TY - JOUR
T1 - An expanded theoretical treatment of iteration-dependent majorize-minimize algorithms
AU - Jacobson, Matthew W.
AU - Fessler, Jeffrey A.
N1 - Funding Information:
Manuscript received May 8, 2006; revised April 29, 2007. This work was supported in part by the National Institutes of Health/NCI under Grant 1P01 CA87634. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Peter C. Doerschuk. M. W. Jacobson is with Xoran Technologies, Inc., Ann Arbor, MI 48103 USA (e-mail: mjacobson@xorantech.com). J. A. Fessler is with The Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122 USA. Digital Object Identifier 10.1109/TIP.2007.904387 1The term MM was coined in [24].The technique has gone by various other names as well, such as optimization transfer, SAGE, and iterative majorization.
PY - 2007/10
Y1 - 2007/10
N2 - The majorize-minimize (MM) optimization technique has received considerable attention in signal and image processing applications, as well as in statistics literature. At each iteration of an MM algorithm, one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate. The next iterate is obtained by minimizing this tangent majorant function, resulting in a sequence of iterates that reduces the cost function monotonically. A well-known special case of MM methods are expectation-maximization algorithms. In this paper, we expand on previous analyses of MM, due to Fessler and Hero, that allowed the tangent majorants to be constructed in iteration-dependent ways. Also, this paper overcomes an error in one of those earlier analyses. There are three main aspects in which our analysis builds upon previous work. First, our treatment relaxes many assumptions related to the structure of the cost function, feasible set, and tangent majorants. For example, the cost function can be nonconvex and the feasible set for the problem can be any convex set. Second, we propose convergence conditions, based on upper curvature bounds, that can be easier to verify than more standard continuity conditions. Furthermore, these conditions allow for considerable design freedom in the iteration-dependent behavior of the algorithm. Finally, we give an original characterization of the local region of convergence of MM algorithms based on connected (e.g., convex) tangent majorants. For such algorithms, cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods. This expanded treatment widens the scope of the MM algorithm designs that can be considered for signal and image processing applications, allows us to verify the convergent behavior of previously published algorithms, and gives a fuller understanding overall of how these algorithms behave.
AB - The majorize-minimize (MM) optimization technique has received considerable attention in signal and image processing applications, as well as in statistics literature. At each iteration of an MM algorithm, one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate. The next iterate is obtained by minimizing this tangent majorant function, resulting in a sequence of iterates that reduces the cost function monotonically. A well-known special case of MM methods are expectation-maximization algorithms. In this paper, we expand on previous analyses of MM, due to Fessler and Hero, that allowed the tangent majorants to be constructed in iteration-dependent ways. Also, this paper overcomes an error in one of those earlier analyses. There are three main aspects in which our analysis builds upon previous work. First, our treatment relaxes many assumptions related to the structure of the cost function, feasible set, and tangent majorants. For example, the cost function can be nonconvex and the feasible set for the problem can be any convex set. Second, we propose convergence conditions, based on upper curvature bounds, that can be easier to verify than more standard continuity conditions. Furthermore, these conditions allow for considerable design freedom in the iteration-dependent behavior of the algorithm. Finally, we give an original characterization of the local region of convergence of MM algorithms based on connected (e.g., convex) tangent majorants. For such algorithms, cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods. This expanded treatment widens the scope of the MM algorithm designs that can be considered for signal and image processing applications, allows us to verify the convergent behavior of previously published algorithms, and gives a fuller understanding overall of how these algorithms behave.
KW - Expectation-maximization (EM)
KW - Majorize-minimize (MM)
KW - Optimization transfer
KW - SAGE
UR - http://www.scopus.com/inward/record.url?scp=34648825731&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=34648825731&partnerID=8YFLogxK
U2 - 10.1109/TIP.2007.904387
DO - 10.1109/TIP.2007.904387
M3 - Article
C2 - 17926925
AN - SCOPUS:34648825731
SN - 1057-7149
VL - 16
SP - 2411
EP - 2422
JO - IEEE Transactions on Image Processing
JF - IEEE Transactions on Image Processing
IS - 10
ER -