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Title: Bayesian Nonparametric Variable Selection as an Exploratory Tool for Finding Genes that Matter
Abstract: High-throughput scientific studies involving no clear a'priori hypothesis are common. For example, a large-scale genomic study of a disease may examine thousands of genes without hypothesizing that any specific gene is responsible for the disease. In these studies, the objective is to explore a large number of possible factors (e.g. genes) in order to identify a small number that will be considered in follow-up studies that tend to be more thorough and on smaller scales. For large-scale studies, we propose a nonparametric Bayesian approach based on random partition models. Our model thus divides the set of candidate factors into several subgroups according to their degrees of relevance, or potential effect, in relation to the outcome of interest. The model allows for a latent rank to be assigned to each factor according to the overall potential importance of its corresponding group. The posterior expectation or mode of these ranks is used to set up a threshold for selecting potentially relevant factors. Using simulated data, we demonstrate that our approach could be quite effective in finding relevant genes compared to several alternative methods. We apply our model to two large-scale studies. The first study involves transcriptome analysis of infection by human cytomegalovirus (HCMV). The objective of the second study is to identify differentially expressed genes between two types of leukemia.
Title: Predicting Positive and Negative Links in Online Social Networks
Abstract: We study online social networks in which relationships can be either positive (indicating relations such as friendship) or negative (indicating relations such as opposition or antagonism). Such a mix of positive and negative links arise in a variety of online settings; we study datasets from Epinions, Slashdot and Wikipedia. We find that the signs of links in the underlying social networks can be predicted with high accuracy, using models that generalize across this diverse range of sites. These models provide insight into some of the fundamental principles that drive the formation of signed links in networks, shedding light on theories of balance and status from social psychology; they also suggest social computing applications by which the attitude of one user toward another can be estimated from evidence provided by their relationships with other members of the surrounding social network.
Title: The Directed Closure Process in Hybrid Social-Information Networks, with an Analysis of Link Formation on Twitter
Abstract: It has often been taken as a working assumption that directed links in information networks are frequently formed by "short-cutting" a two-step path between the source and the destination -- a kind of implicit "link copying" analogous to the process of triadic closure in social networks. Despite the role of this assumption in theoretical models such as preferential attachment, it has received very little direct empirical investigation. Here we develop a formalization and methodology for studying this type of directed closure process, and we provide evidence for its important role in the formation of links on Twitter. We then analyze a sequence of models designed to capture the structural phenomena related to directed closure that we observe in the Twitter data.
Title: Structure-Aware Stochastic Control for Transmission Scheduling
Abstract: In this paper, we consider the problem of real-time transmission scheduling over time-varying channels. We first formulate the transmission scheduling problem as a Markov decision process (MDP) and systematically unravel the structural properties (e.g. concavity in the state-value function and monotonicity in the optimal scheduling policy) exhibited by the optimal solutions. We then propose an online learning algorithm which preserves these structural properties and achieves -optimal solutions for an arbitrarily small . The advantages of the proposed online method are that: (i) it does not require a priori knowledge of the traffic arrival and channel statistics and (ii) it adaptively approximates the state-value functions using piece-wise linear functions and has low storage and computation complexity. We also extend the proposed low-complexity online learning solution to the prioritized data transmission. The simulation results demonstrate that the proposed method achieves significantly better utility (or delay)-energy trade-offs when comparing to existing state-of-art online optimization methods.
Title: Inductive Logic Programming in Databases: from Datalog to DL+log
Abstract: In this paper we address an issue that has been brought to the attention of the database community with the advent of the Semantic Web, i.e. the issue of how ontologies (and semantics conveyed by them) can help solving typical database problems, through a better understanding of KR aspects related to databases. In particular, we investigate this issue from the ILP perspective by considering two database problems, (i) the definition of views and (ii) the definition of constraints, for a database whose schema is represented also by means of an ontology. Both can be reformulated as ILP problems and can benefit from the expressive and deductive power of the KR framework DL+log. We illustrate the application scenarios by means of examples. Keywords: Inductive Logic Programming, Relational Databases, Ontologies, Description Logics, Hybrid Knowledge Representation and Reasoning Systems. Note: To appear in Theory and Practice of Logic Programming (TPLP).
