Inmachinelearning,cccpisextensivelyusedinmany learning algorithms, including sparse support vector machines svms, transductive svms, and sparse principal component analysis. We introduce the concaveconvex procedure cccp which constructs discrete time iterative dynamical systems which are guaranteed to monotonically decrease global optimizationenergy functions. Concave is an adjective that describes a surface that curves inward, or is thinner in the middle than on the edges. We also simplify the problem to a linear program and solve the corresponding cccp in a. An energy efficient design for uav communication with mobile edge computing. The concaveconvex procedure cccp is an iterative algorithm that solves d. The concaveconvex procedure, neural computation 10. The concave convex procedure cccp is a majorizationminimization algorithm that solves d. This implies that cccp can be applied to almost any optimization problem. So this is a dcp difference convex programming which can be solved by convexconcave procedure cccp.
Lingyan fan 1, wu yan 1, xihan chen 2, zhiyong chen 3, qingjiang shi 4. The calibrated cccp approach for the smoothly clipped absolute deviation scad penalty is known to be pathconsistent with probability. I also brie y discuss the majorisationminimisation mm algorithm, where cccp is. A cartoon for the concaveconvex procedure yingzhen li oct 2015 this short note provides an illustration of the concaveconvex procedure cccp yuille and rangarajan, 2003. Pdf we introduce t he concaveconvex procedure cccp which constructs discrete time iterative dynamical systems which are. In mathematics, a concave function is the negative of a convex function. This procedure can be applied to almost any optimization problem, and many existing algorithms can. This procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of 1. In ordinary usage, concave and convex are typically used when referring to glass surfaces, like the lenses of optical viewing equipment. We then employ a concaveconvex procedure cccp 8 to solve the problem. In this paper, we numerically compare the performance of both decodings for certain rate half ldpc codes with code lengths of approximately and 2000. Generalization of concave and convex decomposition in. The concaveconvex procedure cccp is a way to construct discrete time iterative dynamical systems which are guaranteed to monotonically decrease global optimization energy functions.
This procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it. Theres a cave in concave, and caves are inward curvatures. Instance annotation for multiinstance multilabel learning. So just remember, concave has the word cave in it so i always imagine that its kind of caving inwards caving inwards its a little bit of an exaggerated drawing but you get. We investigate the convexconcave procedure ccp, a local heuristic that utilizes the tools of convex optimization to find local optima of difference of convex dc programming problems. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. To solve the convex problem, we employ the pegasos framework of primal subgradient descent, and. The concave convex procedure cccp is a way to construct discretetime iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically. Although the global convergence behavior of cc cp has been well studied, the convergence rate of cccp is still an.
A unified approach for learning the parameters of sum. The concaveconvex procedure neural computation mit press. Sparse graphical models via calibrated concave convex. An energy efficient design for uav communication with. Curved mirrors concave and convex linkedin slideshare. The class of dc problems includes many difficult problems such as the traveling salesman problem. This procedure can be applied to almost any optimization.
The authors in 28 further investigated the global convergence of the cccp method. The concaveconvex procedure neural computation mit. The cccp procedure is closely related to the difference of convex dc methods, which were successfully applied to a lot of different and various nondifferentiable nonconvex optimization problems especially in the largescale setting. The concaveconvex procedure cccp proceedings of the.
The concaveconvex procedure cccp is a way to construct discrete time iterative dynamical systems which are guaranteed to monotonically decrease global optimizationenergy functions. Sma and the concaveconvex procedure cccp, respectively. The concaveconvex procedure cccp jhu computer science. In machine learning, cccp is extensively used in many learning algorithms, including sparse support vector machines svms, transductive svms, and sparse principal component analysis. Lenses concave and convex apparatus selection of lenses, collimated light source, e. The concaveconvex procedure cccp is a way to construct discretetime iterative dynamical systems that are guaranteed to decrease global optimization and.
