Selecting training sets for support vector machines: a review
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Abstract
Support vector machines (SVMs) are a supervised classifier successfully applied in a plethora of reallife applications. However, they suffer from the important shortcomings of their high time and memory training complexities, which depend on the training set size. This issue is especially challenging nowadays, since the amount of data generated every second becomes tremendously large in many domains. This review provides an extensive survey on existing methods for selecting SVM training data from large datasets. We divide the stateoftheart techniques into several categories. They help understand the underlying ideas behind these algorithms, which may be useful in designing new methods to deal with this important problem. The review is complemented with the discussion on the future research pathways which can make SVMs easier to exploit in practice.
Keywords
Support vector machine Training set selection Data reduction Classification1 Introduction
Support vector machine (SVM) (Cortes and Vapnik 1995) is a supervised classifier which has been proved highly effective in solving a wide range of pattern recognition and computer vision problems (AranaDaniel and BayroCorrochano 2006; Cyganek 2008; AranaDaniel et al. 2009; BayroCorrochano and AranaDaniel 2010; Cyganek et al. 2015; Li et al. 2016; Rodan et al. 2016). Nowadays, in the era of big data, the machine learning community faces new challenges concerned with applying SVMs in reallife scenarios, which result from data variety, volume, velocity, and veracity. The amount of data (of varying quality) which is being generated every day grows tremendously in the majority of scientific and engineering domains, including, among others, medical imaging, text categorization, computational biology, genomics and banking. Although it may appear quite beneficial at the first glance—more data could mean more possibilities of extracting and revealing useful underlying knowledge—training SVMs from extremely large and difficult datasets became a pivotal issue due to the high time and memory complexity of the SVM training (Liu et al. 2016; Qiu et al. 2016).
SVM training consists in determining a hyperplane to separate the training data belonging to two classes. Its position is defined with a (usually small) subset of vectors from the training set (\(\varvec{T}\)), called support vectors (SVs). Knowing which vectors are selected as SVs increases the interpretability of the SVM decisions. Though the hyperplane separates the data linearly, SVMs are applicable to nonlinear problems, thanks to mapping the data into higherdimensional spaces, in which they are linearly separable—this mapping is achieved using kernel functions. A crucial drawback of SVMs lies in their high \(O(t^3)\) time and \(O(t^2)\) memory training complexities, where \(t\) is the cardinality of \(\varvec{T}\). This problem has attracted significant attention from the researchers—developed techniques are aimed either at improving the training phase, or at extracting reduced (significantly smaller) SVM training sets from which SVs are likely to be determined. This review summarizes the achievements in this field. To the best of our knowledge, this is the first review of methods devoted to selecting the SVM training sets reported in the literature so far.
1.1 Broader context
To better contextualize this review in the literature, we highlight the main problems related to SVMs which are actively being tackled and should be inevitably resolved—they reach far beyond dealing with large datasets. These problems concern selecting the SVM hyperparameters (Sect. 1.1.1) and learning SVMs from data of questionable quality (Sect. 1.1.2). Both issues, along with selecting SVM training data from large datasets, significantly affect the applicability of the SVM classifier in practice. Addressing them successfully will help exploit this classifier in emerging big data scenarios.
1.1.1 Model selection for SVMs
Model selection for SVMs—being a problem of determining the SVM hyperparameters, including a kernel function and its parameters—is a pivotal, yet computationally expensive task (Gold and Sollich 2003; Ding et al. 2015). Automatic model selection is a crucial issue, since improperly tuned parameters can affect the SVM performance. Although there exist techniques tailored to tune predefined kernels (Tang et al. 2009), the research effort is put into designing algorithms which determine the desired kernels.
Friedrichs and Igel (2005) proposed the covariance matrix adaptation evolution strategy to determine a kernel from a parameterized kernel space. Their experimental study showed that this strategy easily outperforms a standard gridsearch approach for selecting these hyperparameters (which is obviously not scalable for large numbers of parameters). Lessmann et al. (2006) incorporated the model selection criterion into the fitness function of their genetic technique. In the hybrid genetic algorithm (GA), the evolutionary optimization was combined with the gradient descent method (Zhou and Xu 2009). GAs were recently used for the smooth twin parametricmargin SVMs (Wang et al. 2013b). In the latest algorithm by Chou et al. (2014), the SVM parameters were optimized using a fast messy Ali and SmithMiles (2006) explored the possibility of applying rulebased classifiers to generate SVM models. Other interesting approaches include tabu searches (Lebrun et al. 2008), compressionbased techniques (Luxburg et al. 2004), and geneticprogrammingbased systems (Sullivan and Luke 2007). Zhang and Song (2015) noticed that various kernels may perform equally well for a certain dataset, and proposed a multilabel kernel recommendation method built on the data characteristics. An interesting model adaptation, which combines the swarm intelligence with a grid search, was proposed by Kapp et al. (2012).
To speed up the process of model selection for SVMs, a number of parallel algorithms have been proposed (Devos et al. 2014). However, their underpinning approaches are often very simple (Shi and Liu 2012; Ripepi et al. 2015). A promising research direction includes algorithms to construct new kernels tailored for a problem at hand (Lessmann et al. 2006). Such approaches include neurofuzzy systems which construct kernels from scratch (Simiński 2014). This algorithm was used in the preliminary research on parameterless SVMs (Nalepa et al. 2015b). It is worth mentioning that determining the desired SVM model should be coupled with techniques for training SVMs from large datasets (especially for reducing the cardinality of SVM training sets), because the bestperforming kernel may be dependent on the outcome of a training set selection algorithm. This research direction has not been exploited so far, and we believe it will significantly change in the nearest future.
1.1.2 Learning from weaklylabeled, noisy, and poorquality data
Retrieving correctly labeled datasets is an expensive and challenging task, because it may involve repeating experiments or performing timeconsuming annotation procedures (e.g., in the field of medical imaging). Therefore, learning from weaklylabeled data became an important issue. All weaklabel problems are divided into several groups, based on the label characteristics. They include problems with (i) partiallyknown labels (most of the training set vectors are unlabeled and only some of them are labeled), (ii) implicitlyknown labels (training vectors are grouped into bags for which the labels are known^{1}—the labels of the training set vectors are implicit and they are based on their bag membership), and (iii) unknown labels (Li et al. 2013). Other potential issues concerned with the data quality relate to the label and/or feature noise, which can adversely impact the classifier performance. It is especially visible in practical medical applications, in which a majority of diagnostic tests are not 100% accurate, and cannot be considered a gold standard (Frenay and Verleysen 2014) (e.g., there may be discrepancies between the segmentation of the same medical image analyzed by two independent radiology experts). The consequences of the label noise on the behavior of a classifier can be very severe. First, its performance may be significantly deteriorated, the learning requirements can be easily affected (e.g., an appropriate cardinality of the training set can notably increase to compensate mislabeled or noisy data points), the final model can be much more complex than it should be, and the other algorithms (e.g., for feature selection) may be polluted as well. Frenay and Verleysen (2014) indicate that the label noise affects the observed frequencies of medical test results, hence leads to incorrect conclusions on population characteristics.
There exist three main groups of approaches for dealing with noisy sets (manual analysis should not be considered, because it is unacceptably timeconsuming and infeasible for reallife data). First, there are classifiers that are said to be robust against the noise (however, the underlying nature and model of such noise is not considered in these techniques at all) (Duan and Wu 2017). Alternatively, it is possible to build a noise model (typically, it is retrieved in parallel with the learned classifier, and they are finally coupled for the higherquality classification). Such embedded data cleansing was used for SVMs (Xu et al. 2006), also for adversarial label noise (Xiao et al. 2015). The last group encompasses algorithms which filter noisy and/or mislabeled vectors from the input set. Although it appears quite tempting (and natural), since it resembles removing outliers and anomalies from the data, it is not trivial. These filtering algorithms include various graph and ensemblebased methods, and those which detect mislabeled vectors by analyzing their impact on the learning procedure. There are works indicating that evolutionary techniques can effectively detect and remove (or just identify) the noise (Ghoggali and Melgani 2009; Han and Chang 2013). A generic solution for learning SVMs from weak labels was introduced by Li et al. (2013)—labels are subject to the optimization. This is effective, if vectors belonging to the opposite classes form wellseparated clusters in the kernel space, but this assumption may not hold in many scenarios (Cour et al. 2009; Tapaswi et al. 2015).
Handling poor quality data attracts more and more research attention nowadays (Zhu et al. 2014). It concerns not only dealing with noisy and weaklylabeled sets, but also with detecting ambiguous or duplicated data (with overlapping feature values), and outliers (Tsyurmasto et al. 2014; Kourou et al. 2015). As mentioned by Frenay and Verleysen (2014), most of the algorithms make assumptions concerning the data, and are characterized by difficulttotune parameters. These approaches should be validated using a larger number of reallife scenarios to find the real noise characteristics. Evolving labels can significantly improve the SVM performance for weaklylabeled sets, as shown by Kawulok and Nalepa (2015). An interesting research direction involves algorithms which evolve both labels and reduced training sets. Such methods could address the problems of training SVMs from large datasets and coping with lowquality data comprehensively.
1.2 Motivation and goals
The problem of training SVMs from large datasets is becoming increasingly important since the amount of data grows extremely rapidly (note that the term large dataset is very ambiguous in the literature—sizes of such datasets range from hundreds to millions of training vectors). There exist generic training set selection techniques [also referred to as instance selection algorithms in the literature (OlveraLópez et al. 2010)], and those designed for other classifiers [knearest neighbors (Angiulli 2005), neural networks (Reeves and Taylor 1998), and many other (HernandezLeal et al. 2013; Wenyuan et al. 2013)], but—due to the specific characteristics of the SVM training process and operation—the majority of SVM training set selection algorithms are crafted for this classifier. In this review, we summarize the stateoftheart algorithms for selecting SVM training data from large datasets.

