Statistics on Lie groups: using the pseudo-Riemannian framework? - - PowerPoint PPT Presentation

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Statistics on Lie groups: using the pseudo-Riemannian framework? - - PowerPoint PPT Presentation

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Statistics on Lie groups: using the pseudo-Riemannian framework? Nina Miolane, Xavier Pennec Asclepios Team, INRIA Sophia Antipolis Max Ent 2014 Nina Miolane - Statistics on Lie groups Models with Lie groups Articulated models Shape

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Statistics on Lie groups: using the pseudo-Riemannian framework?

Nina Miolane, Xavier Pennec Asclepios Team, INRIA Sophia Antipolis

Max Ent’ 2014 Nina Miolane - Statistics on Lie groups

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Models with Lie groups

Nina Miolane - Statistics on Lie groups

http://www.societyofrobots.com/ 2006, Nature Publishing Group, Genetics

Spherical arm

𝑻𝑷 πŸ’

Patient 3 Reference Patient 1 Patient 2 Patient 4 Patient 5

1 2 3 4 5

οƒ  Statistics

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Articulated models Shape models

Robotics Computational Anatomy Paleontology Computational Medicine

𝑻𝑭(πŸ’)

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Articulated model of spine: Vertebra 𝑆, 𝑒 ∈ 𝑇𝐹(3)

Models with Lie groups

Nina Miolane - Statistics on Lie groups

PCA : Modes 1 to 4

Rotation Matrix translation vector

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Statistics on spines: Mean, PCA… Computational Anatomy : model and analyze the variability of human anatomy (𝑆, 𝑒)

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New statistics on Lie groups?

Average pose of two vertebrae is not a vertebra

Nina Miolane - Statistics on Lie groups

Usual statistics: linear Lie groups: not linear in general

?

π’πŸ‘, π’–πŸ‘ π’πŸ, π’–πŸ π’πŸ, π’–πŸ π’πŸ‘, π’–πŸ‘

(

π‘ΊπŸ+π‘ΊπŸ‘ πŸ‘

,

π’–πŸ+π’–πŸ‘ πŸ‘ )

𝑻𝑭 πŸ’

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?

(π‘ΊπŸ+π‘ΊπŸ‘

πŸ‘

, π’–πŸ+π’–πŸ‘

πŸ‘ )

On Lie group 𝑻𝑭(πŸ’) (space of transformations) Consequence for model Vertebra : 𝑆, 𝑒 ∈ 𝑇𝐹(3) Mean not on 𝑇𝐹(3), thus not a rigid transformation Need new statistics!

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Outline

  • 1. Pseudo-Riemannian structures on Lie groups
  • 2. Algorithm to compute bi-invariant pseudo-metrics
  • 3. Results on selected Lie groups

Nina Miolane - Statistics on Lie groups

Can we use the pseudo-Riemannian framework to define statistics on Lie groups?

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Outline

  • 1. Pseudo-Riemannian structures on Lie groups
  • 2. An algorithm to compute bi-invariant pseudo-metrics
  • 3. Results on selected Lie groups

Nina Miolane - Statistics on Lie groups 10/17/2014 6

Can we use the pseudo-Riemannian framework to define statistics on Lie groups?

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Quadratic Lie group (𝑯,∘, <, >) (Pseudo-) Riemannian manifold (𝑡, <, >) Manifold 𝑡

(Pseudo-) Riemannian structure on Lie groups

Nina Miolane - Statistics on Lie groups

Lie group (𝑯,∘)

Differential structure

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+ algebraic + metrical Metric <, >: collection of positive definite inner products Pseudo-metric <, >: collection of definite inner products

Consistency of the structures requires bi-invariant pseudo-metric

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Group exponential barycenter [1] =group mean

Consistency Computability

Statistics on Lie groups : the mean

οƒ  Requirements for mean of data set

Nina Miolane - Statistics on Lie groups

FrΓ©chet mean: definition with computability conditions FrΓ©chet mean = group mean 𝒉 𝑕𝑗 𝑗

Riemannian structure οƒ  Bi-invariant metric

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π’Š ∘ 𝒉 mean of β„Ž ∘ 𝑕𝑗 𝑗 𝒉 mean of 𝑕𝑗 𝑗

π‘€β„Ž

Conditions

  • n 𝑕𝑗 𝑗

such that 𝒉 exists and is unique?

