Statistics on Lie groups: using the pseudo-Riemannian framework? Nina Miolane, Xavier Pennec Asclepios Team, INRIA Sophia Antipolis
Nina Miolane - Statistics on Lie groups
Max Ent’ 2014
Models with Lie groups Articulated models
Shape models
Robotics
Computational Medicine
𝑺𝑶 𝟑
1
Reference
Spherical arm Patient 1
2
3
4
5
http://www.societyofrobots.com/
Patient 2
Patient 3
Patient 5
Patient 4
Paleontology
Computational Anatomy
𝑺𝑬(𝟑) 2006, Nature Publishing Group, Genetics
Statistics Nina Miolane - Statistics on Lie groups
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Models with Lie groups Articulated model of spine: Vertebra 𝑅, 𝑡 ∈ 𝑆𝐸(3) Rotation Matrix
Statistics on spines: Mean, PCA…
translation vector
(𝑅, 𝑡)
PCA : Modes 1 to 4
Computational Anatomy : model and analyze the variability of human anatomy Nina Miolane - Statistics on Lie groups
10/17/2014
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New statistics on Lie groups? On Lie group 𝑺𝑬(𝟑) (space of transformations)
Consequence for model Vertebra : 𝑅, 𝑡 ∈ 𝑆𝐸(3)
𝐑 𝟏 , 𝒕𝟏
𝐑 𝟐 , 𝒕𝟐 𝐑 𝟏 , 𝒕𝟏
?
𝑹 +𝑹 𝒕 +𝒕 ( 𝟏 𝟐 𝟐, 𝟏 𝟐 𝟐)
(
𝑹𝟏 +𝑹𝟐 𝒕𝟏 +𝒕𝟐 , 𝟐 ) 𝟐
𝐑 𝟐 , 𝒕𝟐 𝑺𝑬 𝟑
Mean not on 𝑆𝐸(3), thus not a rigid transformation Usual statistics: linear Lie groups: not linear in general Nina Miolane - Statistics on Lie groups
?
Average pose of two vertebrae is not a vertebra Need new statistics! 10/17/2014
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Outline Can we use the pseudo-Riemannian framework to define statistics on Lie groups?
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
10/17/2014
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Outline Can we use the pseudo-Riemannian framework to define statistics on Lie groups?
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
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(Pseudo-) Riemannian structure on Lie groups Manifold 𝑴 Differential structure
+ metrical
+ algebraic
Lie group (𝑮,∘)
(Pseudo-) Riemannian manifold (𝑴, ) Metric : collection of positive definite inner products Pseudo-metric : collection of definite inner products
Consistency of the structures requires bi-invariant pseudo-metric
Quadratic Lie group (𝑮,∘, )
Nina Miolane - Statistics on Lie groups
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Statistics on Lie groups : the mean Requirements for
𝒈
mean of data set
Consistency 𝒈 mean of 𝑔𝑖
𝑖
Computability
𝑖
Conditions on 𝑔𝑖 𝑖 such that 𝒈 exists and is unique?
𝐿ℎ
𝒉 ∘ 𝒈 mean of ℎ ∘ 𝑔𝑖
𝑔𝑖
Riemannian structure Bi-invariant metric Fréchet mean = group mean Fréchet mean: definition with computability conditions
𝑖
Group exponential barycenter [1] =group mean [1] Pennec & Arsigny. Exponential barycenter of the canonical Cartan connection and invariant means on Lie groups. (2012)
Nina Miolane - Statistics on Lie groups
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Bi-invariant metric? On all Lie groups? Existence of a bi-invariant metric?
NO! On Lie groups that have a group mean? NO!
Characterization of Lie groups with bi-invariant metric by Cartan [2] : compact & abelian
𝑺𝑬(𝒏): • unique (a.e.) group mean • no bi-invariant metric
Group mean not characterized by a bi-invariant metric [1] [1] Pennec & Arsigny. Exponential barycenter of the canonical Cartan connection and invariant means on Lie groups. (2012) [2] Elie Cartan. La théorie des groups finis et continus et l’analyse situs. (1952)
Nina Miolane - Statistics on Lie groups
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Statistics on Lie groups : the mean Requirements for
𝒈
mean of data set
Consistency 𝒈 mean of 𝑔𝑖
𝑔𝑖
𝑖
Computability Riemannian structure
𝑖
Conditions on 𝑔𝑖 𝑖 such that 𝒈 exists and is unique?
𝐿ℎ
𝒉 ∘ 𝒈 mean of ℎ ∘ 𝑔𝑖
𝑖
Group exponential barycenter [1] =group mean
Fréchet mean: definition with computability conditions
Bi-invariant metric Fréchet mean = group mean Generalize Riemannian to pseudo-Riemannian
Pseudo-Riemannian structure Bi-invariant pseudo-metric
[1] Pennec & Arsigny. Exponential barycenter of the canonical Cartan connection and invariant means on Lie groups. (2012)
Nina Miolane - Statistics on Lie groups
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Bi-invariant pseudo-metrics? On all Lie groups? NO!
