We show that the Laplace approximation of a supremum by L-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem P and its dual P∗ appear naturally when using this simple approximation technique for the value function g of P or its Legendre-Fenchel conjugate g∗. In addition, minimizing the LBF of the dual P∗ is just e...

In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intractable because of two problems: (1) it includes the evaluation of the partiti...