Relative volume of separable bipartite states

Every choice of an orthonormal frame in the d-dimensional Hilbert space of a system corresponds to one set of all mutually commuting density matrices or, equivalently, a classical statistical state space of the system; the quantum state space itself can thus be profitably viewed as an SU(d) orbit of classical state spaces, one for each orthonormal frame. We exploit this connection to study the relative volume of separable states of a bipartite quantum system. While the two-qubit case is studied in considerable analytic detail, for higher dimensional systems we fall back on Monte Carlo. Several new insights seem to emerge from our study.Comment: Essentially the published versio

A genetic prototype learner

Supervised classification problems have received considerable attention from the machine learning community. We propose a novel genetic algorithm based prototype learning system, PLEASE, for this class of problems. Given a set of prototypes for each of the possible classes, the class of an input instance is determined by the prototype nearest to this instance. We assume ordinal attributes and prototypes are represented as sets of feature-value pairs. A genetic algorithm is used to evolve the number of prototypes per class and their positions on the input space as determined by corresponding feature-value pairs. Comparisons with C4.5 on a set of artificial problems of controlled complexity demonstrate the effectiveness of the proposed system.

LEARNING HYPERPLANES THAT CAPTURES THE GEOMETRIC STRUCTURE OF CLASS REGIONS

Most of the decision tree algorithms rely on impurity measures to evaluate the goodness of hyperplanes at each node while learning a decision tree in a top-down fashion. These impurity measures are not differentiable with relation to the hyperplane parameters. Therefore the algorithms for decision tree learning using impurity measures need to use some search techniques for finding the best hyperplane at every node. These impurity measures don’t properly capture the geometric structures of the data. In this paper a Two-Class algorithm for learning oblique decision trees is proposed. Aggravated by this, the algorithm uses a strategy, to evaluate the hyperplanes in such a way that the (linear) geometric structure in the data is taken into consideration. At each node of the decision tree, algorithm finds the clustering hyperplanes for both the classes. The clustering hyperplanes are obtained by solving the generalized Eigen-value problem. Then the data is splitted based on angle bisector and recursively learn the left and right sub-trees of the node. Since, in general, there will be two angle bisectors; one is selected which is better based on an impurity measure gini index. Thus the algorithm combines the ideas of linear tendencies in data and purity of nodes to find better decision trees. This idea leads to small decision trees and better performance

Classification in the Presence of Ordered Classes and Weighted Evaluative Attributes

We are interested in an important family of problems in the interface of the Multi-Attribute Decision-Making and Data Mining fields. This is a special case of the general classification problem, in which records describing entities of interest have been expressed in terms of a number of evaluative attributes. These attributes are associated with weights of importance, and both the data and the classes are ordinal. Our goal is to use historical records and the corresponding decisions to best estimate the class values of new data points in a way consistent with prior classification decisions, without knowledge of the weights of the evaluative attributes. We study three variants of this problem. The first is when all decisions are consistent with a single set of attribute weights (called the separable case.) The second is when all decisions are consistent, but involve two sets of attribute weights corresponding to two decision makers, who determine the classification of the data together (called the two-plane separable case.) The third is when there is some inconsistency in the set of weights that must be accounted for (called the non-separable case.) Furthermore, we examine 2-class problems and also multiple class problems. We propose the Ordinal Boundary method, which has a significant advantage over traditional approaches in multi-class problems. Linear programming (optimization) based approaches provide a promising avenue for dealing with these problems effectively. We present computational results that support this argument

A Novel Feature Extraction Technique of Electronic Nose for Detecting of Wound Infection Based on Phase Space

Rapid and timely monitoring of traumatic inflammation is conducive to doctors? diagnosis and treatment. It has been proved that electronic nose (E-nose) is an effective way to predict the bacterial classes of wound infection by smelling the odor produced by the metabolites, and it has also been found thatthe classification accuracy of E-nose is very different when different feature is extracted and put into the classifier. The gas sensor array of E-nose can be seen as a dynamic system whose response temporally evolves following the concentration of the odors. As the central concept in the analysis of dynamic systems, phase space is the first time to be employed by us to construct the feature matrix of wound infection data in this paper. Dynamic moments, the functions of time delay in phase space, is used as the feature of wound infection. The odors of four different classes of wound (wound uninfected, and infected withP. aeruginosa, E. coliandS. aureus) are used as the original response of E-nose.Experimental results prove that the classification accuracy of test data set is 96.43% when R2 is used as the feature, which is much better than M2P, M3P (other two dynamic moments), maximum value of the steady-state response and maximum value of the first-order derivative (two traditional feature of E-nose)

Improved Compression of the Okamura-Seymour Metric

Let $G=(V,E)$ be an undirected unweighted planar graph. Consider a vector storing the distances from an arbitrary vertex $v$ to all vertices $S = \{ s_1 , s_2 , \ldots , s_k \}$ of a single face in their cyclic order. The pattern of $v$ is obtained by taking the difference between every pair of consecutive values of this vector. In STOC'19, Li and Parter used a VC-dimension argument to show that in planar graphs, the number of distinct patterns, denoted $x$, is only $O(k^3)$. This resulted in a simple compression scheme requiring $\tilde O(\min \{ k^4+|T|, k\cdot |T|\})$ space to encode the distances between $S$ and a subset of terminal vertices $T \subseteq V$. This is known as the Okamura-Seymour metric compression problem. We give an alternative proof of the $x=O(k^3)$ bound that exploits planarity beyond the VC-dimension argument. Namely, our proof relies on cut-cycle duality, as well as on the fact that distances among vertices of $S$ are bounded by $k$. Our method implies the following: (1) An $\tilde{O}(x+k+|T|)$ space compression of the Okamura-Seymour metric, thus improving the compression of Li and Parter to $\tilde O(\min \{k^3+|T|,k \cdot |T| \})$. (2) An optimal $\tilde{O}(k+|T|)$ space compression of the Okamura-Seymour metric, in the case where the vertices of $T$ induce a connected component in $G$. (3) A tight bound of $x = \Theta(k^2)$ for the family of Halin graphs, whereas the VC-dimension argument is limited to showing $x=O(k^3)$

Dynamic ham-sandwich cuts in the plane

We design efficient data structures for dynamically maintaining a ham-sandwich cut of two point sets in the plane subject to insertions and deletions of points in either set. A ham-sandwich cut is a line that simultaneously bisects the cardinality of both point sets. For general point sets, our first data structure supports each operation in O(n1/3+ε) amortized time and O(n4/3+ε) space. Our second data structure performs faster when each point set decomposes into a small number k of subsets in convex position: it supports insertions and deletions in O(logn) time and ham-sandwich queries in O(klog4n) time. In addition, if each point set has convex peeling depth k , then we can maintain the decomposition automatically using O(klogn) time per insertion and deletion. Alternatively, we can view each convex point set as a convex polygon, and we show how to find a ham-sandwich cut that bisects the total areas or total perimeters of these polygons in O(klog4n) time plus the O((kb)polylog(kb)) time required to approximate the root of a polynomial of degree O(k) up to b bits of precision. We also show how to maintain a partition of the plane by two lines into four regions each containing a quarter of the total point count, area, or perimeter in polylogarithmic time.Engineering and Applied Science