Title: Causality and Statistical Learning
Abstract: We review some approaches and philosophies of causal inference coming from sociology, economics, computer science, cognitive science, and statistics
Title: Release ZERO.0.1 of package RefereeToolbox
Abstract: RefereeToolbox is a java package implementing combination operators for fusing evidences. It is downloadable from: http://refereefunction.fredericdambreville.com/releases RefereeToolbox is based on an interpretation of the fusion rules by means of Referee Functions. This approach implies a dissociation between the definition of the combination and its actual implementation, which is common to all referee-based combinations. As a result, RefereeToolbox is designed with the aim to be generic and evolutive.
Title: The role of semantics in mining frequent patterns from knowledge bases in description logics with rules
Abstract: We propose a new method for mining frequent patterns in a language that combines both Semantic Web ontologies and rules. In particular we consider the setting of using a language that combines description logics with DL-safe rules. This setting is important for the practical application of data mining to the Semantic Web. We focus on the relation of the semantics of the representation formalism to the task of frequent pattern discovery, and for the core of our method, we propose an algorithm that exploits the semantics of the combined knowledge base. We have developed a proof-of-concept data mining implementation of this. Using this we have empirically shown that using the combined knowledge base to perform semantic tests can make data mining faster by pruning useless candidate patterns before their evaluation. We have also shown that the quality of the set of patterns produced may be improved: the patterns are more compact, and there are fewer patterns. We conclude that exploiting the semantics of a chosen representation formalism is key to the design and application of (onto-)relational frequent pattern discovery methods. Note: To appear in Theory and Practice of Logic Programming (TPLP)
Title: Near-Optimal Evasion of Convex-Inducing Classifiers
Abstract: Classifiers are often used to detect miscreant activities. We study how an adversary can efficiently query a classifier to elicit information that allows the adversary to evade detection at near-minimal cost. We generalize results of Lowd and Meek (2005) to convex-inducing classifiers. We present algorithms that construct undetected instances of near-minimal cost using only polynomially many queries in the dimension of the space and without reverse engineering the decision boundary.
Title: Targeted Event Detection
Abstract: We consider the problem of event detection based upon a (typically multivariate) data stream characterizing some system. Most of the time the system is quiescent - nothing of interest is happening - but occasionally events of interest occur. The goal of event detection is to raise an alarm as soon as possible after the onset of an event. A simple way of addressing the event detection problem is to look for changes in the data stream and equate `change' with `onset of event'. However, there might be many kinds of changes in the stream that are uninteresting. We assume that we are given a segment of the stream where interesting events have been marked. We propose a method for using these training data to construct a `targeted' detector that is specifically sensitive to changes signaling the onset of interesting events.
Title: A note on the Berman condition
Abstract: It is established that if a time series satisfies the Berman condition, and another related (summability) condition, the result of filtering that series through a certain type of filter also satisfies the two conditions. In particular it follows that if $X_t$ satisfies the two conditions and if $X_t$ and $a_t$ are related by an invertible ARMA model, then the $a_t$ satisfy the two conditions.
Title: Universal Regularizers For Robust Sparse Coding and Modeling
Abstract: Sparse data models, where data is assumed to be well represented as a linear combination of a few elements from a dictionary, have gained considerable attention in recent years, and their use has led to state-of-the-art results in many signal and image processing tasks. It is now well understood that the choice of the sparsity regularization term is critical in the success of such models. Based on a codelength minimization interpretation of sparse coding, and using tools from universal coding theory, we propose a framework for designing sparsity regularization terms which have theoretical and practical advantages when compared to the more standard l0 or l1 ones. The presentation of the framework and theoretical foundations is complemented with examples that show its practical advantages in image denoising, zooming and classification.