We present an extensive review of the stateoftheart methods for selecting SVM training sets. Not only do we report these methods, but we also discuss their potential weaknesses, strengths, and ideas behind them. This will allow for better understanding (i) how to cope with massive reallife sets, and (ii) how to select an appropriate method for a problem at hand.

We believe that this review will notably help in developing new approaches for selecting SVM training sets. The presented taxonomy should be useful in identifying the potential pitfalls of emerging training set selection algorithms, and in determining which techniques could be successfully combined into hybrid algorithms to further boost all available knowledge concerning the \(\varvec{T}\) vectors (e.g., those methods which utilize complementary sources of information in search of valuable training vectors). The literature in this field is very diverse—we hope that this review will clearly highlight the areas which should (or should not) be further explored.
1.3 Structure of the review
Section 2 serves as a short theoretical introduction to SVMs. Section 3 begins with the proposed taxonomy to classify the methods of selecting SVM training data from large datasets. We discuss in detail techniques which help reduce the size of the SVM training sets, and highlight their most important characteristics. Section 4 concludes the review and serves as an outlook to the future work.
2 Theoretical background
Consider a set \(\varvec{T}\) of \(t\) training feature vectors \(\varvec{x}_i\in \mathbb {R}^\mathrm{\mathcal {D}}\), \(i=1,\dots ,t\), and the corresponding class labels \(y_i\in \{+1, 1\}\) (for the binary classification). Vectors with the class label \(+1\) are the positive ones (class \(\mathcal {C}_{+}\)), whereas the others belong to the negative class \(\mathcal {C}_{}\).
2.1 Linear SVMs
2.2 Nonlinear SVMs
Determining the SVM decision hyperplane is a constrained QP optimization problem—see Eqs. (21) and (22). This QP problem can be solved in \(O(t^3)\) time with \(O(t^2)\) memory, where \(t\) is the cardinality of \(\varvec{T}\), using a standard QP solver (Zeng et al. 2008b). It quickly becomes infeasible for massively large, reallife datasets. Although there exist techniques aimed at accelerating the SVM training which include—among others—decompositionbased (Joachims 1999), parallel (Li et al. 2011; Ferragut and Laska 2012) and approximation (Le et al. 2014) approaches, many of them introduce additional memory burden during the optimization (Alamdar et al. 2016). Hence, algorithms for reducing the size of SVM training sets are considered an immediate remedy to the problem of learning SVMs from large datasets. Also, the SVM classification time is linearly dependent on the number of SVs [see Eqs. (17) and (29)—only those vectors for which the Lagrange multipliers are greater than zero contribute to the decision]. Therefore, the number of SVs should be kept low to speed up the classification of incoming (unseen) vectors. The number of SVs indirectly depends on the cardinality of a training set—the smaller the number of vectors in \(\varvec{T}\), the less SVs are determined in the training process. A more extensive background information on SVMs, complemented with numerous examples and analogies which further illustrate the concepts behind SVMs are explained in an excellent tutorial by Burges (1998).
3 Selecting SVM training sets
All algorithms for dealing with training SVMs from large datasets can be divided into two main categories including techniques which (i) speed up the SVM training, and (ii) reduce the size of training sets by selecting candidate vectors (i.e., those vectors which are likely to be annotated as SVs). In the first case, existing techniques are applied to either reduce the complexity of the underlying optimization problem, or to handle the optimization process more efficiently. However, this approach still induces the problem of high memory complexity of the SVM training process which is challenging and has to be endured in big data problems (Guo and Boukir 2015; Wang and Xu 2004). The algorithms from the second category select vectors from \(\varvec{T}\) to form significantly smaller training sets—in this review, we focus on approaches for selecting SVM training sets from large datasets.
This section gathers the algorithms which extract refined SVM training sets in order to reduce the computational and storage burden of the training. We divide these techniques into five main categories: (i) data geometry analysis algorithms (investigating the geometry of \(\varvec{T}\) in search of candidate vectors that should be included into the refined sets \(\varvec{T'}\)’s), (ii) neighborhood analysis methods (exploiting the statistical properties of \(\varvec{T}\) and investigating the local neighborhoods of \(\varvec{T}\) vectors), (iii) evolutionary techniques (evolving refined training sets), (iv) active learning, and (v) random sampling techniques.
3.1 Data geometry analysis methods
The following section discusses approaches which exploit the information about the training set structure to extract SV candidates (i.e., such vectors, which are likely to be selected as SVs in the training process). These vectors are then used to form refined training sets of significantly smaller sizes than the original dataset. All approaches can be divided into two groups—the first encompasses clusteringbased techniques, whereas the second contains the remaining geometrybased algorithms.
3.1.1 Clusteringbased methods
Clusteringbased algorithms have been intensively studied for selecting refined training sets. Lyhyaoui et al. (1999) indicate their theoretical advantages: (i) clusteringbased techniques can always eliminate the useless vectors from \(\varvec{T}\), (ii) they are applicable to multiclass problems, (iii) their cost objectives may be freely established for a given problem. However, these methods suffer from a difficult problem of determining a potentially large number of parameters (the clustering parameters, and the number of vectors annotated as important for each cluster are the most important parameters).
Lyhyaoui et al. (1999) applied the frequencysensitive competitive learning to cluster training set vectors (Scheunders and Backer 1999), with various numbers of centroids for each class. Once centroids are determined, they are further analyzed to extract the most important (critical) centroids. First, each of them is visited and the nearest oppositeclass centroid is found. If two centroids (denoted as the centroids A and B) are the nearest to one another in both senses (thus when the centroid A is the closest centroid for B and vice versa), then they are put into the pool of critical centroids. Finally, the already selected critical centroids are utilized to classify the remaining ones using the 1nearest neighbor algorithm, and the wrongly classified centroids are considered important and annotated as critical (they will most likely lay near the decision hyperplane). The authors developed four different sample selection mechanisms to extract the final vectors which are to be included in the refined training set. These approaches are based on: (i) analysis of the dispersion of the vectors, (ii) the vector’s neighborhood analysis (i.e., the nearest oppositeclass vector of the one added to \(\varvec{T'}\) is added to \(\varvec{T'}\) as well), (iii) the combination of (i) and (ii), and (iv) analysis of the relations between vectors and centroids. The authors concluded that applying different selection algorithms does not drastically influence the classification score (however, the twoclass training set used in the experiments was very small).
The kmeans clustering has been utilized by Barros de Almeida et al. (2000) in their refined training set selection algorithm referred to as SVMKM. In SVMKM, k clusters (where k is a userdefined input parameter of the algorithm) are formed for the entire training set (not for vectors belonging to different classes independently). Then, the oneclass clusters (i.e., those containing vectors belonging to a single class) are disregarded and only their centroids survive in a refined set, whereas all vectors from the heterogeneous clusters (containing vectors from different classes) are appended to \(\varvec{T'}\). It is worth noting that the data distribution may significantly affect the performance of SVMKM (it is suitable for dense datasets and may misbehave for the sparse ones). Also, the value of k should be set with care, since it can easily jeopardize the algorithm behavior.
