Axis-parallel Hyperplanes Research Articles

Red Blue Set Cover problem on axis-parallel hyperplanes and other objects

Given a universe U=R∪B of a finite set of red elements R, and a finite set of blue elements B and a family F of subsets of U, the Red Blue Set Cover problem is to find a subset F′ of F that covers all blue elements of B and minimum number of red elements from R.We prove that the Red Blue Set Cover problem is NP-hard even when R and B respectively are sets of red and blue points in IR2 and the sets in F are defined by axis−parallel lines i.e., every set is a maximal set of points with the same x or y coordinate.We then study the parameterized complexity of a generalization of this problem, where U is a set of points in IRd and F is a collection of set of axis−parallel hyperplanes in IRd under different parameterizations, where d is a constant. For every parameter, we show that the problem is fixed-parameter tractable and also show the existence of a polynomial kernel. We further consider the Red Blue Set Cover problem for some special types of rectangles in IR2.

Red Blue Set Cover Problem on Axis-Parallel Hyperplanes and Other Objects

Red Blue Set Cover Problem on Axis-Parallel Hyperplanes and Other Objects

Separation dimension and degree

AbstractThe separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝd, such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by Alon et al. [J. Graph Theory 2018].

Heterogeneous oblique random forest

Decision trees in random forests use a single feature in non-leaf nodes to split the data. Such splitting results in axis-parallel decision boundaries which may fail to exploit the geometric structure in the data. In oblique decision trees, an oblique hyperplane is employed instead of an axis-parallel hyperplane. Trees with such hyperplanes can better exploit the geometric structure to increase the accuracy of the trees and reduce the depth. The present realizations of oblique decision trees do not evaluate many promising oblique splits to select the best. In this paper, we propose a random forest of heterogeneous oblique decision trees that employ several linear classifiers at each non-leaf node on some top ranked partitions which are obtained via one-vs-all and two-hyperclasses based approaches and ranked based on ideal Gini scores and cluster separability. The oblique hyperplane that optimizes the impurity criterion is then selected as the splitting hyperplane for that node. We benchmark 190 classifiers on 121 UCI datasets. The results show that the oblique random forests proposed in this paper are the top 3 ranked classifiers with the heterogeneous oblique random forest being statistically better than all 189 classifiers in the literature.

A Simple Convolutional Neural Network with Rule Extraction

Classification responses provided by Multi Layer Perceptrons (MLPs) can be explained by means of propositional rules. So far, many rule extraction techniques have been proposed for shallow MLPs, but not for Convolutional Neural Networks (CNNs). To fill this gap, this work presents a new rule extraction method applied to a typical CNN architecture used in Sentiment Analysis (SA). We focus on the textual data on which the CNN is trained with “tweets” of movie reviews. Its architecture includes an input layer representing words by “word embeddings”, a convolutional layer, a max-pooling layer, followed by a fully connected layer. Rule extraction is performed on the fully connected layer, with the help of the Discretized Interpretable Multi Layer Perceptron (DIMLP). This transparent MLP architecture allows us to generate symbolic rules, by precisely locating axis-parallel hyperplanes. Experiments based on cross-validation emphasize that our approach is more accurate than that based on SVMs and decision trees that substitute DIMLPs. Overall, rules reach high fidelity and the discriminative n-grams represented in the antecedents explain the classifications adequately. With several test examples we illustrate the n-grams represented in the activated rules. They present the particularity to contribute to the final classification with a certain intensity.

Tree-based models for survival data with competing risks

Objective. Tree-based models belong to common, assumption-free methods of data analysis. Their application in survival data is narrowed to univariate models, which partition the feature space with axis-parallel hyperplanes, meaning that each internal node involves a single feature. In this paper, I extend the idea of oblique survival tree induction for competing risks by modifying a piecewise-linear criterion function. Additionally, the use of tree-based ensembles to analyze the competing events is proposed.Method and materials. Two types of competing risks trees are proposed: a single event tree designed for analysis of the event of interest and a composite event tree, in which all the competing events are taken into account. The induction process is similar, except that the calculation of the criterion function is minimized for the individual tree nodes generation. These two tree types were also used for building the ensembles with aggregated cumulative incidence functions as an outcome. Nine real data sets, as well as a simulated data set, were taken to assess performance of the models, while detailed analysis was conducted on the basis of follicular cell lymphoma data.Results. The evaluation was focused on two measures: the prediction error expressed by an integrated Brier score (IBS), and the ranked measure of predictive ability calculated as a time-truncated concordance index (C–index). The proposed techniques were compared with the existing approaches of the Fine–Gray subdistribution hazard model, Fine–Gray regression model with backward elimination, and random survival forest for competing risks. The results for both the IBS and the C–index indicated statistically significant differences between these methods (p < .0001).Conclusions. The prediction error of the individual trees was similar to the other methods, but the results of the C–index differ in comparison to the Fine–Gray subdistribution hazard model and the Fine–Gray regression with backward elimination. The ensembles prediction ability was comparable to existing algorithms, but their IBS values were better than either random survival forest or Fine–Gray regression with backward elimination.

