Approximation by group invariant subspaces Davide Barbieri - PowerPoint PPT Presentation

Approximation by group invariant subspaces Davide Barbieri (Universidad Aut´ onoma de Madrid) Joint work with C. Cabrelli, E. Hern´ andez and U. Molter XIV Encuentro Nacional de Analistas A. P. Calder´ on Villa General Belgrano, 22 de Noviembre de 2018

Motivation I: dimensionality reduction Approximation by linear subspaces of finite dimensional data in a vector space: Principal Component Analysis.

Motivation I: dimensionality reduction Approximation by linear subspaces of finite dimensional data in a vector space: Principal Component Analysis.

Motivation I: dimensionality reduction Approximation by linear subspaces of finite dimensional data in a vector space: Principal Component Analysis.

Motivation I: dimensionality reduction Approximation by linear subspaces of finite dimensional data in a vector space: Principal Component Analysis. Approximation by shift-invariant subspaces of data in L 2 ( R d ): Aldroubi, Cabrelli, Hardin and Molter 2007.

Motivation II: symmetries in data - abelian

Motivation II: symmetries in data - abelian

Motivation II: symmetries in data - non abelian

Results For non abelian symmetries on L 2 ( R d ), we will discuss: 1. characterizations of invariant spaces; 2. construction of group Parseval frames; 3. approximation by group invariant subspaces.

Definition of group invariance Let Λ ⊂ R d be a lattice subgroup 1 , and let G ⊂ O ( d ) be a finite group of isometries such that g Λ = Λ for all g ∈ G . 1 That is Λ = A Z d ⊂ R d for A ∈ GL d ( R ).

Definition of group invariance Let Λ ⊂ R d be a lattice subgroup 1 , and let G ⊂ O ( d ) be a finite group of isometries such that g Λ = Λ for all g ∈ G . Let Γ = Λ ⋊ G = { ( k , g ) : k ∈ Λ , g ∈ G } , with composition law ( k , g ) · ( k ′ , g ′ ) = ( gk ′ + k , gg ′ ) . Γ is a crystallographic group, which acts on R d by ( k , g ) x = gx + k . 1 That is Λ = A Z d ⊂ R d for A ∈ GL d ( R ).

Definition of group invariance Let Λ ⊂ R d be a lattice subgroup 1 , and let G ⊂ O ( d ) be a finite group of isometries such that g Λ = Λ for all g ∈ G . Let Γ = Λ ⋊ G = { ( k , g ) : k ∈ Λ , g ∈ G } , with composition law ( k , g ) · ( k ′ , g ′ ) = ( gk ′ + k , gg ′ ) . Γ is a crystallographic group, which acts on R d by ( k , g ) x = gx + k . The corresponding action on L 2 ( R d ) is given by the operators T ( k ) f ( x ) = f ( x − k ) , R ( g ) f ( x ) = f ( g − 1 x ) , for f ∈ L 2 ( R d ) which indeed satisfy T ( k ) R ( g ) T ( k ′ ) R ( g ′ ) = T ( gk ′ + k ) R ( gg ′ ). 1 That is Λ = A Z d ⊂ R d for A ∈ GL d ( R ).

Definition of group invariance Let Λ ⊂ R d be a lattice subgroup 1 , and let G ⊂ O ( d ) be a finite group of isometries such that g Λ = Λ for all g ∈ G . Let Γ = Λ ⋊ G = { ( k , g ) : k ∈ Λ , g ∈ G } , with composition law ( k , g ) · ( k ′ , g ′ ) = ( gk ′ + k , gg ′ ) . Γ is a crystallographic group, which acts on R d by ( k , g ) x = gx + k . The corresponding action on L 2 ( R d ) is given by the operators T ( k ) f ( x ) = f ( x − k ) , R ( g ) f ( x ) = f ( g − 1 x ) , for f ∈ L 2 ( R d ) which indeed satisfy T ( k ) R ( g ) T ( k ′ ) R ( g ′ ) = T ( gk ′ + k ) R ( gg ′ ). A closed subspace V ⊂ L 2 ( R d ) is Γ-invariant if T ( k ) R ( g ) V ⊂ V ∀ k ∈ Λ , g ∈ G . 1 That is Λ = A Z d ⊂ R d for A ∈ GL d ( R ).