In CBSVM, the CF trees are constructed for both classes separately, and SVMs are trained using centroids of the root entries (there is at least one entry in the root, each entry being a cluster, therefore there is at least one centroid) of both trees. If there are too few vectors in this set, then the second level entries of the trees are included in a refined set. Then, the entries positioned near the hyperplane (socalled the low margin clusters) are declustered, and the child entries declustered from the parents are added to \(\varvec{T'}\) along with the nondeclustered parents. Another SVM is finally trained using the centroids of \(\varvec{T'}\) entries—this process is continued until there are no entries to be declustered. Although the method appeared to be wellscalable for large datasets, the authors pointed out that it is currently limited to linear kernels since the hierarchical microclusters will not be isomorphic to highdimensional feature spaces. Also, the algorithm parameters (\(b_\mathrm{CF}\) and \(t_\mathrm{CF}\)) should be selected with care for an analyzed dataset.
In the algorithm proposed by Cervantes et al. (2008), the concept of the minimum enclosing ball (MEB) clustering has been introduced. The MEB of a given set \(S_\mathrm{MEB}\) is the smallest ball enclosing all balls and vectors in \(S_\mathrm{MEB}\). The ball is denoted as \(B(c_B,r_B)\), where \(c_B\) and \(r_B\) are the center and the radius of B. Since finding an optimal ball for a given set is very challenging, the authors proposed to use the \((1+\epsilon )\)—approximation of MEBs. After the MEB clustering, a refined set contains all the vectors from mixedclass clusters, along with centroids of oneclass clusters. After the SVM training, an additional declustering is applied to recover other potentially valuable \(\varvec{T}\) vectors which lay near the decision hyperplane and to append them to \(\varvec{T'}\).
A similar approach (named SebSVM) was proposed by Zeng et al. (2008). Here, the convex hull vectors are selected to form refined training sets in the feature space. This is performed by solving the MEB problem in the feature space: at first, data are mapped into a higherdimensional kernel space, and two MEBs are created (for both classes independently). Based on those MEBs, the convex hull vectors from \(\varvec{T}\) are extracted. Similar to Koggalage and Halgamuge (2004), the safety region is utilized in SebSVM to avoid removing useful vectors from \(\varvec{T'}\).
An interesting technique which combines the kmeans clustering with edge detection within the entire training set has been proposed by Li et al. (2009). In this algorithm, the training set is interpreted as a color image (there are two distinct colors for binary classification denoting two classes). Relying on image processing techniques, a pixel’s neighborhood is scanned to detect strong changes of brightness and color which may correspond to edges. In the edge detection exploited by Li et al. (2009), vectors from \(\varvec{T}\) are analyzed—if at least one neighboring vector is of a different class than the investigated one (\(\varvec{a}\)), then \(\varvec{a}\) survives in \(\varvec{T'}\) (the neighboring vectors are rejected). This process is complemented with the kmeans clustering which aims at finding the centroids from \(\varvec{T}\), which are also appended to the refined training set.
The analysis of convex hulls have been applied in numerous other algorithms for selecting refined training sets (also for e.g., artificial neural networks) (Wang et al. 2007). These approaches include interesting analyses of CHs exploited for the online classifier training (Khosravani et al. 2013; Wang et al. 2013a). In these techniques, SVMs are updated dynamically when new vectors arrive to the system (based on the skeleton samples—being the vertices of convex hulls—extracted either offline or online, when new vectors appear). The authors indicated that the algorithm may not be applicable in the case of noisy datasets, and they suggest to incorporate denoising methods before the offline selection of the \(\varvec{T'}\) vectors (removing noisy vectors in the online update step still requires investigation).
An additional technique introduced by Shen et al. (2016) concerns removing redundant clusters. The initial clusters retrieved using kmeans clustering are further divided into oneclass and heterogeneous clusters. The latter ones are then subclustered to distinguish oneclass inner clusters. The authors point out that SVs will be derived from the heterogeneous clusters with a higher probability, and some \(\varvec{T}\) vectors can be safely deleted from oneclass clusters. Redundant oneclass clusters are removed using the maxmin cluster distance algorithm, and the vectors belonging to these clusters are rejected from \(\varvec{T'}\).
Since clustering techniques may become quite timeconsuming, there appeared approaches which utilize various parallel architectures (e.g., graphics processing units) in order to speed up the \(\varvec{T'}\) selection process (Yuan et al. 2015), and they were applied to reallife problems. Another important issue of these methods which needs to be addressed is a proper selection of their crucial parameters, which can easily affect refined training sets. Finally, in many cases it is still necessary to analyze the entire \(\varvec{T}\) to extract useful information.
3.1.2 Nonclustering methods
Angiulli and Astorino (2010) proposed an interesting technique which utilizes the fast nearest neighbor condensation classification rule (FCNN) (Angiulli 2007). In their algorithm (abbreviated as FCNNSVM), SVMs are coupled with the FCNN—unlike clusteringbased methods, the vector selection criteria are guided by the decision boundary. The FCNN rules start with an initial refined training set composed of the centroids generated for each class independently. Then, for each vector \(\varvec{a}\) in \(\varvec{T'}\), a point belonging to the Voronoi cell (i.e., the Voronoi cell of \(\varvec{a}\) is a set of \(\varvec{T}\) vectors that are positioned closer to \(\varvec{a}\) compared with any other vector in the current \(\varvec{T'}\)) of \(\varvec{a}\), but annotated with an oppositeclass label is included in a refined set. The algorithm continues until there are no more vectors from \(\varvec{T}\) to be appended to \(\varvec{T'}\). Although the algorithm is quite simple, it proved to be efficient and retrieves highquality refined training sets.
3.2 Neighborhood analysis methods
The kNN analysis may become quite computationally intensive. The same authors improved this technique to speed up the computation (Shin and Cho 2003). The improved algorithm is based on a simple observation, that the neighbors of a vector which is positioned near the hyperplane are also situated in its vicinity. This observation has been used to reduce the search space—a significant number of \(\varvec{T}\) vectors can be pruned once some of vectors positioned near the hyperplane are found. The k value notably affects the performance of this technique, therefore it should be carefully tuned (Shin and Cho 2007).
A simple yet effective neighborhood analysis of each \(\varvec{T}\) vector was proposed by Wang et al. (2005). For each training vector, the largest sphere which contains only vectors of the same class is determined, and the number of vectors encompassed by this sphere is verified (\(N_{\varvec{a}}\) for each \(\varvec{a}\)). Then, all \(\varvec{T}\) vectors are sorted ascendingly according to the \(N_{\varvec{a}}\) values—\(t'/2\) vectors with the lowest \(N_{\varvec{a}}\)’s (for each class) are appended to a refined set, since vectors surrounded by the sameclass vectors will most likely not be SVs and can be safely removed from \(\varvec{T'}\). The rejected \(\varvec{T}\) vectors are thus characterized by large \(N_{\varvec{a}}\) values. This approach slightly resembles the MEBbased techniques.
