Quantum Fisher information matrix for unitary processes: closed relation for <Emphasis Type="Italic">SU</Emphasis>(2)

Quantum Fisher information plays a central role in the field of quantum metrology. In this paper, we study the problem of quantum Fisher information of unitary processes. Associated with each parameter \(\theta _i\) of unitary process \(U(\varvec{\theta })\), there exists a unique Hermitian matrix \(M_{\theta _i}=i(U^\dagger \partial _{\theta _i} U)\). Except for some simple cases, such as when the parameter under estimation is an overall multiplicative factor in the Hamiltonian, calculation of these matrices is not an easy task to treat even for estimating a single parameter of qubit systems. Using the Bloch vector \(\varvec{m}_{\theta _i}\), corresponding to each matrix \(M_{\theta _i}\), we find a closed relation for the quantum Fisher information matrix of the SU(2) processes for an arbitrary number of estimation parameters and an arbitrary initial state. We extend our results and present an explicit relation for each vector \(\varvec{m}_{\theta _i}\) for a general Hamiltonian with arbitrary parametrization. We illustrate our results by obtaining the quantum Fisher information matrix of the so-called angle-axis parameters of a general SU(2) process. Using a linear transformation between two different parameter spaces of a unitary process, we provide a way to move from quantum Fisher information of a unitary process in a given parametrization to the one of the other parametrizations. Knowing this linear transformation enables one to calculate the quantum Fisher information of a composite unitary process, i.e., a unitary process resulted from successive action of some simple unitary processes. We apply this method for a spin-half system and obtain the quantum Fisher matrix of the coset parameters in terms of the one of the angle-axis parameters.     

A behavioral characterization of the positive real and bounded real characteristic values in balancing

In passivity preserving and bounded realness preserving model reduction by balanced truncation, an important role is played by the so-called positive real (PR) and bounded real (BR) characteristic values. Both for the positive real as well as the bounded real case, these values are defined in terms of the extremal solutions of the algebraic Riccati associated with the system, more precisely as the square roots of the eigenvalues of the product matrix obtained by multiplying the smallest solution with the inverse of the largest solution of the Riccati equation. In this paper we will establish a representation free characterization of these values in terms of the behavior of the system. We will consider positive realness and bounded realness as special cases of half line dissipativity of the behavior. We will then show that both for the PR and the BR case, the characteristic values coincide with the singular values of the linear operator that assigns to each past trajectory in the input-output behavior its unique maximal supply extracting future continuation. We will explain that the term ‘singular values’ should be interpreted here in a generalized sense, since in our setup the future behavior is only an indefinite inner product space.     

Estimating probabilities under the three-parameter gamma distribution using composite sampling

Composite sampling may be used in industrial or environmental settings for the purpose of quality monitoring and regulation, particularly if the cost of testing samples is high relative to the cost of collecting samples. In such settings, it is often of interest to estimate the proportion of individual sampling units in the population that are above or below a given threshold value, C. We consider estimation of a proportion of the form p=P(X>C) from composite sample data, assuming that X follows a three-parameter gamma distribution. The gamma distribution is useful for modeling skewed data, which arise in many applications, and adding a shift parameter to the usual two-parameter gamma distribution also allows the analyst to model a minimum or baseline level of the response. We propose an estimator of p that is based on maximum likelihood estimates of the parameters α, β, and γ, and an associated variance estimator based on the observed information matrix. Theoretical properties of the estimator are briefly discussed, and simulation results are given to assess the performance of the estimator. We illustrate the proposed estimator using an example of composite sample data from the meat products industry.     

State observer synthesis by measurement results for nonlinear Lipschitz systems with uncertain disturbances

We propose ways to synthesize state observers that ensure that the estimation error is bounded on a finite interval with respect to given sets of initial states and admissible trajectories and also simultaneous H _∞-suppression at every time moment of initial deviations and uncertain deviations bounded in L ₂-norm, external disturbances for non-autonomous continuous Lipschitz systems. Here the gain of the observers depend on the time and are defined based on a numerical solution of optimization problems with differential linear matrix inequalities or numerical solution of the corresponding matrix comparison system. With the example of a single-link manipulator we show that their application for the state estimating of autonomous systems proves to be more efficient (in terms of convergence time and accuracy of the resulting estimates) as compared to observers with constant coefficients obtained with numerical solutions of optimization problems with linear matrix inequalities.     

