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

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 models Robotics Computational Medicine Reference  1 𝑻𝑷 𝟒 Spherical arm  5 Patient 1  2  4 Patient 5  3 http://www.societyofrobots.com/ Patient 4 Patient 3 Patient 2 Paleontology Computational Anatomy 𝑻𝑭(𝟒) 2006, Nature Publishing Group, Genetics  Statistics 10/17/2014 2 Nina Miolane - Statistics on Lie groups

Models with Lie groups Articulated model of spine: Statistics on spines: Vertebra 𝑆, 𝑢 ∈ 𝑇𝐹(3) Mean, PCA… Rotation Matrix translation vector (𝑆, 𝑢) PCA : Modes 1 to 4 Computational Anatomy : model and analyze the variability of human anatomy 10/17/2014 3 Nina Miolane - Statistics on Lie groups

New statistics on Lie groups? On Lie group 𝑻𝑭(𝟒) Consequence for model Vertebra : 𝑆, 𝑢 ∈ 𝑇𝐹(3) (space of transformations) 𝐒 𝟐 , 𝒖 𝟐 𝐒 𝟑 , 𝒖 𝟑 ? ( 𝑺 𝟐 +𝑺 𝟑 , 𝒖 𝟐 +𝒖 𝟑 𝐒 𝟐 , 𝒖 𝟐 𝑺 𝟐 +𝑺 𝟑 𝒖 𝟐 +𝒖 𝟑 𝟑 ) ( 𝟑 ) , 𝟑 𝟑 ? 𝐒 𝟑 , 𝒖 𝟑 𝑻𝑭 𝟒 Mean not on 𝑇𝐹(3) , thus not a Average pose of two vertebrae is rigid transformation not a vertebra Usual statistics: linear Need new statistics! Lie groups: not linear in general 10/17/2014 4 Nina Miolane - Statistics on Lie groups

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 10/17/2014 5 Nina Miolane - Statistics on Lie groups

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

(Pseudo-) Riemannian structure on Lie groups Manifold 𝑵 Differential structure + algebraic + metrical Lie group (Pseudo-) Riemannian manifold (𝑯,∘) (𝑵, <, >) 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 (𝑯,∘, <, >) 10/17/2014 7 Nina Miolane - Statistics on Lie groups

Statistics on Lie groups : the mean 𝒉 𝑕 𝑗 𝑗  Requirements for mean of data set Consistency Computability Riemannian structure 𝒉 mean of 𝑕 𝑗 𝑗  Bi-invariant metric Conditions on 𝑕 𝑗 𝑗 Fréchet mean = group mean 𝑀 ℎ such that exists 𝒉 Fréchet mean: definition with and is computability conditions unique? 𝒊 ∘ 𝒉 mean of ℎ ∘ 𝑕 𝑗 𝑗 Group exponential barycenter [1] =group mean [1] Pennec & Arsigny. Exponential barycenter of the canonical Cartan connection and invariant means on Lie groups. (2012) 10/17/2014 8 Nina Miolane - Statistics on Lie groups

Bi-invariant metric? On all Lie groups? NO! Existence of a bi-invariant metric ? On Lie groups that have a group mean? NO! Characterization of Lie groups with 𝑻𝑭(𝒐) : bi-invariant metric by Cartan [2] : • unique (a.e.) group mean compact & abelian • 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) 10/17/2014 9 Nina Miolane - Statistics on Lie groups

Statistics on Lie groups : the mean 𝒉 𝑕 𝑗 𝑗  Requirements for mean of data set Consistency Computability Riemannian structure 𝒉 mean of 𝑕 𝑗 𝑗 Fréchet mean: definition with Conditions computability conditions on 𝑕 𝑗 𝑗 𝑀 ℎ such that exists 𝒉 Bi-invariant metric and is  Fréchet mean = group mean unique? 𝒊 ∘ 𝒉 mean of ℎ ∘ 𝑕 𝑗 𝑗 Generalize Riemannian to pseudo-Riemannian Pseudo-Riemannian structure Group exponential barycenter [1] =group mean Bi-invariant pseudo-metric [1] Pennec & Arsigny. Exponential barycenter of the canonical Cartan connection and invariant means on Lie groups. (2012) 10/17/2014 10 Nina Miolane - Statistics on Lie groups

Bi-invariant pseudo-metrics? On all Lie groups? NO! Existence of a bi-invariant pseudo-metric ? On Lie groups that have a group mean? 𝑇𝐹(𝑜) : Characterization of Lie groups with • unique (a.e.) group mean bi-invariant pseudo-metric • bi-invariant pseudo-metric ? by Medina & Revoy [3] Group mean characterized by a bi-invariant pseudo-metric ? [3] Medina & Revoy. Groupes de Lie munis de pseudo-metriques bi-invariantes. (1982) 10/17/2014 11 Nina Miolane - Statistics on Lie groups

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

From Lie group to Lie algebra Compute bi-invariant pseudo-metrics on finite dimensional connected Lie group 𝑯 𝑬𝑴 𝒊 . 𝒘 bi-invariant : < 𝒗, 𝒘 > 𝒉 =< 𝑬𝑴 𝒊 . 𝒗, 𝑬𝑴 𝒊 . 𝒘 > 𝑴 𝒊 𝒉 =< 𝐸𝑆 ℎ . 𝑣, 𝐸𝑆 ℎ . 𝑤 > 𝑆 ℎ 𝑕 𝑴 𝒊 where 𝑀 ℎ 𝑕 = ℎ ∘ 𝑕 and 𝑆 ℎ 𝑕 = 𝑕 ∘ ℎ 𝒗 𝒘 𝑴 𝒊 𝒉 = 𝒊 ∘ 𝒉 𝒉 𝑬𝑴 𝒊 . 𝒗 Compute bi-invariant pseudo-metrics on its Lie algebra 𝖍 = 𝐔 𝐟 𝐇, +, . , , 𝖍 bi-invariant: < 𝑦, 𝑧 𝔥 , 𝑨 > 𝑓 +< 𝑧, 𝑦, 𝑨 𝔥 > 𝑓 = 0 Structure of quadratic 𝖍 ? 10/17/2014 13 Nina Miolane - Statistics on Lie groups