Title: Agreement Maintenance Based on Schema and Ontology Change in P2P Environment
Abstract: This paper is concern about developing a semantic agreement maintenance method based on semantic distance by calculating the change of local schema or ontology. This approach is important in dynamic and autonomous environment, in which the current approach assumed that agreement or mapping in static environment. The contribution of this research is to develop a framework based on semantic agreement maintenance approach for P2P environment. This framework based on two level hybrid P2P model architecture, which consist of two peer type: (1) super peer that use to register and manage the other peers, and (2) simple peer, as a simple peer, it exports and shares its contents with others. This research develop a model to maintain the semantic agreement in P2P environment, so the current approach which does not have the mechanism to know the change, since it assumed that ontology and local schema are in the static condition, and it is different in dynamic condition. The main issues are how to calculate the change of local schema or common ontology and the calculation result is used to determine which algorithm in maintaining the agreement. The experiment on the job matching domain in Indonesia have been done to show how far the performance of the approach. From the experiment, the main result are (i) the more change so the F-measure value tend to be decreased, (ii) there is no significant different in F-measure value for various modification type (add, delete, rename), and (iii) the correct choice of algorithm would improve the F-measure value.
Title: Covariance-Adaptive Slice Sampling
Abstract: We describe two slice sampling methods for taking multivariate steps using the crumb framework. These methods use the gradients at rejected proposals to adapt to the local curvature of the log-density surface, a technique that can produce much better proposals when parameters are highly correlated. We evaluate our methods on four distributions and compare their performance to that of a non-adaptive slice sampling method and a Metropolis method. The adaptive methods perform favorably on low-dimensional target distributions with highly-correlated parameters.
Title: Probability distributions with summary graph structure
Abstract: A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of edges that couple node pairs. One important class contains regression graphs. Regression graphs are a type of so-called chain graph and describe stepwise processes, in which at each step single or joint responses are generated given the relevant explanatory variables in their past. For joint densities that result after possible marginalising or conditioning, we introduce summary graphs. These graphs reflect the independence structure implied by the generating process for the reduced set of variables and they preserve the implied independences after additional marginalising and conditioning. They can identify generating dependences that remain unchanged and alert to possibly severe distortions due to direct and indirect confounding. Operators for matrix representations of graphs are used to derive these properties of summary graphs and to translate them into special types of paths in graphs.
Title: Pattern recognition using inverse resonance filtration
Abstract: An approach to textures pattern recognition based on inverse resonance filtration (IRF) is considered. A set of principal resonance harmonics of textured image signal fluctuations eigen harmonic decomposition (EHD) is used for the IRF design. It was shown that EHD is invariant to textured image linear shift. The recognition of texture is made by transfer of its signal into unstructured signal which simple statistical parameters can be used for texture pattern recognition. Anomalous variations of this signal point on foreign objects. Two methods of 2D EHD parameters estimation are considered with the account of texture signal breaks presence. The first method is based on the linear symmetry model that is not sensitive to signal phase jumps. The condition of characteristic polynomial symmetry provides the model stationarity and periodicity. Second method is based on the eigenvalues problem of matrices pencil projection into principal vectors space of singular values decomposition (SVD) of 2D correlation matrix. Two methods of classification of retrieval from textured image foreign objects are offered.
Title: Graphics Processing Units and High-Dimensional Optimization
Abstract: This paper discusses the potential of graphics processing units (GPUs) in high-dimensional optimization problems. A single GPU card with hundreds of arithmetic cores can be inserted in a personal computer and dramatically accelerates many statistical algorithms. To exploit these devices fully, optimization algorithms should reduce to multiple parallel tasks, each accessing a limited amount of data. These criteria favor EM and MM algorithms that separate parameters and data. To a lesser extent block relaxation and coordinate descent and ascent also qualify. We demonstrate the utility of GPUs in nonnegative matrix factorization, PET image reconstruction, and multidimensional scaling. Speedups of 100 fold can easily be attained. Over the next decade, GPUs will fundamentally alter the landscape of computational statistics. It is time for more statisticians to get on-board.
Title: A New Heuristic for Feature Selection by Consistent Biclustering
Abstract: Given a set of data, biclustering aims at finding simultaneous partitions in biclusters of its samples and of the features which are used for representing the samples. Consistent biclusterings allow to obtain correct classifications of the samples from the known classification of the features, and vice versa, and they are very useful for performing supervised classifications. The problem of finding consistent biclusterings can be seen as a feature selection problem, where the features that are not relevant for classification purposes are removed from the set of data, while the total number of features is maximized in order to preserve information. This feature selection problem can be formulated as a linear fractional 0-1 optimization problem. We propose a reformulation of this problem as a bilevel optimization problem, and we present a heuristic algorithm for an efficient solution of the reformulated problem. Computational experiments show that the presented algorithm is able to find better solutions with respect to the ones obtained by employing previously presented heuristic algorithms.
