Subset Selection Problem Research Articles

Multimodal Multiplatform Social Media Event Summarization

Social media platforms are turning into important news sources since they provide real-time information from different perspectives. However, high volume, dynamism, noise, and redundancy exhibited by social media data make it difficult to comprehend the entire content. Recent works emphasize on summarizing the content of either a single social media platform or of a single modality (either textual or visual). However, each platform has its own unique characteristics and user base, which brings to light different aspects of real-world events. This makes it critical as well as challenging to combine textual and visual data from different platforms. In this article, we propose summarization of real-world events with data stemming from different platforms and multiple modalities. We present the use of a Markov Random Fields based similarity measure to link content across multiple platforms. This measure also enables the linking of content across time, which is useful for tracking the evolution of long-running events. For the final content selection, summarization is modeled as a subset selection problem. To handle the complexity of the optimal subset selection, we propose the use of submodular objectives. Facets such as coverage, novelty, and significance are modeled as submodular objectives in a multimodal social media setting. We conduct a series of quantitative and qualitative experiments to illustrate the effectiveness of our approach compared to alternative methods.

A salient dictionary learning framework for activity video summarization via key-frame extraction

Recently, dictionary learning methods for unsupervised video summarization have surpassed traditional video frame clustering approaches. This paper addresses static summarization of videos depicting activities, which possess certain recurrent properties. In this context, a flexible definition of an activity video summary is proposed, as the set of key-frames that can both reconstruct the original, full-length video and simultaneously represent its most salient parts. Both objectives can be jointly optimized across several information modalities. The two criteria are merged into a “salient dictionary” learning task that is proposed as a strict definition of the video summarization problem, encapsulating many existing algorithms. Three specific, novel video summarization methods are derived from this definition: the Numerical, the Greedy and the Genetic Algorithm. In all formulations, the reconstruction term is modeled algebraically as a Column Subset Selection Problem (CSSP), while the saliency term is modeled as an outlier detection problem, a low-rank approximation problem, or a summary dispersion maximization problem. In quantitative evaluation, the Greedy Algorithm seems to provide the best balance between speed and overall performance, with the faster Numerical Algorithm a close second. All the proposed methods outperform a baseline clustering approach and two competing state-of-the-art static video summarization algorithms.

Finding Points in General Position

We study the General Position Subset Selection problem: Given a set of points in the plane, find a maximum-cardinality subset of points in general position. We prove that General Position Subset Selection is NP-hard, APX-hard, and present several fixed-parameter tractability results for the problem as well as a subexponential running time lower bound based on the Exponential Time Hypothesis.

Iterative column subset selection

Dimensionality reduction is often a crucial step for the successful application of machine learning and data mining methods. One way to achieve said reduction is feature selection. Due to the impossibility of labelling many data sets, unsupervised approaches are frequently the only option. The column subset selection problem translates naturally to this purpose and has received considerable attention over the last few years, as it provides simple linear models for low-rank data reconstruction. Recently, it was empirically shown that an iterative algorithm, which can be implemented efficiently, provides better subsets than other state-of-the-art methods. In this paper, we describe this algorithm and provide a more in-depth analysis. We carry out numerous experiments to gain insights on its behaviour and derive a simple bound for the norm recovered by the resulting matrix. To the best of our knowledge, this is the first theoretical result of this kind for this algorithm.

Submodular Sparse Sensing for Gaussian Detection With Correlated Observations

Detection of a signal under noise is a classical signal processing problem. When monitoring spatial phenomena under a fixed budget, i.e., either physical, economical or computational constraints, the selection of a subset of available sensors, referred to as sparse sensing, that meets both the budget and performance requirements is highly desirable. Unfortunately, the subset selection problem for detection under dependent observations is combinatorial in nature and suboptimal subset selection algorithms must be employed. In this work, different from the widely used convex relaxation of the problem, we leverage submodularity, the diminishing returns property, to provide practical near optimal algorithms suitable for large-scale subset selection. This is achieved by means of low-complexity greedy algorithms, which incur a reduced computational complexity compared to their convex counterparts.

Foreground Segmentation with Tree-Structured Sparse RPCA.

