Wavelets on the symmetric group Risi Kondor Walter Dempsey f ( - PowerPoint PPT Presentation

Wavelets on the symmetric group Risi Kondor Walter Dempsey

� � f ( λ ) = f ( σ ) ρ λ ( σ ) σ ∈ S n

� � � � � S 3 � � ������������������������ � � � � � � � � � � � � � � � � � � � � � � � S 3 S 2 S 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � S 3 , 2 = { [123] } S 3 , 1 = { [213] } S 2 , 3 = { [132] } S 2 , 1 = { [312] } S 1 , 3 = { [231] } S 1 , 2 = { [321] } S i 1 ...i k = { σ ∈ S n | σ ( n ) = i 1 , . . . , σ ( n − k +1) = i k } = µ i 1 ...i k S n − k

� � � � �� � � � � � �� � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � . . . ⊂ V − 1 ⊂ V 0 ⊂ V 1 ⊂ V 2 ⊂ . . . ⊂ L 2 ( R ) MRA1. � k V k = { 0 } , MRA2. � k V k = L 2 ( R ), MRA3. for any f ∈ V k and any m ∈ Z , the function f � ( x ) = f ( x − m 2 � k ) is also in V k , MRA4. for any f ∈ V k , the function f � ( x ) = f (2 x ), is in V k +1 . � V 0 . . . . . . V − 1 V − 2 V − 3 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � W − 1 W − 2 W − 3 W − 4

We say that a sequence of spaces V 0 ⊆ V 1 ⊆ . . . ⊆ V n − 1 = R S n forms a left-invariant coset based multiresolution analysis (L-CMRA) for S n if L1. for any f ∈ V k and any τ ∈ S n , we have T τ f ∈ V k , L2. if f ∈ V k , then P i 1 ...i k +1 f ∈ V k +1 , for any i 1 , . . . , i k +1 , and L3. if g ∈ V k +1 , then for any i 1 , . . . , i k +1 there is an f ∈ V k such that P i 1 ...i k +1 f = g .

We say that a sequence of spaces V 0 ⊆ V 1 ⊆ . . . ⊆ V n − 1 = R S n forms a bi-invariant coset based multiresolution analysis (Bi-CMRA) for S n if Bi1. for any f ∈ V k and any τ ∈ S n , we have T τ f ∈ V k and T R τ f ∈ V k Bi2. if f ∈ V k , then P i 1 ...i k +1 f ∈ V k +1 , for any i 1 , . . . , i k +1 ; and Bi3. if g ∈ V k +1 , then for any i 1 , . . . , i k +1 there is an f ∈ V k such that P i 1 ...i k +1 f = g .

Proposition 1 If { M t } t ∈ T n are the adapted left S n –modules of R S n , and V 0 = � t ∈ ν 0 M t for some ν 0 ⊆ T n , then � � ν k = ν 0 ↓ n − k ↑ n , V k = W k = M t , M t , where t ∈ ν k t ∈ ν k +1 \ ν k (1) for any k ∈ { 0 , 1 , . . . , n − 1 } . 1 3 5 6 7 t = 2 4 ∈ T 8 , 8

If ν 0 = { 1 2 3 4 5 } , then � � 1 2 3 4 1 2 3 4 ↑ 5 = ν 1 = 1 2 3 4 5 , 5 1 2 3 � � 1 2 3 4 1 2 3 5 1 2 3 4 1 2 3 ↑ 5 = ν 2 = 1 2 3 4 5 , 5 4 4 5 , 5 , , 1 2 3 1 2 4 1 2 5 � � 1 2 3 4 1 2 3 5 1 2 4 5 1 2 3 1 2 4 1 2 5 4 3 3 1 2 ↑ 5 = ν 3 = 1 2 3 4 5 , 5 4 3 4 5 , 3 5 , 3 4 , 5 5 4 , , , , , , . . . � � 1 2 3 4 1 2 3 5 1 2 4 5 1 3 4 5 1 2 3 1 2 4 1 2 5 1 3 4 1 3 5 1 ↑ 5 = ν 4 = 1 2 3 4 5 , 5 4 3 2 4 5 , 3 5 , 3 4 , 2 5 , 2 4 , . . . , , , , (analog of Haar wavelets)

Proposition 1 Given a set of partitions ν 0 ⊆ Λ n , the corresponding Bi-CMRA comprises the spaces ν k = ν 0 ↓ n − k ↑ n . � � V k = W k = (1) where U λ , U λ , λ ∈ ν k λ ∈ ν k +1 \ ν k Moreover, any system of spaces satisfying Definition ?? is of this form for some ν 0 ⊆ Λ n . λ =

If ν 0 = { (5) } = { } , then � � } ↑ 5 = ν 1 = { , � � } ↑ 5 = ν 2 = { , , , � � } ↑ 5 = ν 3 = { , , , , , � � } ↑ 5 = ν 4 = { , , , , , , .

� � d λ ( t ) / ( n − k )! [ ρ λ ( t ) ( µ − 1 i 1 ...i k σ )] t � ,t σ ∈ µ i 1 ...i k S n − k ψ i 1 ...i kt,t � ( σ ) := 0 otherwise , � � d λ ( t ) / ( n − k )! [ ρ λ ( t ) ( µ − 1 i 1 ...i k σ µ j 1 ...j k )] t � ,t σ ∈ µ i 1 ...i k S n − k µ j 1 ...j k ψ i 1 ...i k j 1 ...j k ,t,t � ( σ ) := 0 otherwise ,

1: function FastLCWT( f, ν , ( i 1 . . . i k )) { 2: if k = n � 1 then return ( Scaling ν ( v ( f ))) 3: 4: end if 5: v � 0 6: for each i k +1 �� { i 1 . . . i k } do if P i 1 ...i k +1 f � = 0 then 7: v � v + Φ i k (FastLCWT( f � i 1 ...i k +1 , ν � n − k − 1 , ( i 1 . . . i k +1 ))) 8: end if 9: 10: end for 11: output Wavelet ν ↓ n − k − 1 ↑ n − k \ ν ( v ) 12: return Scaling ν ( v ) } Complexity: O ( n 2 q � t ∈ ν 1 d λ ( t ) )

to do: • properly implement in SnOB2 • try sparse estimation in wavelet space on data • algorithms for manipulating distributions while maintaining sparsity • think about what projective MRA means on other groups