In a recent paper, Guo and Boukir (2015) extended their ensemble marginbased algorithm—they pointed out that classic bagging trees are not effective in the case of large training sets and the large dimensionality of the input data. They proposed to exploit more powerful ensemble methods including random forests and a very small ensemble referred to as the small votes instance selection (SVIS). In SVIS, the authors decreased the size of the classifier committee. Ensemble classifiers were utilized in other algorithms to tackle realworld problems, e.g., selecting refined training sets from biomedical data (Oh et al. 2011).
Li and Maguire (2011) proposed a method for selecting critical patterns from the input dataset which combines various techniques. First, the surface which passes through all extreme points and encompassing oneclass vectors is created, and then the hyperplane is positioned at the tangent to this surface. The position of the vectors will depend on the curvature of the surface (if it is convex, then all vectors will appear on the same size of the plane). To deal with the overlapping patterns in the input space, the authors enhanced the algorithm with a remedy which removes the class overlap in the set. This strategy is based on the Bayes posterior probability of a vector \(\varvec{a}\) belonging to a class \(\mathcal {C}\), denoted as \(\mathcal {P}(\mathcal {C},\varvec{a})\). For twoclass sets, both \(\mathcal {P}(\mathcal {C}_{+},\varvec{a})\) and \(\mathcal {P}(\mathcal {C}_{},\varvec{a})\) are estimated. If the larger probability is obtained for the class which \(\varvec{a}\) does not belong to, then this vector is removed from the training set. Finally, any duplicated patterns from \(\varvec{T}\) are removed from the dataset during the preprocessing. The authors showed that their algorithm is competitive to four stateoftheart techniques and is applicable to other classifiers.
In a recent paper, Cervantes et al. (2015) incorporated an induction tree to reduce the size of SVM training sets. The main idea behind the proposed technique is to train SVMs using significantly smaller refined training sets, and then to label vectors from \(\varvec{T}\) as those which are close or far from the decision hyperplane. A decision tree is utilized to identify vectors which have similar characteristics to those annotated as SVs. The initial selection of a small subset of \(\varvec{T}\) is accomplished with a very simple heuristics in which the level of dataset imbalance is investigated. The authors classify the incoming dataset (based on two predefined thresholds, \(\tau _{u}=0.1\) and \(\tau _{b}=0.25\), and the imbalance ratio \(\mathcal {I}\) of the dataset, given as \(\mathcal {I}=\frac{\min \left\{ t_{+},t_{}\right\} }{t}\), where \(t_{+}\) and \(t_{}\) denote the numbers of vectors from each class in \(\varvec{T}\)) to one out of the following classes: (i) balanced, (ii) slightly imbalanced (if \(\tau _{b}\le \mathcal {I}\le 0.5\)), (iii) moderately imbalanced (\(\tau _{u}\le \mathcal {I}<\tau _{b}\)), or (iv) highly imbalanced (\(\mathcal {I}<\tau _{u}\)). If a dataset is balanced, then the initial subset is retrieved using random sampling. Otherwise, if a dataset is slightly or moderately imbalanced, the inverse probability proportional to the dataset cardinality is applied (e.g., if \(80\%\) of vectors come from the negative class, hence \(\mathcal {I}=0.2\), then random sampling draws \(80\%\) of positiveclass vectors). If a dataset is highly imbalanced, then all vectors from the less numerous class survive in \(\varvec{T'}\). Based on the decision hyperplane obtained using \(\varvec{T'}\), a decision tree is induced to model the distribution of SVs. This tree is used to retrieve those vectors which were not annotated as SVs, but follow a similar distribution—they are included in \(\varvec{T'}\).
He et al. (2011) introduced a neighborhoodbased rough set model (FARNeM) to search for boundary vectors in \(\varvec{T}\). This model is used to divide the vectors into three regions: (i) the positive region, (ii) the noisy region, and (iii) the boundary region. Additionally, all input data features are partitioned into: (i) strongly relevant features, (ii) weakly relevant and indispensable features, (iii) weakly relevant and redundant features, and (iv) irrelevant ones. The authors find a feature space based on these feature groups, and then look for important \(\varvec{T}\) vectors which should be added to \(\varvec{T'}\). The aim of the feature selection algorithm is to retrieve the minimum number of attributes which characterize the input data as good as all attributes, thus it incrementally increases the subset of attributes until the dependence is not boosted. FARNeM proceeds with the analysis of training set vectors to distinguish between SV candidates (those vectors positioned in the boundary region are probable SVs), useless vectors and noisy ones based on the neighborhood rough set model. The authors use two important thresholds which affect the performance of FARNeM—they should be tuned with care since their improper selection can quite easily jeopardize the algorithm performance.
3.3 Evolutionary methods
Although evolutionary algorithms (EAs) have been shown very effective in solving a wide range of pattern recognition and optimization tasks (Pietruszkiewicz and Imada 2013; Li et al. 2007; Wrona and Pawełczyk 2013; Acampora et al. 2015; Nalepa et al. 2015a), they have not been extensively explored to select refined SVM training sets so far (Kawulok 2007). Nishida and Kurita (2008) proposed a hybrid algorithm (RANSAC–SVM) which couples random sampling, consensus approach (Fischler and Bolles 1981) and a simple evolutionary technique to retrieve \(\varvec{T'}\)’s. In their approach, several refined training sets of a small size are randomly drawn at first. Then—based on the classification scores of SVMs learned using the corresponding refined sets—the best \(\varvec{T'}\) is determined (by means of the best consensus). Additionally, the authors employed a simple GA with a multipoint crossover to further improve the refined sets (pairs of these refined sets are crossed over to form child solutions which inherit random training set vectors from both parents). The entire procedure (including random selection of SVM training sets and their evolution) is repeated multiple times, hence numerous potentially uncorrelated populations are processed.
In the genetic algorithm (GASVM) proposed by Kawulok and Nalepa (2012), a population of individuals (chromosomes), representing refined training sets of a given size, evolves in time. This evolution encompasses standard genetic operators—selection, crossover, and mutation. The fitness of each individual is the area under receiver operating characteristic curve (or the classification accuracy) retrieved for \(\varvec{T}\). Although this algorithm appeared very effective, and outperformed random sampling techniques, it was unclear how to select the size of individuals (which could not be changed later). This issue was tackled in the adaptive genetic algorithm (AGA) suggested by the same authors (Nalepa and Kawulok 2014a)—the size of individuals, along with the population size and the selection scheme, have been adapted on the fly to respond to the evolution progress as best as possible. This adaptation was steered by the parameters set a priori. Hence, improperly tuned parameter values could easily jeopardize the search (e.g., exploiting smaller refined sets and exploring larger ones could have been not balanced). The dynamically adaptive genetic algorithm (DAGA) (Kawulok and Nalepa 2014a) introduced the adaptation scheme which can be updated during the evolution, based on the characteristics of best individuals (i.e., the expected ratio of SVs within the refined sets). The expected ratio has to be determined beforehand, which is nontrivial.
Memetic algorithms (MA) combine EAs with refinement procedures to boost the solutions already found. They can exploit the knowledge attained during the evolution or extracted beforehand. Such techniques have been shown extremely effective in solving numerous challenging problems (Nalepa and Blocho 2016). Nalepa and Kawulok (2014b) proposed the first MA (termed MASVM) for selecting refined SVM training sets. The pool of important vectors (which were selected as SVs during the evolution) is maintained and used to educate the population, and to introduce super individuals—refined sets composed of SVs only. Hence, the knowledge gained dynamically is exploited in MASVM. This algorithm was utilized in the parameterless SVMs proposed by Nalepa et al. (2015b). In the abovementioned algorithms, initial populations were sampled randomly from \(\varvec{T}\).