A Rule Extraction Study from SVM on Sentiment Analysis

A natural way to determine the knowledge embedded within connectionist models is to generate symbolic rules. Nevertheless, extracting rules from Multi Layer Perceptrons (MLPs) is NP-hard. With the advent of social networks, techniques applied to Sentiment Analysis show a growing interest, but rule extraction from connectionist models in this context has been rarely performed because of the very high dimensionality of the input space. To fill the gap we present a case study on rule extraction from ensembles of Neural Networks and Support Vector Machines (SVMs), the purpose being the characterization of the complexity of the rules on two particular Sentiment Analysis problems. Our rule extraction method is based on a special Multi Layer Perceptron architecture for which axis-parallel hyperplanes are precisely located. Two datasets representing movie reviews are transformed into Bag-of-Words vectors and learned by ensembles of neural networks and SVMs. Generated rules from ensembles of MLPs are less accurate and less complex than those extracted from SVMs. Moreover, a clear trade-off appears between rules’ accuracy, complexity and covering. For instance, if rules are too complex, less complex rules can be re-extracted by sacrificing to some extent their accuracy. Finally, rules can be viewed as feature detectors in which very often only one word must be present and a longer list of words must be absent.

NP-completeness and FPT Results for Rectilinear Covering Problems

This paper discusses three rectilinear (that is, axis-parallel) covering prob- lems in d dimensions and their variants. The first problem is the Rectilinear Line Cover where the inputs are n points in R d and a positive integer k ,a nd we are asked to answer if we can cover these n points with at most k lines where these lines are restricted to be axis parallel. We show that this problem has efficient fixed-parameter tractable (FPT) algorithms. The second problem is the Rectilinear k-Links Span- ning Path Problem where the inputs are also n points in R d and a positive integer k but here we are asked to answer if there is a piecewise linear path through these n points having at most k line-segments (links) where these line-segments are axis- parallel. We prove that this second problem is FPT under the assumption that no two line-segments share the same line. The third problem is the Rectilinear Hyper- plane Cover problem and we are asked to cover a set of n points in d dimensions with k axis-parallel hyperplanes of d − 1 dimensions. We also demonstrate this has an FPT-algorithm. Previous to the results above, only conjectures were enunciated over several years on the NP-completeness of the Rectilinear Minimum Link Traveling Salesman Problem ,t heMinimum Link Spanning Path Problem and the Recti- linear Hyperplane Cover. We provide the proof that the Rectilinear Minimum Link Traveling Salesman Problem and the Rectilinear Minimum Link Span- ning Path Problem are NP-complete by a reduction from the One-In-Three 3-SAT problem. The NP-completeness of the Rectilinear Hyperplane Cover problem is proved by a reduction from 3-SAT. This suggests dealing with the intractability just discovered with fixed-parameter tractability. Moreover, if we extend our problems to a finite set of orientations, our approach proves these problems remain FPT.

Inducing NNC-trees with the R/sup 4/-rule

An NNC-Tree is a decision tree (DT) with each non-terminal node containing a nearest neighbor classifier (NNC). Compared with the conventional axis-parallel DTs (APDTs), the NNC-Trees can be more efficient, because the decision boundary made by an NNC is more complex than an axis-parallel hyperplane. Compared with single-layer NNCs, the NNC-Trees can classify given data in a hierarchical structure that is often useful for many applications. This paper proposes an algorithm for inducing NNC-Trees based on the R4-rule, which was proposed by the author for finding the smallest nearest neighbor based multilayer perceptrons (NN-MLPs). There are mainly two contributions here. 1) A heuristic but effective method is given to define the teacher signals (group labels) for the data assigned to each nonterminal node. 2) The R4-rule is modified so that an NNC with proper size can be designed automatically in each nonterminal node. Experiments with several public databases show that the proposed algorithm can produce NNC-Trees effectively and efficiently.

SEPARATING POINTS BY AXIS-PARALLEL LINES

We study the problem of separating n points in the plane, no two of which have the same x- or y-coordinate, using a minimum number of vertical and horizontal lines avoiding the points, so that each cell of the subdivision contains at most one point. Extending previous NP-hardness results due to Freimer et al. we prove that this problem and some variants of it are APX-hard. We give a 2-approximation algorithm for this problem, and a d-approximation algorithm for the d-dimensional variant, in which the points are to be separated using axis-parallel hyperplanes. To this end, we reduce the point separation problem to the rectangle stabbing problem studied by Gaur et al. Their approximation algorithm uses LP-rounding. We present an alternative LP-rounding procedure which also works for the rectangle stabbing problem. We show that the integrality ratio of the LP is exactly 2.