Lie algebra representations Structure of quadratic 𝖍 ? Study adjoint representation of 𝔥 Representation 𝜽 of 𝖍 on 𝑾 : Lie algebra homomorphism 𝜃: 𝔥 ↦ 𝔥𝔪(𝑊) Ex.: Homogeneous representation of 𝑇𝐹 3 on ℝ 4 : 𝜃: 𝑇𝐹 3 ↦ 𝔥𝔪 ℝ 4 1 . 𝑦 s.t. 𝑆 𝑢 1 = 𝑆. 𝑦 + 𝑢 𝑆, 𝑢 ↦ 𝑆 𝑢 0 1 1 0 Subrepresentation: subspace of 𝑊 stable by 𝜃 𝔥 Subrepresentation decomposition: 𝑊 = 𝐶 1 ⊕ 𝔥 … ⊕ 𝔥 𝐶 𝑂 with 𝐶 𝑗 subrepresentations Indecomposable subrepresentation: not a sum of subrepresentations Adjoint representation 𝒃𝒆 of 𝖍 (on itself: 𝑾 = 𝖍 ) 𝑏𝑒: 𝔥 ↦ 𝔥𝔪(𝔥) 𝑦 ↦ 𝑏𝑒 𝑦 = 𝑦,∙ 𝔥 𝔥 ⊕ 𝔥 … ⊕ 𝔥 𝐶 𝑂 𝔥 : decomposition into indecomposable subrepresentations 𝔥 = 𝐶 1 10/17/2014 14 Nina Miolane - Statistics on Lie groups

Structure of quadratic 𝔥 𝔥 𝔥 𝔥 𝔥 𝐶 1 𝐶 𝑗 𝐶 𝑂 𝖍 = 𝐓 ⊕ 𝑻 ∗ 𝖍 = 𝐗 ⊕ 𝐓 ⊕ 𝑻 ∗ 𝖍 =1-dim. 𝖍 =simple 𝑪 𝒋 𝑪 𝒋 𝑪 𝒋 𝑪 𝒋 Th. (Medina & Revoy): Structure of quadratic 𝖍 𝔥 ⊕ 𝔥 … ⊕ 𝔥 B N 𝔥 has indecomposable B 𝑗 𝔥 s.t.: Adjoint representation decomposition 𝔥 = B 1 𝖍 simple or 1-dim. Type (1): 𝑪 𝒋 𝖍 = 𝐗 ⊕ 𝐓 ⊕ 𝐓 ∗ double extension of 𝑋 quadratic by 𝑇 of Type (1) Type (2): 𝑪 𝒋 10/17/2014 15 Nina Miolane - Statistics on Lie groups

Bi-invariant pseudo-metric on quadratic 𝔥 ′ + ⋯ + 𝑐 𝑂 > 𝔥 𝔥 < 𝑐 1 + ⋯ + 𝑐 𝑂 , 𝑐 1 ′ > 𝐶 1 + ⋯ +< 𝑐 𝑂 , 𝑐 𝑂 ′ > 𝐶 𝑂 =< 𝑐 1 , 𝑐 1 𝔥 𝔥 𝔥 𝐶 1 𝐶 𝑗 𝐶 𝑂 𝖍 = 𝐓 ⊕ 𝑻 ∗ 𝖍 = 𝐗 ⊕ 𝐓 ⊕ 𝑻 ∗ 𝖍 =1-dim. 𝖍 =simple 𝑪 𝒋 𝑪 𝒋 𝑪 𝒋 𝑪 𝒋 𝖍 = 𝑳𝒋𝒎𝒎𝒋𝒐𝒉(𝒄, 𝒄 ′ ) < 𝒄, 𝒄′ > 𝑪 𝒋 < 𝒙 + 𝒕 + 𝒈, 𝒙 ′ + 𝒕 ′ + 𝒈 ′ > 𝑪 𝒋 𝖍 = < 𝒙, 𝒙 ′ > 𝑿 + 𝒈 𝒕 ′ + 𝒈′(𝒕) < 𝒄, 𝒄 ′ > 𝑪 𝒋 𝖍 = 𝒄. 𝒄′ < 𝒕 + 𝒈, 𝒕 ′ + 𝒈 ′ > 𝑪 𝒋 𝖍 = 𝒈 𝒕 ′ + 𝒈′(𝒕) 10/17/2014 16 Nina Miolane - Statistics on Lie groups

Algorithm 𝔥 𝔥 𝔥 𝔥 𝐶 1 𝐶 𝑂 𝐶 𝑗 𝖍 = 𝐓 ⊕ 𝑻 ∗ ? 𝖍 = 𝐗 ⊕ 𝐓 ⊕ 𝑻 ∗ ? Else : EXIT 𝖍 =1-dim. ? 𝖍 =simple? 𝑪 𝒋 𝑪 𝒋 𝑪 𝒋 𝑪 𝒋 If algorithm finishes: expression of a bi-invariant pseudo-metric on 𝔥 If EXIT: no bi-invariant pseudo-metric on 𝔥 10/17/2014 17 Nina Miolane - Statistics on Lie groups

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