This theoretical paper is inspired by an integer linear programming (ILP) approach to learning the structure of decomposable models. We intend to represent decomposable models by special zeroone vectors, named characteristic imsets. Our approach leads to the study of a special polytope, defined as the convex hull of all characteristic imsets for chordal graphs, named the chordal graph polytope....

Let X be an arbitrary nonempty set and L a lattice of subsets of X such that ∅, X ∈ L. A L denotes the algebra generated by L, andM L denotes those nonnegative, finite, finitely additive measures on A L . In addition, I L denotes the subset of M L which consists of the nontrivial zeroone valued measures. The paper gives detailed analysis of products of lattices, their associated Wallman spaces,...

Many of the ordinal regression algorithms that have been proposed can be viewed as methods that minimize a convex surrogate of the zeroone, absolute, or squared errors. We provide a theoretical analysis of the risk consistency properties of a rich family of surrogate loss functions, including proportional odds and support vector ordinal regression. For all the surrogates considered, we either p...

We study zero-one laws for random graphs. We focus on the following question that was asked by many: Given a graph property P , is there a language of graphs able to express P while obeying the zeroone law? Our results show that on the one hand there is a (regular) language able to express connectivity and k-colorability for any constant k and still obey the zero-one law. On the other hand we s...

The basic idea of a geometric approach to learning a Bayesian network (BN) structure is to represent every BN structure by a certain vector. If the vector representative is chosen properly, it allows one to re-formulate the task of finding the global maximum of a score over BN structures as an integer linear programming (ILP) problem. Suitable such a zeroone vector representative is the charact...

A sequence of random variables is exchangeable if its joint distribution is invariant under variable permutations. We introduce exchangeable variable models (EVMs) as a novel class of probabilistic models whose basic building blocks are partially exchangeable sequences, a generalization of exchangeable sequences. We prove that a family of tractable EVMs is optimal under zeroone loss for a large...

This paper proposes a convex relaxation of a sparse support vector machine (SVM) based on the perspective relaxation of mixed-integer nonlinear programs. We seek to minimize the zero-norm of the hyperplane normal vector with a standard SVM hinge-loss penalty and extend our approach to a zeroone loss penalty. The relaxation that we propose is a second-order cone formulation that can be efficient...

We first define the coarse-graining of probability measures in terms stochastic kernels. when a measure is part another and say that two coexist if they are both parts single measure. then show any coexist. extend these concepts to observables instruments mention need not discretization an observable as special case have \zeroone next consider finite cases, kernels replaced by matrices. also sa...