Other works on EAs for this task have been reported recently. Fernandes et al. (2015) applied a multiobjective evolutionary technique in order to evolve balanced refined training sets extracted from imbalanced datasets. The objectives were to elaborate diverse and wellperforming classifiers, and to combine them into the classifier ensemble. The experiments performed for several benchmark sets showed that the evolutionary approach is able to outperform other stateoftheart techniques for dealing with large and imbalanced datasets.
Pighetti et al. (2015) enhanced a genetic evolution with the locality sensitive hashing (to find the nearest vector in \(\varvec{T}\) for any generated vector during the optimization) (Gorisse et al. 2010), and used it for tackling multiclass classification problems (oneversusall strategy was exploited). Although the approach is promising, it is unclear when to stop the optimization for multiclass tasks (the authors terminated the evolution once 60 vectors from each category have been retrieved).
Verbiest et al. (2016) recently investigated the performance of different evolutionary techniques for selection of SVM training sets: (i) a standard genetic algorithm, (ii) the adaptive genetic algorithm, which dynamically updates the crossover threshold [only notably different parents can be crossed over (Eshelman 1991)], and (iii) the steady state genetic algorithm [two parents are selected to generate offspring (Cano et al. 2003)]. Interestingly, the fitness involved not only the classification accuracy of the SVM classifier, but also the reduction ratio, indicating how much the input \(\varvec{T}\) has been shrunk. These wrapper techniques were initially used for the kNN classification, and the extensive experimental study clearly proved that they can be easily tailored for SVMs as well.
In their recent paper, Kawulok and Nalepa (2015) showed that evolving both training vectors and labels can be effectively used to handle learning SVMs from weaklylabeled training sets. In their memetic approach, the best individual in a population is an expert, and it is used in the tuition operation. The training set is relabeled if necessary, and the other individuals are refined (the vectors which changed the label during the tuition are replaced). Albeit the algorithm performed very well for mislabeled datasets, its performance deteriorated for correctly labeled \(\varvec{T}\)’s—this issue requires further investigation.
An interesting alternating genetic algorithm (abbreviated as ALGA) for optimizing the SVM model alongside SVM training sets has been proposed by Kawulok et al. (2017). The authors observed that different SVM models (i.e., kernel functions and their hyperparameter values) may be optimal for different training sets. In ALGA, two independent populations (one representing refined training sets, and the other the SVM models) are alternately evolved to solve two optimization problems having a common fitness function (classification accuracy over the validation set obtained using an SVM trained with the best refined training set and kernel function). The alternating process continues as long as at least one of these two subsequent optimization phases manages to improve the average population fitness. The experiments performed for both artificially generated and benchmark datasets revealed that ALGA can effectively select an SVM training set without the necessity to tune the SVM hyperparameters beforehand. Although the authors focused on the radialbasis function (RBF) kernel, this method can be easily tailored to any other kernel function. An interesting research direction would be to enhance ALGA with an additional step of selecting features for highdimensional datasets.
3.4 Active learning methods
In active learning models, vectors are initially not labeled, and the goal of an active learner is to infer a predictor of labels from the input data. It is accomplished in an interactive manner, in which the learner may request a label of a particular vector (this operation is associated with an appropriate cost). Hence, active learning may be interpreted as the process of obtaining labels for unlabeled data, and it can be applied for the entirely unlabeled datasets, as well as for those sets which encompass vectors with missing labels.
An active learning technique for selecting refined sets was proposed by Schohn and Cohn (2000)—they utilized a computationally efficient heuristics to label vectors lying near the SVM decision hyperplane. The authors exploit the selective sampling approach (being a form of active learning), in which learners are presented with a large unlabeled dataset, and are given the opportunity of labeling these vectors themselves (labeling of each vector “costs” some artificial fee). The learners attempt to minimize the error on the data which will appear in the system in the future. In the heuristic algorithm suggested by Schohn and Cohn (2000), one active learning criterion is to search for vectors which are orthogonal to the space spanned by the current refined training set. Additionally, the information about the already known data dimensions is boosted by narrowing the existing margin—only those vectors which are close to the decision hyperplane are effectively retrieved.
SVMs enhanced with active learning algorithms have been successfully applied in many reallife applications (Tong and Koller 2002). Tong and Chang (2001) exploited such techniques in their system to conduct effective relevance feedback^{3} for image retrieval, and proposed the poolbased active learning approach. A pool contains unlabeled \(\varvec{T}\) vectors which are analyzed and appended to \(\varvec{T'}\) if necessary. The classifier is trained using a labeled set (if it is the first feedback round, then the user is asked to label a number of randomly drawn vectors; otherwise, the user labels some pool images which are the closest to the decision boundary).
3.5 Random sampling methods
In random sampling techniques for selecting refined SVM training sets, the \(\varvec{T}\) vectors are drawn randomly, and—based on additional heuristics—are appended to \(\varvec{T'}\)’s or not. The simplicity of such methods makes them straightforward to implement and becomes their biggest advantage in practical scenarios. Also, they appear sufficient in a number of reallife circumstances (when the size of the desired refined sets can be estimated), and they are not dependent on the cardinality of \(\varvec{T}\). However, they can easily misbehave for very large and noisy datasets, since removing mislabeled vectors from \(\varvec{T'}\)’s (affecting the SVM performance) is often quite timeconsuming (Nalepa and Kawulok 2016a).
A simple approach to reduce \(t\) is to sample \(t'\) vectors from \(\varvec{T}\) randomly (Balcázar et al. 2001). In this sampling algorithm, a random subset of \(\varvec{T}\) is drawn according to the weights assigned to the training set vectors^{4} (the higher the weight, the larger the probability of including the corresponding vector in \(\varvec{T'}\)). Then, an SVM classifier is trained using this subset, and \(\varvec{T}\) is analyzed to verify which vectors were correctly classified using the resulting decision hyperplane. The weights of those vectors which were misclassified are doubled so that they are more likely to be selected and included in \(\varvec{T'}\) in the next sampling round. If the number of rounds is sufficiently large, then the important vectors (hopefully including SVs) will have higher weights than the other vectors, and a refined set will be composed of these SVs. The “optimal” size of \(\varvec{T'}\) is not known beforehand, thus the number of sampled vectors should be determined carefully (usually in a timeconsuming trialanderror fashion). This becomes a significant drawback of this algorithm especially in the case of massively large datasets. Also, random sampling approaches may ignore important (and useful) relations which occur within the dataset—if these training set features were exploited during the execution, the convergence time of such techniques could be greatly reduced (e.g., only the vectors lying near the boundary of oneclass vector groups could be sampled because they would likely influence the position of the SVM hyperplane).
3.6 Summary of the SVM training set reduction methods

