By exploiting the property that the RBM log-likelihood function is the difference of convex functions, we formulate a stochastic variant of the difference of convex functions (DC) programming to minimize the negative log-likelihood. Interestingly, the traditional contrastive divergence algorithm is a special case of the above formulation and the hyperparameters of the two algorithms can be chos...

Being motivated by the work of Cochran et al. on the Gel'fand triple E] (L 2) E] , we are led to nd elementary functions to replace the exponential generating functions G and G 1== for the characterization of generalized and test functions. We deene the Legendre transformù for u in C +;log and the L-function Lu when u 2 C +;log is (log, exp)-convex. We show that u is equivalent to Lu. The dual ...

We discuss in this paper a method of finding skyline or non-dominated points in a set P of nP points with respect to a set S of nS sites. A point pi ∈ P is non-dominated if and only if for each pj ∈ P , j 6= i, there exists at least one point s ∈ S that is closer to pi than pj . We reduce this problem of determining non-dominated points to the problem of finding sites that have non-empty cells ...

Given an input consisting of an n-vertex convex polygon with k hole vertices or an n-vertex planar straight line graph (PSLG) with k holes and/or reflex vertices inside the convex hull, the parameterized minimum number convex partition (MNCP) problem asks for a partition into a minimum number of convex pieces. We give a fixedparameter tractable algorithm for this problem that runs in the follow...

In this paper we develop the concept of a convex polygon-offset distance function. Using offset as a notion of distance, we show how to compute the corresponding nearestand furthest-site Voronoi diagrams of point sites in the plane. We provide nearoptimal deterministic O(n(log n + log m) + m)-time algorithms, where n is the number of points and m is the complexity of the underlying polygon, for...

In 1957 Chandler Davis proved a theorem that a rotationally invariant function on symmetric matrices is convex if and only if it is convex on the diagonal matrices. We generalize this result to groups acting nonlinearly on convex subsets of arbitrary vector spaces thereby understanding the abstract mechanism behind the classical theorem. We apply the new theorem to a problem from the mathematic...

In many machine learning problems such as the dual form of SVM, the objective function to be minimized is convex but not strongly convex. This fact causes difficulties in obtaining the complexity of some commonly used optimization algorithms. In this paper, we proved the global linear convergence on a wide range of algorithms when they are applied to some non-strongly convex problems. In partic...

We study two properties of random high dimensional sections of convex bodies. In the first part of the paper we estimate the central section function |K ∩F⊥| n−k for random F ∈ Gn,k and K ⊂ R n a centrally symmetric isotropic convex body. This partially answers a question raised by V. Milman and A. Pajor (see [MP], p.88). In the second part we show that every symmetric convex body has random hi...

In many machine learning problems such as the dual form of SVM, the objective function to be minimized is convex but not strongly convex. This fact causes difficulties in obtaining the complexity of some commonly used optimization algorithms. In this paper, we proved the global linear convergence on a wide range of algorithms when they are applied to some non-strongly convex problems. In partic...