[1] Pennec & Arsigny. Exponential barycenter of the canonical Cartan connection and invariant means on Lie groups. (2012)

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Characterization of Lie groups with bi-invariant metric by Cartan [2] : compact & abelian

Bi-invariant metric?

Existence of a bi-invariant metric? On Lie groups that have a group mean?

Nina Miolane - Statistics on Lie groups

οƒ  Group mean not characterized by a bi-invariant metric [1]

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  • unique (a.e.) group mean
  • no bi-invariant metric

𝑻𝑭(𝒐): On all Lie groups? NO! NO!

[2] Elie Cartan. La thΓ©orie des groups finis et continus et l’analyse situs. (1952) [1] Pennec & Arsigny. Exponential barycenter of the canonical Cartan connection and invariant means on Lie groups. (2012)

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Consistency Computability

Statistics on Lie groups : the mean

οƒ  Requirements for mean of data set

Nina Miolane - Statistics on Lie groups

Group exponential barycenter [1] =group mean

FrΓ©chet mean: definition with computability conditions

Bi-invariant metric

οƒ  FrΓ©chet mean = group mean 𝒉 𝑕𝑗 𝑗

Riemannian structure

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π’Š ∘ 𝒉 mean of β„Ž ∘ 𝑕𝑗 𝑗 𝒉 mean of 𝑕𝑗 𝑗

π‘€β„Ž

Conditions

  • n 𝑕𝑗 𝑗

such that 𝒉 exists and is unique?

[1] Pennec & Arsigny. Exponential barycenter of the canonical Cartan connection and invariant means on Lie groups. (2012)

Pseudo-Riemannian structure Bi-invariant pseudo-metric

Generalize Riemannian to pseudo-Riemannian

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Characterization of Lie groups with bi-invariant pseudo-metric by Medina & Revoy [3]

Bi-invariant pseudo-metrics?

Existence of a bi-invariant pseudo-metric? On Lie groups that have a group mean?

Nina Miolane - Statistics on Lie groups

Group mean characterized by a bi-invariant pseudo-metric ?

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  • unique (a.e.) group mean
  • bi-invariant pseudo-metric ?

𝑇𝐹(π‘œ):

[3] Medina & Revoy. Groupes de Lie munis de pseudo-metriques bi-invariantes. (1982)

On all Lie groups? NO!

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Outline

  • 1. Pseudo-Riemannian structures on Lie groups
  • 2. An algorithm to compute bi-invariant pseudo-metrics
  • 3. Results on selected Lie groups

Nina Miolane - Statistics on Lie groups 10/17/2014 12

Can we use the pseudo-Riemannian framework to define statistics on Lie groups?

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From Lie group to Lie algebra

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Compute bi-invariant pseudo-metrics on finite dimensional connected Lie group 𝑯 Compute bi-invariant pseudo-metrics on its Lie algebra 𝖍 = π”πŸπ‡, +, . , , 𝖍

bi-invariant: < 𝒗, π’˜ >𝒉=< π‘¬π‘΄π’Š. 𝒗, π‘¬π‘΄π’Š. π’˜ >π‘΄π’Šπ’‰=< πΈπ‘†β„Ž. 𝑣, πΈπ‘†β„Ž. 𝑀 >π‘†β„Žπ‘• where π‘€β„Žπ‘• = β„Ž ∘ 𝑕 and π‘†β„Žπ‘• = 𝑕 ∘ β„Ž

𝒗 π’˜ π‘¬π‘΄π’Š. 𝒗 π‘¬π‘΄π’Š. π’˜ π‘΄π’Š 𝒉 π‘΄π’Šπ’‰ = π’Š ∘ 𝒉

bi-invariant: < 𝑦, 𝑧 π”₯, 𝑨 >𝑓+< 𝑧, 𝑦, 𝑨 π”₯ >𝑓= 0

Structure of quadratic 𝖍 ?