Type—indicates whether the method is onepass or iterative. In the iterative approaches, the initial refined training set is gradually improved in order to include better vectors from \(\varvec{T}\). Such methods encompass algorithms which (i) keep enhancing the refined sets of a given (constant) size, and those which (ii) decrease or (iii) increase refined sets to boost their quality.

Source—being the underlying source of information concerning a training set. The knowledge extracted from this source is then used for generating refined sets during the optimization process. We distinguish five possible sources of information which can be utilized for this purpose—they are summarized in Table 1. Note that there exist methods which exploit several sources of information.

Randomized—shows whether the algorithm is randomized or deterministic.

Dependent on \(t\)—shows whether the algorithm is dependent on the cardinality of \(\varvec{T}\). If so, it may require analyzing the entire training set which is often not possible in massively large reallife datasets. Hence, the techniques which are independent from \(t\) should be preferred in practical applications.

Data—indicates which types of datasets were used to validate the corresponding algorithm (A—artificially generated, B—benchmark, R—reallife datasets).

Maximum \(t\)—indicates (roughly) the maximum size of the dataset for which the method was tested in the referenced paper. As mentioned in Sect. 1, the term large dataset is quite ambiguous in the literature (the cardinality of large sets may vary from hundreds to millions of \(\varvec{T}\) vectors).
Sources of information used for generating SVM refined training sets
Source  Description 

Local  The local neighborhood of a training vector is analyzed, and based on the outcome of this analysis, this vector may be classified as important (i.e., likely to be selected as a SV), or useless. The latter vectors are usually not appended to the refined sets 
Global  The global layout and characteristics of a training set are investigated in search of important vectors which should be included in the refined set 
Wrapper  In this approach, the SVM classifier is trained using a retrieved refined set (or sets, in the case of populationbased algorithms), and the quality of this refined set is quantified using the classification performance of the learned SVM—see Sect. 3.7 for details (commonly, all vectors from \(\varvec{T}\), or a subset of \(\varvec{T}\) vectors are classified to verify the SVM performance—in the latter case, wrapper techniques may be independent from the cardinality of the entire training set). The classifier is often treated as a “black box” in these wrapper techniques 
SVM  As in wrapper techniques, the SVM classifier is learned using the refined training set at first. However, the \(\varvec{T}\) vectors are assessed based on the specific features of the trained SVM (e.g., the vectors selected as SVs in the training process are considered important) 
Theory  The theory behind the SVM classifier or the desired training data characteristics are used to estimate the importance of a given training vector. Contrary to the “SVM” information source, the classifier is not trained before the training set vectors can be assessed in this case 
3.7 Assessing SVM training set selection algorithms
Assessing the quality of emerging SVM training set selection algorithms is a difficult and multifold task. These techniques can be investigated both quantitatively and qualitatively (e.g., by visualizing the extracted refined sets together with SVs and verifying whether they form any specific geometrical patterns). In this section, we discuss the quantitative measures which are used to assess new and existing training set selection algorithms alongside the standard experimental setup and datasets (together with their characteristics) that are usually adopted in the experiments. Finally, we present several practical applications in which various algorithms for selecting refined SVM training sets have been utilized.
3.7.1 Quantitative measures

Classification performance of an SVM trained using a refined set (\(\uparrow \)) The performance of classifiers (including SVMs) is assessed based on ratios derived from the number of (a) correctly classified positiveclass vectors—true positives (TP), (b) correctly classified negativeclass vectors—true negatives (TN), (c) incorrectly classified negativeclass vectors—false positives (FP), and (d) incorrectly classified positiveclass vectors—false negatives (FN) obtained for a test set which was not used during the training (see Table 2). Utilizing the unseen dataset allows for verifying the generalization capabilities of a classifier.
The derived ratios include, among others, the true positive rate:and the false positive rate:$$\begin{aligned} \mathrm{TPR}=\frac{\mathrm{TP}}{\mathrm{TP+FN}} \end{aligned}$$(45)TPR and FPR are often presented in a form of receiver operating characteristic (ROC) curves (Fawcett 2006). Each point in this curve is the performance of an SVM for a given decision threshold (Yu et al. 2015). Calculating the area under this curve (AUC) reduces a ROC curve into a single scalar value representing the classifier performance (the higher the AUC values, the better, and \(0\le \mathrm{AUC} \le 1\)). The area under the ROC curve and the accuracy (ACC):$$\begin{aligned} \mathrm{FPR}=\frac{\mathrm{FP}}{\mathrm{FP+TN}.} \end{aligned}$$(46)are the most widely used measures exploited to quantify the performance of training set selection algorithms (the classification performance of an SVM trained using a refined set should be maximized). Other common measures include precision, recall and the Fmeasure (Khosravani et al. 2013).$$\begin{aligned} \mathrm{ACC}=\frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{TP}+\mathrm{TN}+\mathrm{FP}+\mathrm{FN}} \end{aligned}$$(47)  Size of the refined training set (\(\downarrow \)) The main objective of training set selection algorithms is to minimize the cardinality of the training set (ideally without decaying the SVM classification performance). Hence, the number of vectors in the refined sets elaborated using such approaches is almost always investigated. To make this measure easier to interpret for datasets of different sizes, it is very often presented as the reduction rate (\(\mathcal {R}\)):where \(t'\) is the cardinality of the refined training set, and \(t\) is the size of the original dataset. This reduction rate should be maximized.$$\begin{aligned} \mathcal {R}=\frac{t}{t'}, \end{aligned}$$(48)

The number of support vectors (\(\downarrow \)) As already mentioned, the number of SVs influences (linearly) the SVM classification time. Therefore, it should be minimized to speed up the operation of a trained classifier.

The percentage of vectors in a refined training set selected as support vectors (\(\uparrow \) Determining the desired cardinality of refined sets is often a critical step in training set selection algorithms. Such refined sets should be small and should include important vectors which are likely to be selected as SVs during the SVM training. In several works, the percentage of vectors in refined training sets selected as SVs has been investigated (Nalepa and Kawulok 2016a; Verbiest et al. 2016). This percentage should be maximized to keep the number of “useless” vectors in a refined set as small as possible. However, this measure can easily become misleading—selecting all training set vectors as SVs can be a sign of overfitting and lack of generalization capabilities.

