128,817 research outputs found
Combinatorial generation via permutation languages. VI. Binary trees
In this paper we propose a notion of pattern avoidance in binary trees that
generalizes the avoidance of contiguous tree patterns studied by Rowland and
non-contiguous tree patterns studied by Dairyko, Pudwell, Tyner, and Wynn.
Specifically, we propose algorithms for generating different classes of binary
trees that are characterized by avoiding one or more of these generalized
patterns. This is achieved by applying the recent
Hartung-Hoang-M\"utze-Williams generation framework, by encoding binary trees
via permutations. In particular, we establish a one-to-one correspondence
between tree patterns and certain mesh permutation patterns. We also conduct a
systematic investigation of all tree patterns on at most 5 vertices, and we
establish bijections between pattern-avoiding binary trees and other
combinatorial objects, in particular pattern-avoiding lattice paths and set
partitions
Structured Learning of Tree Potentials in CRF for Image Segmentation
We propose a new approach to image segmentation, which exploits the
advantages of both conditional random fields (CRFs) and decision trees. In the
literature, the potential functions of CRFs are mostly defined as a linear
combination of some pre-defined parametric models, and then methods like
structured support vector machines (SSVMs) are applied to learn those linear
coefficients. We instead formulate the unary and pairwise potentials as
nonparametric forests---ensembles of decision trees, and learn the ensemble
parameters and the trees in a unified optimization problem within the
large-margin framework. In this fashion, we easily achieve nonlinear learning
of potential functions on both unary and pairwise terms in CRFs. Moreover, we
learn class-wise decision trees for each object that appears in the image. Due
to the rich structure and flexibility of decision trees, our approach is
powerful in modelling complex data likelihoods and label relationships. The
resulting optimization problem is very challenging because it can have
exponentially many variables and constraints. We show that this challenging
optimization can be efficiently solved by combining a modified column
generation and cutting-planes techniques. Experimental results on both binary
(Graz-02, Weizmann horse, Oxford flower) and multi-class (MSRC-21, PASCAL VOC
2012) segmentation datasets demonstrate the power of the learned nonlinear
nonparametric potentials.Comment: 10 pages. Appearing in IEEE Transactions on Neural Networks and
Learning System
msBP: An R package to perform Bayesian nonparametric inference using multiscale Bernstein polynomials mixtures
msBP is an R package that implements a new method to perform Bayesian multiscale nonparametric inference introduced by Canale and Dunson (2016). The method, based on mixtures of multiscale beta dictionary densities, overcomes the drawbacks of Pólya trees and inherits many of the advantages of Dirichlet process mixture models. The key idea is that an infinitely-deep binary tree is introduced, with a beta dictionary density assigned to each node of the tree. Using a multiscale stick-breaking characterization, stochastically decreasing weights are assigned to each node. The result is an infinite mixture model. The package msBP implements a series of basic functions to deal with this family of priors such as random densities and numbers generation, creation and manipulation of binary tree objects, and generic functions to plot and print the results. In addition, it implements the Gibbs samplers for posterior computation to perform multiscale density estimation and multiscale testing of group differences described in Canale and Dunson (2016)
On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms
Uniquely closable skeletons of lambda terms are Motzkin-trees that
predetermine the unique closed lambda term that can be obtained by labeling
their leaves with de Bruijn indices. Likewise, uniquely typable skeletons of
closed lambda terms predetermine the unique simply-typed lambda term that can
be obtained by labeling their leaves with de Bruijn indices.
We derive, through a sequence of logic program transformations, efficient
code for their combinatorial generation and study their statistical properties.
As a result, we obtain context-free grammars describing closable and uniquely
closable skeletons of lambda terms, opening the door for their in-depth study
with tools from analytic combinatorics.
Our empirical study of the more difficult case of (uniquely) typable terms
reveals some interesting open problems about their density and asymptotic
behavior.
As a connection between the two classes of terms, we also show that uniquely
typable closed lambda term skeletons of size are in a bijection with
binary trees of size .Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
ローテーションの回数に基づく2分木間の距離
A certain distance between two binary trees is defined. The distance is the minimal number of the rotations changing from tree A to tree B.If the distance between tree A and tree B is 1, we say that the two trees are connected. Binary trees are expressed by the codewords as the computer representations, and ranked by the lexicographic generation of codewords. An algorithm that makes the table of the connection using the rank is proposed. The distance is expressed by the minimal path length on the graph made from the table
Interactive segmentation based on component-trees
International audienceComponent-trees associate to a discrete grey-level image a descriptive data structure induced by the inclusion relation between the binary components obtained at successive level-sets. This article presents an original interactive segmen- tation methodology based on component-trees. It consists of the extraction of a subset of the image component-tree, enabling the generation of a binary object which fits at best (with respect to the grey-level structure of the image) a given binary target selected beforehand in the image. A proof of the algorithmic efficiency of this methodological scheme is proposed. Concrete application examples on magnetic resonance imaging (MRI) data emphasise its actual computational efficiency and its usefulness for interactive segmentation of real images
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