Background subtraction is a fundamental video analysis technique that consists of creation of a background model that allows distinguishing foreground pixels. We present a new method in which the image sequence is assumed to be made up of the sum of a low-rank background matrix and a dynamic tree-structured sparse matrix. The decomposition task is then solved using our approximated Robust Principal Component Analysis (ARPCA) method which is an extension to the RPCA that can handle camera motion and noise. Our model dynamically estimates the support of the foreground regions via a superpixel generation step, so that spatial coherence can be imposed on these regions. Unlike conventional smoothness constraints such as MRF, our method is able to obtain crisp and meaningful foreground regions, and in general, handles large dynamic background motion better. To reduce the dimensionality and the curse of scale that is persistent in the RPCA-based methods, we model the background via Column Subset Selection Problem, that reduces the order of complexity and hence decreases computation time. Comprehensive evaluation on four benchmark datasets demonstrate the effectiveness of our method in outperforming state-of-the-art alternatives.

Alpha Matting With KL-Divergence-Based Sparse Sampling.

In this paper, we present a new sampling-based alpha matting approach for the accurate estimation of foreground and background layers of an image. Previous sampling-based methods typically rely on certain heuristics in collecting representative samples from known regions, and thus their performance deteriorates if the underlying assumptions are not satisfied. To alleviate this, we take an entirely new approach and formulate sampling as a sparse subset selection problem where we propose to pick a small set of candidate samples that best explains the unknown pixels. Moreover, we describe a new dissimilarity measure for comparing two samples which is based on KL-divergence between the distributions of features extracted in the vicinity of the samples. The proposed framework is general and could be easily extended to video matting by additionally taking temporal information into account in the sampling process. Evaluation on standard benchmark data sets for image and video matting demonstrates that our approach provides more accurate results compared with the state-of-the-art methods.

A Theory and Algorithms for Combinatorial Reoptimization

Many real-life applications involve systems that change dynamically over time. Thus, throughout the continuous operation of such a system, it is required to compute solutions for new problem instances, derived from previous instances. Since the transition from one solution to another incurs some cost, a natural goal is to have the solution for the new instance close to the original one (under a certain distance measure). In this paper we develop a general framework for combinatorial repotimization, encompassing classical objective functions as well as the goal of minimizing the transition cost from one solution to the other. Formally, we say that $$\mathcal{A}$$ is an $$(r, \rho )$$ -reapproximation algorithm if it achieves a $$\rho $$ -approximation for the optimization problem, while paying a transition cost that is at most r times the minimum required for solving the problem optimally. Using our model we derive reoptimization and reapproximation algorithms for several classes of combinatorial reoptimization problems. This includes a fully polynomial time $$(1+\varepsilon _1, 1+\varepsilon _2)$$ -reapproximation scheme for the wide class of DP-benevolent problems introduced by Woeginger (Proceedings of Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, 1999), a (1, 3)-reapproximation algorithm for the metric k-Center problem, and (1, 1)-reoptimization algorithm for polynomially solvable subset-selection problems. Thus, we distinguish here for the first time between classes of reoptimization problems by their hardness status with respect to the objective of minimizing transition costs, while guaranteeing a good approximation for the underlying optimization problem.

An estimation of distribution algorithm for public transport driver scheduling

Public transport driver scheduling is a process of selecting a set of duties for the drivers of vehicles to form a number of legal driver shifts. The problem usually has two objectives which are minimising both the total number of shifts and the total shift cost, while taking into account some constraints related to labour and company rules. A commonly used approach is firstly to generate a large set of feasible shifts by domain-specific heuristics, and then to select a subset to form the final schedule by an integer programming method. This paper presents an estimation of distribution algorithm (EDA) to deal with the subset selection problem which is NP-hard. To obtain a candidate schedules, the EDA applies a number of rules, with each rule corresponding to a particular way of selecting a shift. Computational results from some real-world instances of drive scheduling demonstrate the availability of this approach.

An improved lower bound for general position subset selection

In the General Position Subset Selection (GPSS) problem, the goal is to find the largest possible subset of a set of points such that no three of its members are collinear. If $s_{\mathrm{GPSS}}$ is the size of the optimal solution, $\sqrt{s_{\mathrm{GPSS}}}$ is the current best guarantee for the size of the solution obtained using a polynomial time algorithm. In this paper we present an algorithm for GPSS to improve this bound based on the number of collinear pairs of points. We experimentally evaluate this and few other GPSS algorithms; the result of these experiments suggests further opportunities for obtaining tighter lower bounds for GPSS.