Existence of a bi-invariant pseudo-metric?
On Lie groups that have a group mean? Characterization of Lie groups with bi-invariant pseudo-metric by Medina & Revoy [3]
𝑆𝐸(𝑛): • unique (a.e.) group mean • bi-invariant pseudo-metric ?
Group mean characterized by a bi-invariant pseudo-metric ? [3] Medina & Revoy. Groupes de Lie munis de pseudo-metriques bi-invariantes. (1982)
Nina Miolane - Statistics on Lie groups
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Outline Can we use the pseudo-Riemannian framework to define statistics on Lie groups?
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
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From Lie group to Lie algebra Compute bi-invariant pseudo-metrics on finite dimensional connected Lie group 𝑮 𝑫𝑳𝒉 . 𝒗
𝒗
𝒖 𝒈
bi-invariant: < 𝒖, 𝒗 >𝒈 =< 𝑫𝑳𝒉 . 𝒖, 𝑫𝑳𝒉 . 𝒗 >𝑳𝒉 𝒈 =< 𝐷𝑅ℎ . 𝑢, 𝐷𝑅ℎ . 𝑣 >𝑅ℎ 𝑔 where 𝐿ℎ 𝑔 = ℎ ∘ 𝑔 and 𝑅ℎ 𝑔 = 𝑔 ∘ ℎ
𝑳𝒉 𝑳𝒉 𝒈 = 𝒉 ∘ 𝒈
𝑫𝑳𝒉 . 𝒖
Compute bi-invariant pseudo-metrics on its Lie algebra 𝖌 = 𝐓𝐞 𝐆, +, . , , bi-invariant: < 𝑥, 𝑦 𝔤 , 𝑧 >𝑒 +< 𝑦, 𝑥, 𝑧
𝔤
𝖌
>𝑒 = 0
Structure of quadratic 𝖌 ? Nina Miolane - Statistics on Lie groups
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Lie algebra representations Structure of quadratic 𝖌? Study adjoint representation of 𝔤 Representation 𝜼 of 𝖌 on 𝑽: Lie algebra homomorphism 𝜂: 𝔤 ↦ 𝔤𝔩(𝑉) Ex.: Homogeneous representation of 𝑆𝐸 3 on ℝ4 :
𝜂: 𝑆𝐸 3 ↦ 𝔤𝔩 ℝ4 𝑅 𝑡 𝑅, 𝑡 ↦ 0 1
s.t.
𝑅 0
𝑥 𝑡 𝑅. 𝑥 + 𝑡 . = 1 1 1
Subrepresentation: subspace of 𝑉 stable by 𝜂 𝔤 Subrepresentation decomposition: 𝑉 = 𝐵1 ⊕𝔤 … ⊕𝔤 𝐵𝑁 with 𝐵𝑖 subrepresentations Indecomposable subrepresentation: not a sum of subrepresentations Adjoint representation 𝒂𝒅 of 𝖌 (on itself: 𝑽 = 𝖌 ) 𝔤
𝑎𝑑: 𝔤 ↦ 𝔤𝔩(𝔤) 𝑥 ↦ 𝑎𝑑 𝑥 = 𝑥,∙
𝔤
𝔤
𝔤 = 𝐵1 ⊕𝔤 … ⊕𝔤 𝐵𝑁 : decomposition into indecomposable subrepresentations Nina Miolane - Statistics on Lie groups
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Structure of quadratic 𝔤 𝔤 𝐵1 𝔤
𝐵𝑁 𝔤
𝐵𝑖 𝔤
𝑩𝒊 𝖌 =1-dim.
𝑩𝒊 𝖌 =simple
𝑩𝒊 𝖌 = 𝐒 ⊕ 𝑺∗
𝑩𝒊 𝖌 = 𝐖 ⊕ 𝐒 ⊕ 𝑺∗
Th. (Medina & Revoy): Structure of quadratic 𝖌 𝔤 𝔤 𝔤 Adjoint representation decomposition 𝔤 = B1 ⊕𝔤 … ⊕𝔤 BN has indecomposable B𝑖 s.t.: 𝖌 Type (1): 𝑩𝒊 simple or 1-dim. 𝖌 Type (2): 𝑩𝒊 = 𝐖 ⊕ 𝐒 ⊕ 𝐒 ∗ double extension of 𝑊 quadratic by 𝑆 of Type (1)
Nina Miolane - Statistics on Lie groups
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Bi-invariant pseudo-metric on quadratic 𝔤 < 𝑏1 + ⋯ + 𝑏𝑁 , 𝑏1′ + ⋯ + 𝑏𝑁 >𝔤 =< 𝑏1 , 𝑏1′ >𝐵1 + ⋯ +< 𝑏𝑁 , 𝑏𝑁′ >𝐵𝑁
𝔤 𝐵1 𝔤
𝐵𝑁 𝔤
𝐵𝑖 𝔤
𝑩𝒊 𝖌 =1-dim.