Title: Computational Methods in Bayesian Statistics
Abstract: This paper focuses on utilizing two different Bayesian methods to deal with a variety of toy problems which occur in data analysis. In particular we implement the Variational Bayesian and Nested Sampling methods to tackle the problems of polynomial selection and Gaussian Mixture Models, comparing the algorithms in terms of processing speed and accuracy. In the problems tackled here it is the Variational Bayesian algorithms which are the faster though both results give similar results.
Title: Linear Time Feature Selection for Regularized Least-Squares
Abstract: We propose a novel algorithm for greedy forward feature selection for regularized least-squares (RLS) regression and classification, also known as the least-squares support vector machine or ridge regression. The algorithm, which we call greedy RLS, starts from the empty feature set, and on each iteration adds the feature whose addition provides the best leave-one-out cross-validation performance. Our method is considerably faster than the previously proposed ones, since its time complexity is linear in the number of training examples, the number of features in the original data set, and the desired size of the set of selected features. Therefore, as a side effect we obtain a new training algorithm for learning sparse linear RLS predictors which can be used for large scale learning. This speed is possible due to matrix calculus based short-cuts for leave-one-out and feature addition. We experimentally demonstrate the scalability of our algorithm and its ability to find good quality feature sets.
Title: Sliding window approach based Text Binarisation from Complex Textual images
Abstract: Text binarisation process classifies individual pixels as text or background in the textual images. Binarization is necessary to bridge the gap between localization and recognition by OCR. This paper presents Sliding window method to binarise text from textual images with textured background. Suitable preprocessing techniques are applied first to increase the contrast of the image and blur the background noises due to textured background. Then Edges are detected by iterative thresholding. Subsequently formed edge boxes are analyzed to remove unwanted edges due to complex background and binarised by sliding window approach based character size uniformity check algorithm. The proposed method has been applied on localized region from heterogeneous textual images and compared with Otsu, Niblack methods and shown encouraging performance of the proposed method.
Title: Connected Spatial Networks over Random Points and a Route-Length Statistic
Abstract: We review mathematically tractable models for connected networks on random points in the plane, emphasizing the class of proximity graphs which deserves to be better known to applied probabilists and statisticians. We introduce and motivate a particular statistic $R$ measuring shortness of routes in a network. We illustrate, via Monte Carlo in part, the trade-off between normalized network length and $R$ in a one-parameter family of proximity graphs. How close this family comes to the optimal trade-off over all possible networks remains an intriguing open question. The paper is a write-up of a talk developed by the first author during 2007--2009.
Title: Modelling and simulating retail management practices: a first approach
Abstract: Multi-agent systems offer a new and exciting way of understanding the world of work. We apply agent-based modeling and simulation to investigate a set of problems in a retail context. Specifically, we are working to understand the relationship between people management practices on the shop-floor and retail performance. Despite the fact we are working within a relatively novel and complex domain, it is clear that using an agent-based approach offers great potential for improving organizational capabilities in the future. Our multi-disciplinary research team has worked closely with one of the UK's top ten retailers to collect data and build an understanding of shop-floor operations and the key actors in a department (customers, staff, and managers). Based on this case study we have built and tested our first version of a retail branch agent-based simulation model where we have focused on how we can simulate the effects of people management practices on customer satisfaction and sales. In our experiments we have looked at employee development and cashier empowerment as two examples of shop floor management practices. In this paper we describe the underlying conceptual ideas and the features of our simulation model. We present a selection of experiments we have conducted in order to validate our simulation model and to show its potential for answering "what-if" questions in a retail context. We also introduce a novel performance measure which we have created to quantify customers' satisfaction with service, based on their individual shopping experiences.