An improved lower bound for general position subset selection

In the general position subset selection (GPSS) problem, the goal is to find the largest possible subset of a set of points, such that no three of its members are collinear. If s is the size the optimal solution, the square root of s is the current best guarantee for the size of the solution obtained using a polynomial time algorithm. In this paper, we present an algorithm for GPSS to improve this bound based on the number of collinear pairs of points. We experimentally evaluate this and few other GPSS algorithms; the result of these experiments suggests further opportunities for obtaining tighter lower bounds for GPSS.

OUP accepted manuscript

We propose a scalable algorithmic framework for exact Bayesian variable selection and model averaging in linear models under the assumption that the Gram matrix is block-diagonal, and as a heuristic for exploring the model space for general designs. In block-diagonal designs our approach returns the most probable model of any given size without resorting to numerical integration. The algorithm also provides a novel and efficient solution to the frequentist best subset selection problem for block-diagonal designs. Posterior probabilities for any number of models are obtained by evaluating a single one-dimensional integral, and other quantities of interest such as variable inclusion probabilities and model-averaged regression estimates are obtained by an adaptive, deterministic one-dimensional numerical integration. The overall computational cost scales linearly with the number of blocks, which can be processed in parallel, and exponentially with the block size, rendering it most adequate in situations where predictors are organized in many moderately-sized blocks. For general designs, we approximate the Gram matrix by a block-diagonal matrix using spectral clustering and propose an iterative algorithm that capitalizes on the block-diagonal algorithms to explore efficiently the model space. All methods proposed in this paper are implemented in the R library mombf.

Data analytics: feature extraction for application with small sample in classification algorithms

This paper focuses on improving the classification accuracy for supervised learning in areas of application with very few training data and with extremely available high dimensionality. This paper proposes a framework which acts as a decision support system incorporating both feature selection and feature extraction to improvise the classification accuracy. The feature selection technique comprises redundancy elimination and relevance analysis. Feature subset selection problems eliminate features which are redundant by using correlation-based maximum spanning tree. But, the eliminated features may contain useful information which may contribute in determining the target or class labels. The principal components are extracted from the eliminated features and they are complemented with the selected features to perform classification. The superiority of the proposed method over other feature selection methods, in terms of computational complexity and classification accuracy, is established extensively on various datasets.

PIECEWISE-LINEAR APPROXIMATION FOR FEATURE SUBSET SELECTION IN A SEQUENTIAL LOGIT MODEL

This paper concerns a method of selecting a subset of features for a sequential logit model. Tanaka and Nakagawa (2014) proposed a mixed integer quadratic optimization formulation for solving the problem based on a quadratic approximation of the logistic loss function. However, since there is a significant gap between the logistic loss function and its quadratic approximation, their formulation may fail to find a good subset of features. To overcome this drawback, we apply a piecewise-linear approximation to the logistic loss function. Accordingly, we frame the feature subset selection problem of minimizing an information criterion as a mixed integer linear optimization problem. The computational results demonstrate that our piecewise-linear approximation approach found a better subset of features than the quadratic approximation approach.

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

We consider in this paper a class of semi-continuous quadratic programming problems which arises in many real-world applications such as production planning, portfolio selection and subset selection in regression. We propose a lift-and-convexification approach to derive an equivalent reformulation of the original problem. This lift-and-convexification approach lifts the quadratic term involving $x$ only in the original objective function $f(x,y)$ to a quadratic function of both $x$ and $y$ and convexifies this equivalent objective function. While the continuous relaxation of our new reformulation attains the same tight bound as achieved by the continuous relaxation of the well known perspective reformulation, the new reformulation also retains the linearly constrained quadratic programming structure of the original mix-integer problem. This prominent feature improves the performance of branch-and-bound algorithms by providing the same tightness at the root node as the state-of-the-art perspective reformulation and offering much faster processing time at children nodes. We further combine the lift-and-convexification approach and the quadratic convex reformulation approach in the literature to form an even tighter reformulation. Promising results from our computational tests in both portfolio selection and subset selection problems numerically verify the benefits from these theoretical features of our new reformulations.