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Lie algebra representations

Representation 𝜽 of 𝖍 on 𝑾: Lie algebra homomorphism πœƒ: π”₯ ↦ π”₯π”ͺ(π‘Š) Ex.: Homogeneous representation of 𝑇𝐹 3 on ℝ4:

Nina Miolane - Statistics on Lie groups

π”₯ = 𝐢1

π”₯ βŠ•π”₯ … βŠ•π”₯ 𝐢𝑂 π”₯ : decomposition into indecomposable subrepresentations

πœƒ: 𝑇𝐹 3 ↦ π”₯π”ͺ ℝ4 𝑆, 𝑒 ↦ 𝑆 𝑒 1

s.t. 𝑆

𝑒 1 . 𝑦 1 = 𝑆. 𝑦 + 𝑒 1 𝑏𝑒: π”₯ ↦ π”₯π”ͺ(π”₯) 𝑦 ↦ 𝑏𝑒 𝑦 = 𝑦,βˆ™ π”₯

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Structure of quadratic 𝖍? Study adjoint representation of π”₯ Subrepresentation: subspace of π‘Š stable by πœƒ π”₯ Subrepresentation decomposition: π‘Š = 𝐢1 βŠ•π”₯ … βŠ•π”₯ 𝐢𝑂 with 𝐢𝑗 subrepresentations Indecomposable subrepresentation: not a sum of subrepresentations Adjoint representation 𝒃𝒆 of 𝖍 (on itself: 𝑾 = 𝖍 )

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π‘ͺ𝒋

𝖍 = 𝐓 βŠ• π‘»βˆ—

Structure of quadratic π”₯

  • Th. (Medina & Revoy): Structure of quadratic 𝖍

Adjoint representation decomposition π”₯ = B1

π”₯ βŠ•π”₯ … βŠ•π”₯ BN π”₯ has indecomposable B𝑗 π”₯ s.t.:

Type (1): π‘ͺ𝒋

𝖍 simple or 1-dim.

Type (2): π‘ͺ𝒋

𝖍= 𝐗 βŠ• 𝐓 βŠ• π“βˆ— double extension of 𝑋 quadratic by 𝑇 of Type (1)

Nina Miolane - Statistics on Lie groups

𝐢𝑗

π”₯

𝐢1

π”₯

𝐢𝑂

π”₯

π‘ͺ𝒋

𝖍 =1-dim.

π‘ͺ𝒋

𝖍 =simple

π‘ͺ𝒋

𝖍 = 𝐗 βŠ• 𝐓 βŠ• π‘»βˆ—

π”₯

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π‘ͺ𝒋

𝖍 = 𝐓 βŠ• π‘»βˆ—

Bi-invariant pseudo-metric on quadratic π”₯

Nina Miolane - Statistics on Lie groups

𝐢𝑗

π”₯

𝐢1

π”₯

𝐢𝑂

π”₯

π‘ͺ𝒋

𝖍 =1-dim.

π‘ͺ𝒋

𝖍 =simple

π‘ͺ𝒋

𝖍 = 𝐗 βŠ• 𝐓 βŠ• π‘»βˆ—

< 𝒕 + π’ˆ, 𝒕′ + π’ˆβ€² >π‘ͺ𝒋

𝖍= π’ˆ 𝒕′ + π’ˆβ€²(𝒕)

< 𝒄, 𝒄′ >π‘ͺ𝒋

𝖍= π‘³π’‹π’Žπ’Žπ’‹π’π’‰(𝒄, 𝒄′)

< 𝒄, 𝒄′ >π‘ͺ𝒋

𝖍= 𝒄. 𝒄′

< 𝒙 + 𝒕 + π’ˆ, 𝒙′ + 𝒕′ + π’ˆβ€² >π‘ͺ𝒋

𝖍 = < 𝒙, 𝒙′ >𝑿+ π’ˆ 𝒕′ + π’ˆβ€²(𝒕)

π”₯

< 𝑐1 + β‹― + 𝑐𝑂, 𝑐1

β€² + β‹― + 𝑐𝑂 >π”₯

=< 𝑐1, 𝑐1

β€² >𝐢1 + β‹― +< 𝑐𝑂, 𝑐𝑂 β€² >𝐢𝑂

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Algorithm

Nina Miolane - Statistics on Lie groups

𝐢𝑗

π”₯

𝐢1

π”₯

𝐢𝑂

π”₯

π‘ͺ𝒋

𝖍 =1-dim. ?