𝑩𝒊 𝖌 =simple
𝑩𝒊 𝖌 = 𝐒 ⊕ 𝑺∗
𝑩𝒊 𝖌 = 𝐖 ⊕ 𝐒 ⊕ 𝑺∗
< 𝒃, 𝒃′ >𝑩𝖌 = 𝑲𝒊𝒍𝒍𝒊𝒏𝒈(𝒃, 𝒃′ )
< 𝒘 + 𝒔 + 𝒇, 𝒘′ + 𝒔′ + 𝒇′ >𝑩𝖌 = < 𝒘, 𝒘′ >𝑾 + 𝒇 𝒔′ + 𝒇′(𝒔)
𝒊
𝒊
< 𝒃, 𝒃′ >𝑩𝖌 = 𝒃. 𝒃′ 𝒊
< 𝒔 + 𝒇, 𝒔′ + 𝒇′ >𝑩𝖌 = 𝒇 𝒔′ + 𝒇′(𝒔) 𝒊
Nina Miolane - Statistics on Lie groups
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Algorithm 𝔤 𝐵1 𝔤
𝐵𝑁 𝔤
𝐵𝑖 𝔤
𝑩𝒊 𝖌 =1-dim. ?
𝑩𝒊 𝖌 =simple? 𝑩𝒊 𝖌 = 𝐒 ⊕ 𝑺∗ ?
𝑩𝒊 𝖌 = 𝐖 ⊕ 𝐒 ⊕ 𝑺∗ ?
Else : EXIT
If algorithm finishes: expression of a bi-invariant pseudo-metric on 𝔤 If EXIT: no bi-invariant pseudo-metric on 𝔤
Nina Miolane - Statistics on Lie groups
10/17/2014
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Outline Can we use the pseudo-Riemannian framework to define statistics on Lie groups?
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
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Rigid transformations SE(n) Lie group: 𝑆𝐸 𝑛 =
𝑅, 𝑡 ∈ 𝑆𝑂 𝑛 ⋉ ℝ𝑛
Lie algebra: 𝔰𝔢 𝑛 = { 𝐴, 𝑢 ∈ 𝑆𝑘𝑒𝑤 𝑛 ⊕ ℝ𝑛 }
𝖘𝖊(𝟑)
𝖘𝖊(𝟏) = ℝ
𝖘𝖊(𝟐)
𝑩𝟏 = 𝖘𝖊 𝟏 = ℝ
𝑩𝟏 = 𝖘𝖊(𝟐)
EXIT
𝑩𝟏 = 1-dim. ′
< 𝒃, 𝒃 >𝔰𝔢 𝟏 = 𝒃. 𝒃′
No bi-invariant pseudo-metric
Nina Miolane - Statistics on Lie groups
𝑩𝟏 = 𝖘𝖊(𝟑)
𝑩𝟏 = 𝑺 ⊕ 𝑺 ∗ i.e. 𝖘𝖊 𝟑 = 𝑺𝒌𝒆𝒘 𝟑 ⊕ ℝ𝟑 < 𝒂 + 𝒖, 𝒂′ + 𝒖′ >𝔰𝔢 𝟑 = 𝒂𝑻 . 𝒖′ + 𝒂′𝑻 . 𝒖
𝖘𝖊(𝒏) 𝑩𝟏 = 𝖘𝖊(𝒏)
EXIT No bi-invariant pseudo-metric
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Other Lie groups: ST(n), H, UT(n) Scalings and Translations
Heisenberg
Scaled Upper Unitriangular matrices
𝔰𝔱(𝑛)
𝔥
𝔲𝔱 𝑛
𝔰𝔱(𝑛)
𝔥
EXIT
𝔡
EXIT
𝖉 = 1-dim.
𝔥
EXIT
No bi-invariant pseudo-metric for 𝒏 ≠ 𝟎 Nina Miolane - Statistics on Lie groups
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Conclusion Quadratic Lie group with:
Bi-invariant metric Characterization by Cartan
1-dim.
compact
Bi-invariant pseudo-metric Characterization by Medina & Revoy
1-dim.
simple
𝐒 ⊕ 𝑺∗
𝐖 ⊕ 𝐒 ⊕ 𝑺∗
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 𝐺
Riemannian?
Pseudo-Riemannian?
NO
NO
Most Lie groups with group mean without bi-invariant metric [1]
Most Lie groups with group mean without bi-invariant pseudo-metric
Yet another geometric structure? Nina Miolane - Statistics on Lie groups
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1 Patient 1
Patient 2
Reference
5 2
3
4
Patient 5
Patient 3 Patient 4
Thank you for your attention
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