Efficient Computation of Expectations under Spanning Tree Distributions

Abstract We give a general framework for inference in spanning tree models. We propose unified algorithms for the important cases of first-order expectations and second-order expectations in edge-factored, non-projective spanning-tree models. Our algorithms exploit a fundamental connection between gradients and expectations, which allows us to derive efficient algorithms. These algorithms are easy to implement with or without automatic differentiation software. We motivate the development of our framework with several cautionary tales of previous research, which has developed numerous inefficient algorithms for computing expectations and their gradients. We demonstrate how our framework efficiently computes several quantities with known algorithms, including the expected attachment score, entropy, and generalized expectation criteria. As a bonus, we give algorithms for quantities that are missing in the literature, including the KL divergence. In all cases, our approach matches the efficiency of existing algorithms and, in several cases, reduces the runtime complexity by a factor of the sentence length. We validate the implementation of our framework through runtime experiments. We find our algorithms are up to 15 and 9 times faster than previous algorithms for computing the Shannon entropy and the gradient of the generalized expectation objective, respectively.

Completely monotonic functions related to the special functions

In this paper, we discuss the completely monotonic functions and their relation to some of the famous special functions such as (Gamma, Kumar, Parabolic cylinder, Gauss hypergeometric, MacDonald, Whittaker and Generalized Mittag-Leffler) function. In addition, the relationship of the completely monotonic integrations with absolute progress under conditions of convergence such as transformations (Hankel, Lambert, Stieltjes and Laplace). We will found other modes of composite functions given in terms of non-negative power chains and integrative transformations of completely monotonic non-negative functions, the state of integrative transform functions with a homogeneous nucleus of the first order, and the logarithmically completely monotonic functions. The importance of the row of completely monotonic functions that are associated with the transformation of the Stieltjes defined as a class of special functions regression functions. Some of the oscillations of these functions resulting from completely monotonic functions are not decreasing or convex, but most of them are completely monotonic functions.

The distributive solutions for some partial differential equations of second order

This research studies the distributive solutions for some partial differential equations of second order. We study specially the distributive solutions for Laplas equation, Heat equation, wave equations and schrodinger equation. We introduce the fundamental solutions for precedent equations and inference the distributive solutions by using the convolution of distributions concept. For that we use some of lemmas and theorems with proofs, specially for Laplas equation. And precedent some of concepts, defintions and remarks.