majorizationminimizationtekniikoita

Majorization-minimization (MM) algorithms are a class of iterative optimization methods used to find local minima of objective functions. The core idea behind MM algorithms is to construct a sequence of simpler surrogate functions that are easier to minimize than the original objective function. At each iteration, the algorithm minimizes a surrogate function that "majorizes" the original objective function, meaning it lies above or touches the original function at the current iterate.

The MM algorithm works by repeatedly minimizing a surrogate function, denoted as g(x), that majorizes the objective function f(x) at the current point x_k. This means g(x_k) = f(x_k) and g(x) >= f(x) for all x. The next iterate, x_{k+1}, is then chosen as the minimum of g(x). Since g(x) majorizes f(x), minimizing g(x) guarantees that f(x_{k+1}) <= f(x_k), ensuring convergence to a stationary point. The effectiveness of an MM algorithm depends heavily on the choice of the surrogate function and the ease with which it can be minimized.

MM algorithms have found wide applications in various fields, including statistical estimation, signal processing, and machine learning. They are particularly useful when the objective function is non-convex or difficult to optimize directly using gradient-based methods. Examples of problems where MM algorithms are applied include maximum likelihood estimation with latent variables, parameter estimation in mixture models, and sparse signal recovery. The algorithms are known for their simplicity and guaranteed convergence properties, although they may converge slowly depending on the problem and the chosen surrogate.