π‘ͺ𝒋

𝖍 =simple? π‘ͺ𝒋 𝖍 = 𝐓 βŠ• π‘»βˆ—?

π‘ͺ𝒋

𝖍 = 𝐗 βŠ• 𝐓 βŠ• π‘»βˆ—? Else : EXIT

π”₯ If algorithm finishes: expression of a bi-invariant pseudo-metric on π”₯ If EXIT: no bi-invariant pseudo-metric on π”₯

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Outline

  • 1. Pseudo-Riemannian structures on Lie groups
  • 2. An algorithm to compute bi-invariant pseudo-metrics
  • 3. Results on selected Lie groups

Nina Miolane - Statistics on Lie groups 10/17/2014 18

Can we use the pseudo-Riemannian framework to define statistics on Lie groups?

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Rigid transformations SE(n)

Nina Miolane - Statistics on Lie groups

𝖙𝖋(πŸ’) π‘ͺ𝟐 = 𝖙𝖋(πŸ’) π‘ͺ𝟐 = 𝑻 βŠ• π‘»βˆ— i.e. 𝖙𝖋 πŸ’ = 𝑻𝒍𝒇𝒙 πŸ’ βŠ• β„πŸ’

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Lie group: 𝑇𝐹 π‘œ = 𝑆, 𝑒 ∈ 𝑇𝑃 π‘œ ⋉ β„π‘œ Lie algebra: 𝔱𝔣 π‘œ = { 𝐡, 𝑣 ∈ 𝑇𝑙𝑓π‘₯ π‘œ βŠ• β„π‘œ}

𝖙𝖋(πŸ‘) 𝖙𝖋(𝟐) = ℝ 𝖙𝖋(𝒐) π‘ͺ𝟐 = 𝖙𝖋(πŸ‘) π‘ͺ𝟐 = 𝖙𝖋 𝟐 = ℝ π‘ͺ𝟐 = 𝖙𝖋(𝒐) π‘ͺ𝟐 = 1-dim.

EXIT EXIT

< 𝒃 + 𝒗, 𝒃′ + 𝒗′ >𝔱𝔣 πŸ’ = 𝒃𝑼. 𝒗′ + 𝒃′𝑼. 𝒗 < 𝒄, 𝒄′ >𝔱𝔣 𝟐 = 𝒄. 𝒄′

No bi-invariant pseudo-metric No bi-invariant pseudo-metric

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Other Lie groups: ST(n), H, UT(n)

Nina Miolane - Statistics on Lie groups 10/17/2014 20

Scalings and Translations Scaled Upper Unitriangular matrices Heisenberg No bi-invariant pseudo-metric for 𝒐 β‰  𝟏

𝔦 𝔦 𝔦 𝔱𝔲(π‘œ) 𝔱𝔲(π‘œ) 𝔳𝔲 π‘œ 𝔒

EXIT EXIT EXIT

π–Š = 1-dim.

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Conclusion

Nina Miolane - Statistics on Lie groups

Quadratic Lie group with:

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Characterization by Cartan Characterization by Medina & Revoy

𝐓 βŠ• π‘»βˆ— 1-dim. compact 𝐗 βŠ• 𝐓 βŠ• π‘»βˆ— 1-dim. simple

Bi-invariant metric Bi-invariant pseudo-metric

From Riemannian to pseudo-Riemannian: add the double extension structure

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Conclusion for statistics on Lie groups

Group exponential barycenter [1] =group mean

Existence and uniqueness conditions? Add a geometric structure on 𝐻 Riemannian? Pseudo-Riemannian?

NO

Most Lie groups with group mean without bi-invariant metric [1]

NO

Most Lie groups with group mean without bi-invariant pseudo-metric Yet another geometric structure?

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Thank you for your attention

Nina Miolane - Statistics on Lie groups

Patient 3 Reference Patient 1 Patient 2 Patient 4 Patient 5 1 2 3 4 5

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