Alternative linear discriminant classifier

Fisher linear discriminant analysis (FLDA) finds a set of optimal discriminating vectors by maximizing Fisher criterion, i.e., the ratio of the between scatter to the within scatter. One of its major disadvantages is that the number of its discriminating vectors capable to be found is bounded from above by C-1 for C-class problem. In this paper for binary-class problem, we propose alternative FLDA to breakthrough this limitation by only replacing the original between scatter with a new scatter measure. The experimental results show that our approach give impressive recognition performances compared to both the Fisher approach and linear SVM.     

Ginisupport vector machines for segmental minimum Bayes risk decoding of continuous speech

We describe the use of support vector machines (SVMs) for continuous speech recognition by incorporating them in segmental minimum Bayes risk decoding. Lattice cutting is used to convert the Automatic Speech Recognition search space into sequences of smaller recognition problems. SVMs are then trained as discriminative models over each of these problems and used in a rescoring framework. We pose the estimation of a posterior distribution over hypotheses in these regions of acoustic confusion as a logistic regression problem. We also show that GiniSVMs can be used as an approximation technique to estimate the parameters of the logistic regression problem. On a small vocabulary recognition task we show that the use of GiniSVMs can improve the performance of a well trained hidden Markov model system trained under the Maximum Mutual Information criterion. We also find that it is possible to derive reliable confidence scores over the GiniSVM hypotheses and that these can be used to good effect in hypothesis combination. We discuss the problems that we expect to encounter in extending this approach to large vocabulary continuous speech recognition and describe initial investigation of constrained estimation techniques to derive feature spaces for SVMs.     

Numerical solution for the Cauchy problem in nonlinear 1-D-thermoelasticity

We approximate the Cauchy problem by a problem in a bounded domain Ω _R =(?R,R) withR>0 sufficiently large, and the boundary conditions on ?Ω _R are imposed in terms of the far field behavior of solutions to the Cauchy problem. Then we solve this approximate problem by the finite element method for the spatial variable and the difference method for the time variable. Moreover a coupled numerical scheme for the Cauchy problem is presented. The error estimates are established.     

Time-triggered runtime verification

The goal of runtime verification is to monitor the behavior of a system to check its conformance to a set of desirable logical properties. The literature of runtime verification mostly focuses on event-triggered solutions, where a monitor is invoked when an event of interest occurs (e.g., change in the value of some variable). At invocation, the monitor evaluates the set of properties of the system that are affected by the occurrence of the event. This constant invocation introduces two major defects to the system under scrutiny at run time: (1) significant overhead, and (2) unpredictability of behavior. These defects are serious obstacles when applying runtime verification on safety-critical systems that are time-sensitive by nature. To circumvent the aforementioned defects in runtime verification, in this article, we introduce a novel time-triggered approach, where the monitor takes samples from the system with a constant frequency, in order to analyze the system’s health. We describe the formal semantics of time-triggered monitoring and discuss how to optimize the sampling period using minimum auxiliary memory. We show that such optimization is NP-complete and consequently introduce a mapping to Integer Linear Programming. Experiments on a real-time benchmark suite show that our approach introduces bounded overhead and effectively reduces the involvement of the monitor at run time by using negligible auxiliary memory. We also show that in some cases it is even possible to reduce the overall overhead of runtime verification by using our time-triggered approach when the structure of the system allows choosing a long enough sampling period.     

On the estimation of independent binomial random variables using occurrence and sequential information

We re-visit² the age-old problem of estimating the parameters of a distribution from its observations. Traditionally, scientists and statisticians have attempted to obtain strong estimates by ‘extracting’ the information contained in the observations taken as a set. However, generally speaking, the information contained in the sequence in which the observations have appeared, has been ignored—i.e., except to consider dependence information as in the case of Markov models and n-gram statistics. In this paper, we present results which, to the best of our knowledge, are the first reported results, which consider how estimation can be enhanced by utilizing both the information in the observations and in their sequence of appearance. The strategy, known as sequence based estimation (SBE) works as follows. We first quickly allude to the results pertaining to computing the maximum likelihood estimates (MLE) of the data when the samples are taken individually. We then derive the corresponding MLE results when the samples are taken two-at-a-time, and then extend these for the cases when they are processed three-at-a-time, four-at-a-time etc. In each case, we also experimentally demonstrate the convergence of the corresponding estimates. We then suggest various avenues for future research, including those by which these estimates can be fused to yield a superior overall cumulative estimate of the parameter of the distribution, in pattern recognition (PR), and in other internet and compression applications. We believe that our new estimates have great potential for practitioners, especially when the cardinality of the observation set is small.     

