟𝑜𝑗𝐴𝑓𝑓 𝑥0…𝑥𝑘 (𝑦𝑗)𝑗

Ordering points: energy on the flag manifold

Ordering vectors in GL(n): minimal Area under the curve

Σ = 𝑑𝑖𝑎𝑔(𝜎12, … 𝜎𝑛

2) 𝜎𝑜𝑢𝑡2 𝑘 = 𝜎𝑘+1

2 +⋯𝜎𝑛2

𝐴𝑈𝐶 𝑘 = 𝜎𝑜𝑢𝑡2 𝑖𝑘

𝑖=0 = 𝑖 𝜎𝑖2𝑘

𝑖=0 + 𝑘 + 1 𝜎𝑖2𝑛

𝑖=𝑘+1

minimal for diagonal coordinate system with 𝜎1 ≥ 𝜎2 … ≥ 𝜎𝑛

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