Matrix Factorization and Lifting Palle Jorgensen and Myung-Sin Song - PowerPoint PPT Presentation

Matrix Factorization and Lifting Palle Jorgensen and Myung-Sin Song The University of Iowa, Southern Illinois University Edwardsville January 7, 2011

Introduction ◮ Develop and refine a procedure which uses factorization of families of matrix-valued function. ◮ Use the concept of “time-signal” widely allowing for systems of numbers indexed by pixels. ◮ Allow a practical procedure for breaking down an overall process into small processes in signal processing algorithms. The factorization of matrix functions accomplishes this.

Operations on Time-Signals Definition N = the principal N th root Let N ∈ Z + be given and set ζ N := e i 2 π of 1. Set N − 1 ( A N g )( z ) := 1 � g ( ζ k N z ) . (1) N k = 0 Here the two versions of the operator A N represent transformations in sequence spaces. But by Fourier-duality, this turns into associated actions on spaces of functions, so functions defined on T . Note the summation in (1) is over the cyclic group Z N = Z / N Z (= { 0 , 1 , · · · , N − 1 } ) .

Operations on Time-Signals Definition Let N ∈ Z + be given. If F is a vector valued function defined on T , by ( f k ) we mean the corresponding coordinate functions. The same for G and ( g k ) . If f is a scalar valued function, we denote the corresponding multiplication operator by M f . Two systems of functions F = ( f k ) k ∈ Z N G = ( g k ) k ∈ Z N and are said to be a perfect reconstruction filter iff � M g k A N M f k = I (see Fig. 1) (2) k ∈ Z N where the operator I on the RHS in (2) is the identity operator.

�� �� �� �� �� �� �� �� �� �� �� � �� Operations on Time-Signals In the engineering lingo, e.g. (2) is expressed as follows: g 0 f 0 g 1 f 1 Input Output . . . . . . . . . f N-1 g N-1 down-sampling up-sampling Figure: Perfect reconstruction in subband filtering as used in signal and image processing. Input is broken down into frequency bands, processes and then assembled (synthesis). Perfect reconstruction of output is desired.

Groups of Matrix Functions Let the set of all orthogonal N − filters be denoted OF N and the set, all unitary matrix functions by UM F . Definition Let U be an N × N matrix-function and let F = ( f k ) k ∈ Z N be a function system. Set G ( z ) := U ( z N ) F ( z ) , (3) or equivalently � U k , j ( z N ) f j ( z ) . g k ( z ) = (4) j ∈ Z N

Groups of Matrix Functions Lemma Let N ∈ Z + be given and let F = ( f j ) j ∈ Z + be a function system. Then F ∈ OF N if and only if the operators S j satisfy S ∗ j S k = δ j , k I � S j S ∗ j = I , j ∈ Z N where I denotes the identity operator in H = L 2 ( T ) ; compare with Fig. 1.

Group Actions (i) Outline how the entire processing system in Fig. 1 may be encoded into a representation of a certain C ∗ -algebra, an algebra on N generators and two relations, called Cuntz-relations, or generalized Cuntz-relations. (ii) State our first results regarding factorization in (infinite-dimensional) groups of functions taking values in some Lie group G ; matrix-functions for short.

Group Actions We will be making use of the special vector b ∈ F 2 ( N ) ,   1 z    z 2  b ( z ) = ;    .  .   .   z N − 1 Let ( S j f )( z ) = z j f ( z N ) (5) be the Cuntz-representation.

Group Actions Lemma Let N ∈ Z + be fixed, N > 1 and let A = ( A j , k ) be an N × N matrix-function with A j , k ∈ L 2 ( T ) . Then the following two conditions are equivalent: (i) For F = ( f j ) ∈ F 2 ( N ) , we have F ( z ) = A ( z N ) b ( z ) . (ii) A i , j = S ∗ j f i where the operators S i are from the Cuntz-relations (5).

Group Actions Proof. (i) ⇒ (ii). Writing out the matrix-operation in (i), we get A i , j ( z N ) z j = � � f i ( z ) = ( S j A i , j )( z ) . (6) j j Using S ∗ j S k = δ j , k I , we get A i , j = S ∗ j f i which is (ii). Conversely, assuming (ii) and using � j S i S ∗ j = I , we get � j S j A i , j = f i which is equivalent to (i) by the computation in (6) above.

Factorizations Sketch the first step in the general conclusions about factorization. The size of the problem has two parts: (a) The matrix size, i.e., the size of N where we consider N × N matrices. (b) The number of factors in our factorizations. To illustrate the idea, we begin with consideration of the case when N = 2 and the number of factors is also two.

Factorizations Lemma Let � A � B A = C D be a 2 × 2 matrix-function and let � f 0 ( z ) = A ( z 2 ) + z B ( z 2 ) f 1 ( z ) = C ( z 2 ) + z D ( z 2 ) . Let L and U be scalar functions. Then the following are equivalent: (i) � 1 � � 1 � � A � 0 U B = . L 1 0 1 C D (ii) U = S ∗ 1 f 0 and L = S ∗ 0 f 1 .