An information measure for class discrimination

This article describes a separability measure for class discrimination. This measure is based on the Fisher information measure for estimating the mixing proportion of two classes. The Fisher information measure not only provides a means to assess quantitatively the information content in the features for separating classes, but also gives the lower bound for the variance of any unbiased estimate of the mixing proportion based on observations of the features. Unlike most commonly used separability measures, this measure is not dependent on the form of the probability distribution of the features and does not imply a specific estimation procedure. This is important because the probability distribution function that describes the data for a given class does not have simple analytic forms, such as a Gaussian. Results of applying this measure to compare the information content provided by three LANDSAT-derived feature vectors for the purpose of separating small grains from other crops are presented.     

Decision algorithms for multiplayer noncooperative games of incomplete information

Extending the complexity results of Reif [1,2] for two player games of incomplete information, this paper (see also [3]) presents algorithms for deciding the outcome for various classes of multiplayer games of incomplete information, i.e., deciding whether or not a team has a winning strategy for a particular game. Our companion paper, [4] shows that these algorithms are indeed asymptotically optimal by providing matching lower bounds. The classes of games to which our algorithms are applicable include games which were not previously known to be decidable. We apply our algorithms to provide alternative upper bounds, and new time-space trade-offs on the complexity of multiperson alternating Turing machines [3]. We analyze the algorithms to characterize the space complexity of multiplayer games in terms of the complexity of deterministic computation on Turing machines.In hierarchical multiplayer games, each additional clique (subset of players with the same information) increases the complexity of the outcome problem by a further exponential. We show that an S(n) space bounded k-player game of incomplete information has a deterministic time upper bound of k + 1 repeated exponentials of S(n). Furthermore, S(n) space bounded k-player blindfold games have a deterministic space upper bound of k repeated exponentials of S(n). This paper proves that this exponential blow-up can occur.We also show that time bounded games do not exhibit such hierarchy. A T(n) time bounded blindfold multiplayer game, as well as a T(n) time bounded multiplayer game of incomplete information, has a deterministic space bound of T(n).     

Synthesis of State Unknown Inputs Observers for Nonlinear Lipschitz Systems with Uncertain Disturbances

We propose methods to synthesize observers for the state and unknown input influences that ensure that estimation error is finite time bounded with respect to given sets of initial states and admissible trajectories or suppress initial deviations and uncertain bounded in L∞-norm external disturbances for time-varying continuous Lipschitz systems. Here gain coefficients of the observers depend on time and are determined based on numerical solutions of optimization problems with differential linear matrix inequalities or numerical solutions of the corresponding matrix comparison system. With the example of an electric drive system with elastic transmission of motion we show that their application for state estimation and unknown inputs for time-invariant systems proves to be more efficient (with respect to convergence time and accuracy of the resulting estimates) compared to observers with constant coefficients obtained based on numerical solutions of optimization problems with linear matrix inequalities.     

Theoretical approach to characterize the non-Markovianity and diffusion through the influx of the information

In this paper, we study the Fisher information for a quantum system consisting of two identical qubits, each of them locally interacting with a bosonic reservoir in the same environment for non-Markovian open, dissipative quantum system. Based on the influx of the information, we propose an information-theoretical approach for characterizing the time-dependent memory effect of environment and diffusion function under the effect of the physical parameters. More precisely, an interesting monotonic relation between the time derivative of quantum Fisher information (QFI) and diffusion function behavior is observed during the time evolution. The phenomenon is that the QFI, namely the precision of estimation, changes dramatically with the environment structure. The dependence of the physical parameters shows that the increasing in the temperature will damage the amount of the QFI with respect of the ratio between the reservoir cutoff frequency and the system oscillation frequency.

     

Kalman filtering for multiple time-delay systems

This paper is to study the linear minimum variance estimation for discrete-time systems with instantaneous and l-time delayed measurements by using re-organized innovation analysis. A simple approach to the problem is presented in this paper. It is shown that the derived estimator involves solving l+1 different standard Kalman filtering with the same dimension as the original system.     