Training set selection, SVM training and classification times (\(\downarrow \)) In all stateoftheart approaches, the execution time of a training set selection algorithm should be minimized. Also, the SVM training and classification times are to be minimized (these times are correlated with the size of a refined training set and the number of determined SVs).
 Combined quality measure (\(\uparrow \)) Although the abovementioned measures are usually investigated separately, this approach becomes infeasible in several practical scenarios, e.g., in realtime systems in which a trained SVM should work extremely fast even if it delivers slightly worse results (i.e., minimizing the number of SVs may be more important than maximizing the classification accuracy). In such cases, the problem of selecting SVM training sets can be considered as a two (or multi) objective optimization problem: the first objective is to maximize the classification accuracy of an SVM trained with a refined set, and the second is to minimize the number of SVs. Nalepa (2016) transformed these two objectives into a single quality functionwhere \(\mathrm{AUC}^\mathrm{B}\) denotes the best (the largest) AUC obtained for the test set (note that AUC can be replaced by any other classification performance measure in this formula), \(s^\mathrm{B}\) is the best (the lowest) number of determined SVs across the investigated training set selection algorithms, and q denotes the importance of the first objective (\(0 < q \le 1\)). The largest Q value is retrieved for the best training set selection algorithm.$$\begin{aligned} Q(\mathrm{AUC},s)=q \cdot \frac{\mathrm{AUC}}{\mathrm{AUC}^\mathrm{B}} + (1  q) \cdot \frac{s^\mathrm{B}}{s}, \end{aligned}$$(49)
Predicted versus real conditions—bold text shows the erroneous classification
Predicted  

Positive  Negative  
Real  Positive  True positive  False negative 
Negative  False positive  True negative 
Summary of the stateoftheart methods for reducing the cardinality of SVM training sets
Algorithm  Year  References  Type  Source  Rand.?  Dep. \(t\)?  Data  Max. \(t\) 

Data geometry analysis methods  
Clusteringbased methods  
Clustering and 1nearest neighbor cluster analysis  1999  Lyhyaoui et al. (1999)  Onepass  Global  No  Yes  AR  500 
Analysis of kmeans clusters in the training set  2000  Barros de Almeida et al. (2000)  Onepass  Global  Yes  Yes  A  \(10^3\) 
Hierarchical microclustering  2003  Yu et al. (2003)  Iterative  Global, SVM  No  Yes  AR  \(5\times 10^6\) 
Clustering and processing of the crisp clusters  2004  Koggalage and Halgamuge (2004)  Iterative  Global  Yes  Yes  B  \(5\times 10^3\) 
Analysis of similarities between training vectors  2004  Wang and Xu (2004)  Onepass  Global  No  Yes  A  \(10^5\) 
Analysis of Hausdorff distances between vectors and convex hulls  2007  Wang et al. (2007)  Onepass  Global  No  Yes  B  615 
Minimum enclosing ball clustering  2008  Cervantes et al. (2008)  Onepass  Global, SVM  Yes  Yes  ABR  \(5\times 10^5\) 
Hierarchical clustering and mutual cluster analysis  2008  Wang and Shi (2008)  Onepass  Global  No  Yes  AB  \(3.7\times 10^3\) 
Smallest enclosing ball support vector machines  2008  Zeng et al. (2008)  Onepass  Global  No  Yes  B  \(5\times 10^5\) 
Edge detection and analysis of clusters  2009  Li et al. (2009)  Onepass  Local, global  Yes  Yes  ABR  \(5\times 10^3\) 
Analysis of convexconcave hulls  2012  Onepass  Global  No  Yes  AB  \(2.5\times 10^5\)  
Analysis of convex hulls  2013  Wang et al. (2013a)  Iterative  Global  Yes  Yes  ABR  \(1.6 \times 10^5\) 
Modified analysis of convex hulls  2013  Khosravani et al. (2013)  Iterative  Global  No  Yes  A  \(4 \times 10^3\) 
Analysis of cluster boundaries and cluster independencies  2016  Shen et al. (2016)  Iterative  Global, theory  Yes  Yes  ABR  \(5.7\times 10^5\) 
Nonclustering methods  
Extraction of boundary data using the Mahalanobis distance  2001  Abe and Inoue (2001)  Onepass  Global  No  Yes  B  800 
\(\beta \)skeleton analysis of the training set  2002  Zhang and King (2002)  Onepass  Global  No  Yes  B  \(4.4\times 10^3\) 
Nearest neighbor condensation  2010  Angiulli and Astorino (2010)  Iterative  Global  No  Yes  ABR  \(10^6\) 
Neighborhood analysis methods  
Pattern selection using knearest neighbors  2002  Shin and Cho (2002)  Onepass  Local  No  Yes  A  600 
Analysis of the vector’s neighborhood heterogeneity  2003  Shin and Cho (2003)  Onepass  Local  Yes  Yes  AB  \(1.3 \times 10^4\) 
Spherebased neighborhood analysis of training vectors  2005  Wang et al. (2005)  Onepass  Local  No  Yes  B  547 
Analysis of the vector’s neighborhood properties  2007  Shin and Cho (2007)  Iterative  Local  Yes  Yes  ABR  \(1.2 \times 10^4\) 
Analysis of ensemble margins of training vectors  2010  Guo et al. (2010)  Onepass  Theory  No  Yes  AB  \(2 \times 10^3\) 
Neighborhoodbased rough set model  2011  He et al. (2011)  Onepass  Local  No  Yes  AB  \(5\times 10^3\) 
Extreme vectors selection using the data boundaries  2011  Li (2011)  Onepass  Local  No  Yes  AB  \(1.4\times 10^5\) 
Borderedge pattern selection  2011  Li and Maguire (2011)  Onepass  Local  No  Yes  AB  \(5 \times 10^4\) 
Improved analysis of ensemble margins of each vector  2015  Guo and Boukir (2015)  Onepass  Theory  No  Yes  AB  \(10^4\) 
Decision trees  2015  Cervantes et al. (2015)  Onepass  Local, SVM  Yes  No  AB  \(10^6\) 
Evolutionary methods  
Initial work on genetic algorithms to select \(\varvec{T'}\)’s  2007  Kawulok (2007)  Iterative  Wrapper  Yes  No  R  \(2.9\times 10^5\) 
Random sampling enhanced with crossover  2008  Nishida and Kurita (2008)  Iterative  Wrapper  Yes  No  AB  \(6 \times 10^4\) 
Genetic algorithm (GASVM)  2012  Kawulok and Nalepa (2012)  Iterative  Wrapper  Yes  No  AR  \(7\times 10^6\) 
Adaptive genetic algorithm (AGA)  2014  Nalepa and Kawulok (2014a)  Iterative  Wrapper  Yes  No  ABR  \(3\times 10^5\) 
Dynamically adaptive genetic algorithm (DAGA)  2014  Kawulok and Nalepa (2014a)  Iterative  Wrapper, SVM  Yes  No  ABR  \(9\cdot 10^4\) 
Memetic algorithm (MASVM)  2014  Nalepa and Kawulok (2014b)  Iterative  Wrapper, SVM  Yes  No  ABR  \(4\times 10^6\) 
Multiobjective evolutionary sampling for imbalanced data  2015  Fernandes et al. (2015)  Iterative  Wrapper  Yes  No  B  1484 
Multiobjective evolutionary algorithm  2015  Pighetti et al. (2015)  Iterative  Wrapper  Yes  No  R  \(3 \times 10^4\) 
Memetic algorithm for evolving training sets and labels  2015  Kawulok and Nalepa (2015)  Iterative  Wrapper, SVM  Yes  No  ABR  \(4\times 10^6\) 
Parameterless SVMs  2015  Nalepa et al. (2015b)  Iterative  Wrapper, SVM  Yes  No  AB  \(4\times 10^6\) 
Evolutionary wrapper approaches for training set selection  2016  Verbiest et al. (2016)  Iterative  Wrapper  Yes  No  B  1728 
Adaptive memetic algorithm enhanced with geometry analysis (PCA\(^2\)MA)  2016  Nalepa and Kawulok (2016a)  Iterative  Wrapper, SVM  Yes  Yes  ABR  \(4\times 10^6\) 
An alternating genetic algorithm for selecting SVM model and training set  2017  Kawulok et al. (2017)  Iterative  Wrapper  Yes  No  AB  \(2.7\times 10^4\) 
Active learning methods  
Active learningbased heuristic algorithm  2000  Schohn and Cohn (2000)  Iterative  SVM  Yes  Yes  BR  \(6\times 10^3\) 
Poolbased active learning  2001  Tong and Chang (2001)  Iterative  Wrapper, theory  Yes  No  R  \(2 \times 10^3\) 
Guided active learning  2002  Tong and Koller (2002)  Iterative  Wrapper, theory  Yes  No  BR  \(3.3 \times 10^3\) 
Ensemble learning with active example selection  2011  Oh et al. (2011)  Iterative  Wrapper  Yes  No  R  768 
Random sampling methods  
Random sampling  2001  Balcázar et al. (2001)  Iterative  Wrapper  Yes  No  –  – 
3.7.2 Standard experimental setup