The concaveconvex procedure cccp is a majorizationminimization algorithm that solves d. In this paper, we generalize cccp algorithm for kikuchi free energy and present a new cccp ncccp algorithm which includes a highdimensional free vector. The surface may be either convex bulging outward or concave recessed inward. It can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of cccp. Cccp algorithms to minimize the bethe free energy of 3sat problem. Concaveconvex procedure the cccp is one of powerful optimization algorithms for minimizing nonconvex functions that can be decomposed as a sum of convex and concave functions. Curvedspherical mirror a curved mirror is a mirror with a curved reflecting surface. Smithkettle well eye research institute, san francisco, ca. The authors in study the positioning problem in the presence of clock imperfections for a twtoa based technique and propose a linear least squares based approach to solve the problem. The concaveconvex procedure the concaveconvex procedure yuille, a. Yuille and anand rangarajan smithkettlewell eye research institute.
Pdf the concaveconvex procedure cccp is a way to construct. Variations and extension of the convexconcave procedure. In this paper we use cccp concave convex procedure algorithm to study 3sat problem and we get better results than bp algorithm that. A cccp algorithm illustrated for convex minus convex.
We extend the standard procedure in two major ways and describe several variations. This procedure can be applied to almost any optimization problem and. A curved mirror is a mirror with a curved reflecting surface. The concaveconvex procedure cccp conference paper pdf available in advances in neural information processing systems 15. Theorem 1 shows that any function, subject to weak conditions, can be expressed as the sum of a convex and concave part this decomposition is not unique. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex. Finding a mnemonic device for concave is easy enough. Iterative decoding based on the concaveconvex procedure. In this paper we present several extensions of and variations on the convexconcave procedure ccp, a powerful heuristic method used to. The concaveconvex procedure cccp is a way to construct. Concaveconvex procedure cccp has been widely used to solve nonconvex d.
We present a unified approach for learning the parameters of sumproduct networks spns. The proposed approaches work well in some scenarios, e. This paper proposes a calibrated concave convex procedure calibrated cccp for highdimensional graphical model selection. A proof of convergence of the concaveconvex procedure. Variations and extension of the convexconcave procedure 265 123. The most common nonspherical type are parabolic reflectors, found in optical devices such as reflecting telescopes that need.
Variations and extensions of the convexconcave procedure. The class of dc problems is very general, and includes difficult problems such as the traveling salesman problem. Weve talked a lot about convex lenses, so i thought i would do a quick video on concave lenses, although there arent as many combinations of what a concave lens can do. In particular, we need to solve a sequence of second order cone programs to. Equivalently, a function is convex if its epigraph the set of points on or above the graph of the function is a convex set. We introduce the concaveconvex procedure cccp which constructs discrete time iterative dynamical systems which are guaranteed to. We prove that any complete and decomposable spn is equivalent to a mixture of trees where each tree corresponds to a product of univariate distributions.
Yuillie and rangarajan in 30 proposed a method for solving smooth dc programs that is called the concaveconvex procedure cccp, a variant of dca applied to smooth dc programs 27. Shibuya et al iterative decoding based on the concaveconvex procedure 47 ing enables the possibility to compare the performances of cccp and bp decodings for relatively long codes. Penalty convexconcave procedure for source localization problem darya ismailova and wusheng lu department of electrical and computer engineering university of victoria, victoria, bc, canada email. A concaveconvex procedure for tdoa based positioning. Concave lenses video geometric optics khan academy. The concaveconvex procedure cccp is a way to construct discretetime iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically. The other is to apply the constrained concaveconvex procedure cccp, which can also be interpreted as iteratively updating support instances and solving a convex problem. Ramp loss nonparallel support vector machine for pattern.
We investigate the convexconcave procedure, a local heuristic that utilizes the tools of convex optimization to find local optima of difference of convex dc programming problems. I studied an example and found that cccp can descend into a local minimum. Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are sometimes used in optical devices. This procedure can be applied to almost any optimization problem, and many existing. Cccp is defined as concaveconvex procedure somewhat frequently. In mathematics, a realvalued function defined on an ndimensional interval is called convex or convex downward or concave upward if the line segment between any two points on the graph of the function lies above or on the graph. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The concave convex procedure cccp is an iterative algorithm that solves d. They should try to describe any images they form in terms of magnification, upright or inverted, real or. Concaveconvex procedure how is concaveconvex procedure. Concave convex procedure cccp proposed by yuille is known to be an algorithm which guarantees to monotonically decrease both free energies. We apply our method to a challenging trackingbyassignment problem of a variable number of divisible objects.