Shift-invariant spaces I

Shift-invariant spaces I Let Λ ⊥ ⊂ R d be the annihilator 2 lattice of Λ, and let Ω ⊂ R d be | Ω ∩ (Ω + s ) | = 0 for 0 � = s ∈ Λ ⊥ , and | R d \ � s ∈ Λ ⊥ Ω + s | = 0. 2 If Λ = A Z d , then Λ ⊥ = ( A t ) − 1 Z d .

Shift-invariant spaces I Let Λ ⊥ ⊂ R d be the annihilator 2 lattice of Λ, and let Ω ⊂ R d be | Ω ∩ (Ω + s ) | = 0 for 0 � = s ∈ Λ ⊥ , and | R d \ � s ∈ Λ ⊥ Ω + s | = 0. The map T : L 2 ( R d ) → L 2 (Ω , ℓ 2 (Λ ⊥ )) is the surjective isometry T [ f ]( ω ) = { � f ( ω + s ) } s ∈ Λ ⊥ . 2 If Λ = A Z d , then Λ ⊥ = ( A t ) − 1 Z d .

Shift-invariant spaces I Let Λ ⊥ ⊂ R d be the annihilator 2 lattice of Λ, and let Ω ⊂ R d be | Ω ∩ (Ω + s ) | = 0 for 0 � = s ∈ Λ ⊥ , and | R d \ � s ∈ Λ ⊥ Ω + s | = 0. The map T : L 2 ( R d ) → L 2 (Ω , ℓ 2 (Λ ⊥ )) is the surjective isometry T [ f ]( ω ) = { � f ( ω + s ) } s ∈ Λ ⊥ . Since T [ T ( k ) f ]( ω ) = e − 2 π ik ω T [ f ]( ω ), it is equivalent to have ◮ V ⊂ L 2 ( R d ) is Λ-invariant: f ∈ V ⇒ T ( k ) f ∈ V for all k ∈ Λ ◮ T [ V ] is invariant under multiplication by e − 2 π ik ω for all k ∈ Λ 2 If Λ = A Z d , then Λ ⊥ = ( A t ) − 1 Z d .

Shift-invariant spaces I Let Λ ⊥ ⊂ R d be the annihilator 2 lattice of Λ, and let Ω ⊂ R d be | Ω ∩ (Ω + s ) | = 0 for 0 � = s ∈ Λ ⊥ , and | R d \ � s ∈ Λ ⊥ Ω + s | = 0. The map T : L 2 ( R d ) → L 2 (Ω , ℓ 2 (Λ ⊥ )) is the surjective isometry T [ f ]( ω ) = { � f ( ω + s ) } s ∈ Λ ⊥ . Since T [ T ( k ) f ]( ω ) = e − 2 π ik ω T [ f ]( ω ), it is equivalent to have ◮ V ⊂ L 2 ( R d ) is Λ-invariant: f ∈ V ⇒ T ( k ) f ∈ V for all k ∈ Λ ◮ T [ V ] is invariant under multiplication by e − 2 π ik ω for all k ∈ Λ If V is Λ-invariant, there exists Φ = { φ i } i ∈ N ⊂ L 2 ( R d ) such that L 2 ( R d ) . V = span { T ( k ) φ i : k ∈ Λ , i ∈ N } Thus L 2 (Ω ,ℓ 2 (Λ ⊥ )) T [ V ] = span { e − 2 π ik · T [ φ i ] : k ∈ Λ , i ∈ N } 2 If Λ = A Z d , then Λ ⊥ = ( A t ) − 1 Z d .

Shift-invariant spaces I The map T : L 2 ( R d ) → L 2 (Ω , ℓ 2 (Λ ⊥ )) is the surjective isometry T [ f ]( ω ) = { � f ( ω + s ) } s ∈ Λ ⊥ . Since T [ T ( k ) f ]( ω ) = e − 2 π ik ω T [ f ]( ω ), it is equivalent to have ◮ V ⊂ L 2 ( R d ) is Λ-invariant: f ∈ V ⇒ T ( k ) f ∈ V for all k ∈ Λ ◮ T [ V ] is invariant under multiplication by e − 2 π ik ω for all k ∈ Λ If V is Λ-invariant, there exists Φ = { φ i } i ∈ N ⊂ L 2 ( R d ) such that L 2 ( R d ) . V = span { T ( k ) φ i : k ∈ Λ , i ∈ N } Thus L 2 (Ω ,ℓ 2 (Λ ⊥ )) T [ V ] = span { e − 2 π ik · T [ φ i ] : k ∈ Λ , i ∈ N }

