We introduce the Concave-Convex procedure (CCCP) which constructs discretetime iterative dynamical systems which are guaranteed to monotonically decrease global optimization/energy functions. It can be applied to (almost) any optimization problem and many existing algorithms can be interpreted in terms of CCCP. In particular, we prove relationships to some applications of Legendre transform techniques. We then illustrate CCCP by applications to Potts models, linear assignment, EM algorithms, and Generalized Iterative Scaling (GIS). CCCP can be used both as a new way to understand existing optimization algorithms and as a procedure for generating new algorithms. 1 Introduction There is a lot of interest in designing discrete time dynamical systems for inference and learning (see, for example, [10], [3], [7], [13]).

A new approach for clustering is proposed. This method is based on an analogy to a physical model; the ferromagnetic Potts model at thermal equilibrium is used as an analog computer for this hard optimization problem. We do not assume any structure of the underlying distributionof the data. Phase space of the Potts model is divided into three regions; ferromagnetic, super-paramagnetic and paramagnetic phases. The region of interest is that corresponding to the super-paramagnetic one, where domains of aligned spins appear.