Generalized information criterion for the AR model

This paper studies the generalized information criterion (GIC) for the problem of subset selection in the autoregressive (AR) model under the condition that some of the parameters are irrelevant to the AR model. We prove that the resulting estimator obtained by minimizing the GIC is consistent in subset selection under the mild conditions. Here, the consistency means that the probability of obtaining true sub-model increases to one asymptotically. The results hold when the numbers of the relevant and irrelevant parameters increase to infinity and when the minimum of the absolute values of nonzero parameters decreases toward zero as the sample size increases. A practical method to reduce the computation time is introduced by applying the penalized estimation, which is useful when the order of AR model is very large. Simulation studies are presented to confirm the theoretical results for finite samples. As a real life example, we apply the GIC to annual total sunspot numbers.

APPLICATIONS OF GRAVITATIONAL SEARCH ALGORITHM IN ENGINEERING

Gravitational search algorithm (GSA) is a nature-inspired conceptual framework with roots in gravitational kinematics, a branch of physics that models the motion of masses moving under the influence of gravity. In a recent article the authors reviewed the principles of GSA. This article presents a review of applications of GSA in engineering including combinatorial optimization problems, economic load dispatch problem, economic and emission dispatch problem, optimal power flow problem, optimal reactive power dispatch problem, energy management system problem, clustering and classification problem, feature subset selection problem, parameter identification, training neural networks, traveling salesman problem, filter design and communication systems, unit commitment problem and multiobjective optimization problems.

Subset selection in quantile regression analysis via alternative Bayesian information criteria and heuristic optimization

ABSTRACTSubset selection is an extensively studied problem in statistical learning. Especially it becomes popular for regression analysis. This problem has considerable attention for generalized linear models as well as other types of regression methods. Quantile regression is one of the most used types of regression method. In this article, we consider subset selection problem for quantile regression analysis with adopting some recent Bayesian information criteria. We also utilized heuristic optimization during selection process. Simulation and real data application results demonstrate the capability of the mentioned information criteria. According to results, these information criteria can determine the true models effectively in quantile regression models.

Creating an acceptable consensus ranking for group decision making

This paper examines the problem of combining a set of ordinal rankings to form an acceptable consensus ranking. The objective of traditional group decision making problem is to determine the Minimum Violation Ranking. Motived by the applications of adjusted consensus in recent years, we study this problem from a new perspective, for obtaining an acceptable consensus ranking for group decision making. In this paper, every voter ranks a set of alternatives respectively, and we know the acceptability index, which represents the minimum adjustments that are allowed for each voter. The problem is to find the Minimum Acceptable Violation Ranking (MAVR) which minimizes the sum of voter’s unacceptable violations. Besides, we develop a branch and bound ranking algorithm to solve this problem. The suggested improvement include: (1) analysing the ranking preference by two ways: pairwise preference and ranking-based preference; (2) constructing the lower bound and upper bound, which exclude at most half of the feasible solutions in each iteration process. Furthermore, the effectiveness and efficiency of this algorithm are verified with an example and numerical experiments. Finally, we discuss two extensions of the basic MAVR problem: the Minimum Weighted Acceptable Violation problem, whose voters are accompanied with a set of weights or multiples, and the Minimum Hierarchy Acceptable Violation problem, which uses hierarchical acceptability indexes. In addition, our results can be applied to other ranking and subset selection problems in which provide consensus rankings over the alternatives.

Optimal column subset selection for image classification by genetic algorithms

Many problems in operations research can be solved by combinatorial optimization. Fixed-length subset selection is a family of combinatorial optimization problems that involve selection of a set of unique objects from a larger superset. Feature selection, p-median problem, and column subset selection problem are three examples of hard problems that involve search for fixed-length subsets. Due to their high complexity, exact algorithms are often infeasible to solve real-world instances of these problems and approximate methods based on various heuristic and metaheuristic (e.g. nature-inspired) approaches are often employed. Selecting column subsets from massive data matrices is an important technique useful for construction of compressed representations and low rank approximations of high-dimensional data. Search for an optimal subset of exactly k columns of a matrix, \(A^{m\times n}\), \(k < n\), is a well-known hard optimization problem with practical implications for data processing and mining. It can be used for unsupervised feature selection, dimensionality reduction, data visualization, and so on. A compressed representation of raw real-world data can contribute, for example, to reduction of algorithm training times in supervised learning, to elimination of overfitting in classification and regression, to facilitation of better data understanding, and to many other benefits. This paper proposes a novel genetic algorithm for the column subset selection problem and evaluates it in a series of computational experiments with image classification. The evaluation shows that the proposed modifications improve the results obtained by artificial evolution.