Title: Multi-Agent Simulation and Management Practices
Abstract: Intelligent agents offer a new and exciting way of understanding the world of work. Agent-Based Simulation (ABS), one way of using intelligent agents, carries great potential for progressing our understanding of management practices and how they link to retail performance. We have developed simulation models based on research by a multi-disciplinary team of economists, work psychologists and computer scientists. We will discuss our experiences of implementing these concepts working with a well-known retail department store. There is no doubt that management practices are linked to the performance of an organisation (Reynolds et al., 2005; Wall & Wood, 2005). Best practices have been developed, but when it comes down to the actual application of these guidelines considerable ambiguity remains regarding their effectiveness within particular contexts (Siebers et al., forthcoming a). Most Operational Research (OR) methods can only be used as analysis tools once management practices have been implemented. Often they are not very useful for giving answers to speculative 'what-if' questions, particularly when one is interested in the development of the system over time rather than just the state of the system at a certain point in time. Simulation can be used to analyse the operation of dynamic and stochastic systems. ABS is particularly useful when complex interactions between system entities exist, such as autonomous decision making or negotiation. In an ABS model the researcher explicitly describes the decision process of simulated actors at the micro level. Structures emerge at the macro level as a result of the actions of the agents and their interactions with other agents and the environment. 3 We will show how ABS experiments can deal with testing and optimising management practices such as training, empowerment or teamwork. Hence, questions such as "will staff setting their own break times improve performance?" can be investigated.
Title: Optimisation of a Crossdocking Distribution Centre Simulation Model
Abstract: This paper reports on continuing research into the modelling of an order picking process within a Crossdocking distribution centre using Simulation Optimisation. The aim of this project is to optimise a discrete event simulation model and to understand factors that affect finding its optimal performance. Our initial investigation revealed that the precision of the selected simulation output performance measure and the number of replications required for the evaluation of the optimisation objective function through simulation influences the ability of the optimisation technique. We experimented with Common Random Numbers, in order to improve the precision of our simulation output performance measure, and intended to use the number of replications utilised for this purpose as the initial number of replications for the optimisation of our Crossdocking distribution centre simulation model. Our results demonstrate that we can improve the precision of our selected simulation output performance measure value using Common Random Numbers at various levels of replications. Furthermore, after optimising our Crossdocking distribution centre simulation model, we are able to achieve optimal performance using fewer simulations runs for the simulation model which uses Common Random Numbers as compared to the simulation model which does not use Common Random Numbers.
Title: A Formal Approach to Modeling the Memory of a Living Organism
Abstract: We consider a living organism as an observer of the evolution of its environment recording sensory information about the state space X of the environment in real time. Sensory information is sampled and then processed on two levels. On the biological level, the organism serves as an evaluation mechanism of the subjective relevance of the incoming data to the observer: the observer assigns excitation values to events in X it could recognize using its sensory equipment. On the algorithmic level, sensory input is used for updating a database, the memory of the observer whose purpose is to serve as a geometric/combinatorial model of X, whose nodes are weighted by the excitation values produced by the evaluation mechanism. These values serve as a guidance system for deciding how the database should transform as observation data mounts. We define a searching problem for the proposed model and discuss the model's flexibility and its computational efficiency, as well as the possibility of implementing it as a dynamic network of neuron-like units. We show how various easily observable properties of the human memory and thought process can be explained within the framework of this model. These include: reasoning (with efficiency bounds), errors, temporary and permanent loss of information. We are also able to define general learning problems in terms of the new model, such as the language acquisition problem.
Title: Bayesian Nonparametric Inference of Switching Linear Dynamical Systems
Abstract: Many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes. We consider two such models: the switching linear dynamical system (SLDS) and the switching vector autoregressive (VAR) process. Our Bayesian nonparametric approach utilizes a hierarchical Dirichlet process prior to learn an unknown number of persistent, smooth dynamical modes. We additionally employ automatic relevance determination to infer a sparse set of dynamic dependencies allowing us to learn SLDS with varying state dimension or switching VAR processes with varying autoregressive order. We develop a sampling algorithm that combines a truncated approximation to the Dirichlet process with efficient joint sampling of the mode and state sequences. The utility and flexibility of our model are demonstrated on synthetic data, sequences of dancing honey bees, the IBOVESPA stock index, and a maneuvering target tracking application.