Shift-invariant spaces I The map T : L 2 ( R d ) → L 2 (Ω , ℓ 2 (Λ ⊥ )) is the surjective isometry T [ f ]( ω ) = { � f ( ω + s ) } s ∈ Λ ⊥ . Since T [ T ( k ) f ]( ω ) = e − 2 π ik ω T [ f ]( ω ), it is equivalent to have ◮ V ⊂ L 2 ( R d ) is Λ-invariant: f ∈ V ⇒ T ( k ) f ∈ V for all k ∈ Λ ◮ T [ V ] is invariant under multiplication by e − 2 π ik ω for all k ∈ Λ If V is Λ-invariant, there exists Φ = { φ i } i ∈ N ⊂ L 2 ( R d ) such that L 2 ( R d ) . V = span { T ( k ) φ i : k ∈ Λ , i ∈ N } Thus L 2 (Ω ,ℓ 2 (Λ ⊥ )) T [ V ] = span { e − 2 π ik · T [ φ i ] : k ∈ Λ , i ∈ N } so, we have that f ∈ V if and only if, for a.e. ω ∈ Ω, ℓ 2 (Λ ⊥ ) . T [ f ]( ω ) ∈ span {T [ φ i ]( ω ) : i ∈ N }

Shift-invariant spaces I The map T : L 2 ( R d ) → L 2 (Ω , ℓ 2 (Λ ⊥ )) is the surjective isometry T [ f ]( ω ) = { � f ( ω + s ) } s ∈ Λ ⊥ . If V is Λ-invariant, there exists Φ = { φ i } i ∈ N ⊂ L 2 ( R d ) such that L 2 ( R d ) . V = span { T ( k ) φ i : k ∈ Λ , i ∈ N } Thus L 2 (Ω ,ℓ 2 (Λ ⊥ )) T [ V ] = span { e − 2 π ik · T [ φ i ] : k ∈ Λ , i ∈ N } so, we have that f ∈ V if and only if, for a.e. ω ∈ Ω, ℓ 2 (Λ ⊥ ) . T [ f ]( ω ) ∈ span {T [ φ i ]( ω ) : i ∈ N }

Shift-invariant spaces I The map T : L 2 ( R d ) → L 2 (Ω , ℓ 2 (Λ ⊥ )) is the surjective isometry T [ f ]( ω ) = { � f ( ω + s ) } s ∈ Λ ⊥ . If V is Λ-invariant, there exists Φ = { φ i } i ∈ N ⊂ L 2 ( R d ) such that L 2 ( R d ) . V = span { T ( k ) φ i : k ∈ Λ , i ∈ N } Thus L 2 (Ω ,ℓ 2 (Λ ⊥ )) T [ V ] = span { e − 2 π ik · T [ φ i ] : k ∈ Λ , i ∈ N } so, we have that f ∈ V if and only if, for a.e. ω ∈ Ω, ℓ 2 (Λ ⊥ ) . T [ f ]( ω ) ∈ span {T [ φ i ]( ω ) : i ∈ N } The range function J of V is the measurable map J : Ω → { closed subspaces of ℓ 2 (Λ ⊥ ) } given by ℓ 2 (Λ ⊥ ) . J ( ω ) = span {T ( φ i )( ω ) : i ∈ N }

Shift-invariant spaces I The map T : L 2 ( R d ) → L 2 (Ω , ℓ 2 (Λ ⊥ )) is the surjective isometry T [ f ]( ω ) = { � f ( ω + s ) } s ∈ Λ ⊥ . If V is Λ-invariant, there exists Φ = { φ i } i ∈ N ⊂ L 2 ( R d ) such that L 2 ( R d ) . V = span { T ( k ) φ i : k ∈ Λ , i ∈ N } Thus L 2 (Ω ,ℓ 2 (Λ ⊥ )) T [ V ] = span { e − 2 π ik · T [ φ i ] : k ∈ Λ , i ∈ N } so, we have that f ∈ V if and only if, for a.e. ω ∈ Ω, ℓ 2 (Λ ⊥ ) . T [ f ]( ω ) ∈ span {T [ φ i ]( ω ) : i ∈ N } The range function J of V is the measurable map J : Ω → { closed subspaces of ℓ 2 (Λ ⊥ ) } given by ℓ 2 (Λ ⊥ ) . J ( ω ) = span {T ( φ i )( ω ) : i ∈ N }

Γ-invariance Γ-invariance = Λ-invariance + G -invariance

Γ-invariance Γ-invariance = Λ-invariance + G -invariance Characterize Γ-invariance ⇐ ⇒ characterize G -invariance for shift-invariant spaces.