Notational Conventions Lemma Let f 0 , f 1 , · · · , f N − 1 be a system of N complex functions. (For the present purpose, we only need to assume that each f j is in L ∞ ( T ) .) Then the following three conditions are equivalent: (i) The functions f j satisfy � f j , f k � N ( z ) = δ j , k 1 , ∀ z ∈ T , module-orthogonality . (7) (ii) The operator S f j satisfy the Cuntz-relations � S ∗ f j S f k = δ j , k I L 2 ( T ) , and (8) � N − 1 j = 0 S f j S ∗ f j = I L 2 ( T ) . (iii) With ζ N := e i 2 π N , form the matrix function M N ( z ) = ( f j ( ζ k N z )) j , k ∈ Z N . (9)

Notational Conventions Then M N is a unitary matrix-function. Let N ∈ Z + be given and fixed. The following terminology will be used: GL N (pol): the N × N polynomial matrix function A such that A − 1 is also polynomial. SL N (pol) := { A ∈ GL N (pol) ; detA ≡ 1 } . (10)

Notational Conventions Our work on matrix functions gives the following: Theorem (Sweldens [SwRo91]) Let A ∈ SL 2 (pol), then there are l , p ∈ Z + , K ∈ C \ { 0 } and polynomial functions U 1 , . . . , U p , L 1 , . . . , L p such that � K � � 1 � � � 0 U 1 ( z ) 1 0 A ( z ) = z l · · · (11) K − 1 0 0 1 L 1 ( z ) 1 � 1 � � � U p ( z ) 1 0 . 0 1 L p ( z ) 1

Divisibility and Residues for Matrix-functions The 2 × 2 case To highlight the general ideas, we begin with some details worked out in the 2 × 2 case; see equation (11). First note that from the setting in above Theorem, we may n ∈ H a n z n assume that matrix entries have the form f H ( z ) := � but with H ⊂ { 0 , 1 , 2 , · · · } , i.e., f H ( z ) = a 0 + a 1 z + · · · . This facilitates our use of the Euclidean algorithm. Specifically, if f and g are polynomials (i.e., H ⊂ { 0 , 1 , 2 , · · · } ) and if deg ( g ) ≤ deg ( f ) , the Euclidean algorithm yields f ( z ) = g ( z ) q ( z ) + r ( z ) (12) with deg ( r ) < deg ( g ) . We shall write q = quot ( g , f ) , and r = rem ( g , f ) . (13)

Divisibility and Residues for Matrix-functions Since � K � � 1 � � 1 K 2 U � � K � 0 U 0 = , (14) K − 1 K − 1 0 0 1 0 1 0 we may assume that the factor � K � 0 K − 1 0 from the equation (11) factorization occurs on the rightmost place.

Divisibility and Residues for Matrix-functions The 3 × 3 case In the definition of A ∈ SL 3 ( pol ) , it is understood that A ( z ) has detA ( z ) ≡ 1 and that the entries of the inverse matrix A ( z ) − 1 are again polynomials. Note that if L , M , U and V are polynomials, then the four matrices         1 0 0 1 0 0 1 U 0 1 0 U  ,  , L 1 0 0 1 0 0 1 V and 0 1 0       0 M 1 L 0 1 0 0 1 0 0 1 (15) are in SL 3 ( pol ) since

Divisibility and Residues for Matrix-functions − 1     1 0 0 1 0 0 = − L L 1 0 1 0 and (16)     0 M 1 LM − M 1 − 1     1 U 0 1 − U UV  . 0 1 V = 0 1 − V (17)    0 0 1 0 0 1 Theorem Let A ∈ SL 3 ( pol ) ; then the conclusion in Theorem 8 carries over with the modification that the alternating upper and lower triangular matrix-functions now have the form (15) or (16)-(17) where the functions L j , M j , U j and V j , j = 1 , 2 , · · · are polynomials.

Divisibility and Residues for Matrix-functions The N × N case Theorem Let N ∈ Z + , N > 1 , be given and fixed. Let A ∈ SL N ( pol ) ; then the conclusions in Theorem 8 carry over with the modification that the alternative factors in the product are upper and lower triangular matrix-functions in SL N ( pol ) . We may take the lower triangular matrix-factors L = ( L i , j ) i , j ∈ Z N of the form

Divisibility and Residues for Matrix-functions  1 0 0 0 0 0 0 0  0 1 0 0 0 0 0 0     L p 0 1 0 0 0 0 0     0 L p + 1 0 1 0 0 0 0     0 0 . 0 1 0 0 0     0 0 0 . 0 1 0 0     0 0 0 0 . 0 1 0   0 0 0 0 0 L N − 1 0 1 polynomial entries � L i , i ≡ 1 , (18) L i , j ( z ) = δ i − j , p L i ( z );

Divisibility and Residues for Matrix-functions and the upper triangular factors of the form U = ( U i , j ) i , j ∈ Z N with � U i , i ≡ 1 , (19) L i , j ( z ) = δ i − j , p U i ( z ) .