Title: Adaptive Submodularity: Theory and Applications in Active Learning and Stochastic Optimization
Abstract: Solving stochastic optimization problems under partial observability, where one needs to adaptively make decisions with uncertain outcomes, is a fundamental but notoriously difficult challenge. In this paper, we introduce the concept of adaptive submodularity, generalizing submodular set functions to adaptive policies. We prove that if a problem satisfies this property, a simple adaptive greedy algorithm is guaranteed to be competitive with the optimal policy. In addition to providing performance guarantees for both stochastic maximization and coverage, adaptive submodularity can be exploited to drastically speed up the greedy algorithm by using lazy evaluations. We illustrate the usefulness of the concept by giving several examples of adaptive submodular objectives arising in diverse applications including sensor placement, viral marketing and active learning. Proving adaptive submodularity for these problems allows us to recover existing results in these applications as special cases, improve approximation guarantees and handle natural generalizations.
Title: On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary
Abstract: Among the many ways to model signals, a recent approach that draws considerable attention is sparse representation modeling. In this model, the signal is assumed to be generated as a random linear combination of a few atoms from a pre-specified dictionary. In this work we analyze two Bayesian denoising algorithms -- the Maximum-Aposteriori Probability (MAP) and the Minimum-Mean-Squared-Error (MMSE) estimators, under the assumption that the dictionary is unitary. It is well known that both these estimators lead to a scalar shrinkage on the transformed coefficients, albeit with a different response curve. In this work we start by deriving closed-form expressions for these shrinkage curves and then analyze their performance. Upper bounds on the MAP and the MMSE estimation errors are derived. We tie these to the error obtained by a so-called oracle estimator, where the support is given, establishing a worst-case gain-factor between the MAP/MMSE estimation errors and the oracle's performance. These denoising algorithms are demonstrated on synthetic signals and on true data (images).
Title: The Projected GSURE for Automatic Parameter Tuning in Iterative Shrinkage Methods
Abstract: Linear inverse problems are very common in signal and image processing. Many algorithms that aim at solving such problems include unknown parameters that need tuning. In this work we focus on optimally selecting such parameters in iterative shrinkage methods for image deblurring and image zooming. Our work uses the projected Generalized Stein Unbiased Risk Estimator (GSURE) for determining the threshold value lambda and the iterations number K in these algorithms. The proposed parameter selection is shown to handle any degradation operator, including ill-posed and even rectangular ones. This is achieved by using GSURE on the projected expected error. We further propose an efficient greedy parameter setting scheme, that tunes the parameter while iterating without impairing the resulting deblurring performance. Finally, we provide extensive comparisons to conventional methods for parameter selection, showing the superiority of the use of the projected GSURE.
Title: Colouring and breaking sticks: random distributions and heterogeneous clustering
Abstract: We begin by reviewing some probabilistic results about the Dirichlet Process and its close relatives, focussing on their implications for statistical modelling and analysis. We then introduce a class of simple mixture models in which clusters are of different `colours', with statistical characteristics that are constant within colours, but different between colours. Thus cluster identities are exchangeable only within colours. The basic form of our model is a variant on the familiar Dirichlet process, and we find that much of the standard modelling and computational machinery associated with the Dirichlet process may be readily adapted to our generalisation. The methodology is illustrated with an application to the partially-parametric clustering of gene expression profiles.
Title: System-theoretic approach to image interest point detection
Abstract: Interest point detection is a common task in various computer vision applications. Although a big variety of detector are developed so far computational efficiency of interest point based image analysis remains to be the problem. Current paper proposes a system-theoretic approach to interest point detection. Starting from the analysis of interdependency between detector and descriptor it is shown that given a descriptor it is possible to introduce to notion of detector redundancy. Furthermore for each detector it is possible to construct its irredundant and equivalent modification. Modified detector possesses lower computational complexity and is preferable. It is also shown that several known approaches to reduce computational complexity of image registration can be generalized in terms of proposed theory.
Title: MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems
Abstract: CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.
Title: A Comprehensive Review of Image Enhancement Techniques