Minimum disparity computation via the iteratively reweighted least integrated squares algorithms

Minimum disparity estimation is appealing in that the estimates it provides are simultaneously robust and efficient. This paper presents a family of algorithms called iteratively reweighted least integrated squares for minimum disparity computation. This family of algorithms, indexed by a real parameter α, approximates the disparity measure by quadratic functions, in a form of integrated weighted squared errors, and minimizes the quadratic functions conveniently by using weighted least squares linear regression algorithms. Among all potential values of α, we advocate the use of α=1 from the consideration of robust estimation, which results in an algorithm similar in spirit to the Fisher scoring method for maximum likelihood computation. Numerical studies show that the new algorithms, especially the one that uses α=1, give competitive or better performance over the other algorithms available in the literature.     

On heuristics for minimum length rectilinear partitions

The problem of partitioning a rectilinear figure into rectangles with minimum length is NP-hard and has bounded heuristics. In this paper we study a related problem,Elimination Problem (EP), in which a rectilinear figure is partitioned into a set of rectilinear figures containing no concave vertices of a fixed direction with minimum length. We show that a heuristic for EP within a factor of 4 from optimal can be computed in timeO(n ²), wheren is the number of vertices of the input figure, and a variant of this heuristic, within a factor of 6 from optimal, can be computed in timeO(n logn). As an application, we give a bounded heuristic for the problem of partitioning a rectilinear figure into histograms of a fixed direction with minimum length. An auxiliary result is that an optimal rectangular partition of a monotonic histogram can be computed in timeO(n ²), using a known speed-up technique in dynamic programming.     

The theoretical analysis of FDA and applications

Representation and embedding are usually the two necessary phases in designing a classifier. Fisher discriminant analysis (FDA) is regarded as seeking a direction for which the projected samples are well separated. In this paper, we analyze FDA in terms of representation and embedding. The main contribution is that we prove that the general framework of FDA is based on the simplest and most intuitive FDA with zero within-class variance and therefore the mechanism of FDA is clearly illustrated. Based on our analysis, ε-insensitive SVM regression can be viewed as a soft FDA with ε-insensitive within-class variance and L₁ norm penalty. To verify this viewpoint, several real classification experiments are conducted to demonstrate that the performance of the regression-based classification technique is comparable to regular FDA and SVM.     

Near optimal interval observers bundle for uncertain bioreactors

In this paper we design an interval observer for the estimation of unmeasured variables of uncertain bioreactors. The observer is based on a bounded error observer, as proposed in [Lemesle, V., & Gouzé, J.-L. (2005). Hybrid bounded error observers for uncertain bioreactor models. Bioprocess and Biosystems Engineering, 27, 311-318], that makes use of a loose approximation of the bacterial kinetics. We first show how to generate guaranteed upper and lower bounds on the state, provided that known intervals for the initial condition and the uncertainties are available. These so-called framers depend on a tuning gain. They can be run in parallel and the envelope provides the best estimate. An optimality criterion is introduced leading to the definition of an optimal observer. We show that this criterion provides directly a gain set containing the best framers. The method is applied to the estimation of the total biomass of an industrial wastewater treatment plant, demonstrating its efficiency.     

Determining the K-best solutions of knapsack problems

It is well-known that knapsack problems arise as subproblems of a number of large-scale integer optimization problems. In order to solve these large problems, it is necessary to solve the subproblems efficiently, and for many of them it can be useful to determine the K-best solutions. In this paper, a branch-and-bound method for the unbounded knapsack problem described in the literature is extended to determine the K-best solutions of unbounded and bounded knapsack problems. We show that the proposed extension determines exactly the K-best solutions and we solve important classical instances using high values of K.     

Spanning Trees and the Complexity of Flood-Filling Games

We consider problems related to the combinatorial game (Free-) Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. We show that the minimum number of moves required to flood any given graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T. This result is then applied to give two polynomial-time algorithms for flood-filling problems. Firstly, we can compute in polynomial time the minimum number of moves required to flood a graph with only a polynomial number of connected subgraphs. Secondly, given any coloured connected graph and a subset of the vertices of bounded size, the number of moves required to connect this subset can be computed in polynomial time.