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
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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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