sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... sparse image representation via combined transforms - Convex ...
108 CHAPTER 5. ITERATIVE METHODS diagonal matrix and its transpose as follows: à = ⎛ ⎜ ⎝ ⎞ ⎛ T 1 T 2 1 . .. ⎟ 2 A ⎜ ⎠ ⎝ T m T T 1 T T 2 ⎞ ⎟ ⎠ . .. T T m = ⎛ ⎜ ⎝ ⎞ I + S 1 I ··· I I I + S 2 . , . .. . ⎟ ⎠ I ··· I + S m where S i = T i D i Ti T , i =1, 2,... ,m. In the remainder of this section, we consider what a good preconditioner for à should be. We considered three possible preconditioners: 1. a preconditioner from complete Cholesky factorization, 2. a preconditioner from sparse incomplete factorization of the inverse [9], 3. a diagonal block preconditioner [4, 5, 31]. The main result is that we found the preconditioner 1 and preconditioner 2 are not “optimal”. Here “optimal” means that the matrix-vector multiplication with the matrix associated with the preconditioner L −H should still have fast algorithms, and these fast algorithms should be based on fast algorithms for matrix-vector multiplication for the matrices T 1 ,T 2 ,... ,T m . So the block diagonal preconditioner is the only one we are going to use. We describe the results about the block diagonal preconditioner here, and postpone the discussion about the preconditioners in case 1 and 2 to Section 5.4. The optimal block diagonal preconditioner for à is ⎛ ⎞ (I + S 1 ) −1/2 (I + S 2 ) −1/2 ⎜ . . (5.7) .. ⎟ ⎝ ⎠ (I + S m ) −1/2 A striking result is due to Demmel [78, Page 168, Theorem 10.5.3]. The key idea is that
5.2. LSQR 109 among all the block diagonal preconditioners, the one that takes the Cholesky factorizer of the diagonal submatrix as its diagonal submatrix is nearly optimal. Here “nearly” means that the resulting condition number cannot be larger than m times the best achievable condition number by using a block diagonal preconditioner. Obviously, we have fast algorithms to multiply with matrix (I + S i ) −1/2 ,i=1, 2,... ,m. 5.2 LSQR 5.2.1 What is LSQR? LSQR [117, 116] solves the following two least-squares (LS) problems, depending on whether the damping parameter α is zero or not. [N] When the damping parameter is equal to zero (α = 0), solve Ax = b or minimize x ‖b − Ax‖ 2 . [R] When the damping parameter is not equal to zero (α ≠0): minimize x b ∥ ∥( 0 ) ( A − αI ) x ∥ 2 2 . Here [N] is a nonregularized problem and [R] is a regularized problem. The problem in (5.1) can be rewritten as follows: ⎡ ⎛ ⎢ ⎣ 2ΦT Φ+λ ⎜ ⎝ ¯ρ ′′ (x k 1 ) ⎞⎤ . .. ¯ρ ′′ (x k N ) ⎛ ⎟⎥ ⎠⎦ d =2ΦT (y − Φx k ) − λ ⎜ ⎝ ¯ρ ′ (x k 1 ) ⎞ . ⎟ ⎠ , ¯ρ ′ (x k N ) where d is the Newton direction that we want to solve for and the remaining variables are defined in the previous chapter. The above equation is equivalent to solving an LS problem: (LS) : ] [ ]∥ Φ y − Φxk ∥∥∥∥ minimize ∥ d − , d ∥[ D(x k ) −D −1 (x k )g(x k ) 2
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108 CHAPTER 5. ITERATIVE METHODS<br />
diagonal matrix and its transpose as follows:<br />
à =<br />
⎛<br />
⎜<br />
⎝<br />
⎞ ⎛<br />
T 1 T 2 1<br />
. .. ⎟ 2 A ⎜<br />
⎠ ⎝<br />
T m<br />
T T 1<br />
T T 2<br />
⎞<br />
⎟<br />
⎠<br />
. ..<br />
T T m<br />
=<br />
⎛<br />
⎜<br />
⎝<br />
⎞<br />
I + S 1 I ··· I<br />
I I + S 2<br />
. ,<br />
.<br />
.. . ⎟<br />
⎠<br />
I ··· I + S m<br />
where S i = T i D i Ti T , i =1, 2,... ,m. In the remainder of this section, we consider what a<br />
good preconditioner for à should be.<br />
We considered three possible preconditioners:<br />
1. a preconditioner from complete Cholesky factorization,<br />
2. a preconditioner from <strong>sparse</strong> incomplete factorization of the inverse [9],<br />
3. a diagonal block preconditioner [4, 5, 31].<br />
The main result is that we found the preconditioner 1 and preconditioner 2 are not “optimal”.<br />
Here “optimal” means that the matrix-vector multiplication with the matrix associated<br />
with the preconditioner L −H should still have fast algorithms, and these fast algorithms<br />
should be based on fast algorithms for matrix-vector multiplication for the matrices<br />
T 1 ,T 2 ,... ,T m . So the block diagonal preconditioner is the only one we are going to use.<br />
We describe the results about the block diagonal preconditioner here, and postpone the<br />
discussion about the preconditioners in case 1 and 2 to Section 5.4.<br />
The optimal block diagonal preconditioner for à is<br />
⎛<br />
⎞<br />
(I + S 1 ) −1/2 (I + S 2 ) −1/2 ⎜<br />
. . (5.7)<br />
.. ⎟<br />
⎝<br />
⎠<br />
(I + S m ) −1/2<br />
A striking result is due to Demmel [78, Page 168, Theorem 10.5.3]. The key idea is that
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