Sensitivity analysis The impact of the most important components of a new algorithm on its overall performance is verified in the sensitivity analysis. Usually, one (or more) components are enabled (the other components are disabled), and the experiments are repeated for each configuration.

Comparison with other training set selection algorithms and SVMs trained using the entire set The comparison with the state of the art is always crucial for new training set selection algorithms. Also, they are commonly compared qualitatively and quantitatively (using the measures discussed in the previous section) with SVMs trained using the entire set—without any training set selection applied (however, it may be impossible due to the cardinality of this set) and other techniques from the literature (very often from different categories).
3.7.3 Datasets and practical applications

Artificially generated datasets Vectors in artificial datasets are usually generated to follow a known distribution (e.g., the Gaussian distribution). Therefore, the underlying data characteristics are known (which is not always achievable in the case of benchmark and reallife sets). Additionally, artificially generated sets are often straightforward to visualize. Such datasets are used to understand the behavior of new training set selection algorithms (e.g., whether the vectors in the refined sets are positioned near the decision hyperplane or whether there are any vectors that could be removed from the refined sets as they are not selected as SVs). Several artificially generated datasets are available at http://sun.aei.polsl.pl/~jnalepa/SVM/ (see example datasets in Fig. 10—white and black pixels visualize vectors from the positive \(\varvec{T}_{+}\) and negative \(\varvec{T}_{}\) classes; training set vectors are grouped into clusters in the \(\alpha \) versions of these 2D sets).
 Benchmark datasets Such datasets (of different characteristics) are exploited to compare the performance of training set selection algorithms (benchmark sets were used in more than \(70\%\) of papers presented in this review). These datasets can be downloaded from the following repositories:In Table 4, we gather the characteristics of ten most frequently used (in the analyzed papers) benchmark sets alongside the repository name (the same dataset can be often downloaded from more than one repository). For multiclass sets (e.g., Yeast), pairwise coupling is performed—the multiclass classification problem is decomposed into twoclass problems and the majority voting principle is used (a number of binary SVMs vote for the final class label for an incoming vector). Although the sizes of these benchmark datasets are not very large, they are widely used in the literature to compare training set selection algorithms (also thanks to a welldefined experimental protocol which is often presented at a repository website—it makes the comparisons much easier).

UC Irvine (UCI) machine learning repositoryhttps://archive.ics.uci.edu/ml/index.php^{5}.

Knowledge Extraction based on Evolutionary Learning (KEEL) repository: http://www.keel.es/.

LibSVM repository: https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/.

 Practical applications and reallife datasets Although the amount of generated data is steadily growing nowadays and the size of training sets became a real obstacle in exploiting SVMs in practice, only less than \(45\%\) of all investigated training set selection algorithms were tested using reallife datasets.^{6} The most interesting practical scenarios in which training set selection algorithms have been tested and utilized include:

Handwritten digits classification In several works (Shin and Cho 2003, 2007; Nishida and Kurita 2008), the authors tackled the multiclass handwritten digits classification problem and exploited the MNIST dataset (http://yann.lecun.com/exdb/mnist/). They applied SVM training set selection algorithms to retrieve useful data from \(6\times 10^5\) training images (handwritten digits belonging to 10 classes, see Fig. 11). Important applications of the automated analysis of the digitalized handwritten text include bank check processing, postal address identification, analysis of historical documents or biometric authentication.

Skin detection and segmentation Detecting pixels representing human skin in color images (which is a preliminary step of the skin region segmentation process whose aim is to determine the boundaries of skin regions) is a difficult and important pattern recognition task. Its applications include content filtering, hand and face detection and tracking, humancomputer interaction and many more (Kawulok et al. 2014). Kawulok and Nalepa (2012) generated the Skin dataset of skin and nonskin pixels (in the YC\(_{b}\)C\(_{r}\) color space; \(4\times 10^6\) pixels in total)—they exploited images from the ECU face and skin detection database elaborated by Phung et al. (2005) (see example images in Fig. 12—note that skin pixels expose different color and intensity characteristics), and used this set to test their several SVM training set selection algorithms (Nalepa and Kawulok 2014a, b, 2016a; Kawulok and Nalepa 2014a; Nalepa 2016).

Hand pose estimation Kawulok and Nalepa (2014b) applied SVMs to recognize hand poses based on the shape context descriptors (Belongie et al. 2002). In their approach, vectors of differences between two hand shapes are classified to determine whether they represent the same pose (hence, the class decision is indirect). The authors showed that training sets can become very large even for a relatively small number of gestures (i.e., for n gestures, \(\frac{n!}{2\cdot (n2)!}\) feature vectors are obtained). To make SVMs applicable in this scenario, a genetic technique was utilized for selecting refined SVM training sets (Kawulok and Nalepa 2012).

Face detection Kawulok (2007) and Wang et al. (2013a) verified their SVM training set selection algorithms in the face detection problem—Wang et al. (2013a) exploited a dataset with almost 3500 images, whereas Kawulok (2007) used 1000 images from the famous Feret database presented by Phillips et al. (1998). Face detection is a pattern recognition task aimed at determining whether or not an input image contains a human face. Face detection algorithms are being exploited in surveillance systems, human–computer interaction and entertainment applications, human gait characterization, gender classification and many more (Paul and Haque 2013).

Detection of deceptive facial expressions Facial image analysis is an active topic—new research directions focus on facial dynamics recognition and understanding for deception detection, behavioral analysis and diagnosis of psychological disorders. Kawulok et al. (2016) used fast smile intensity detectors to elaborate textural facial features that are fed into the SVM classification pipeline to distinguish between posed and spontaneous expressions in video sequences from the UvANEMO database containing 1240 sequences, including 643 posed and 597 spontaneous smiles (Dibeklioğlu et al. 2012)—see examples in Fig. 13. Since these features are extracted for each frame (also those which are neutral, without any features exposing the smile characteristics), SVM training sets may become very large and often contain “useless” vectors. To deal with these issues, the authors utilized their memetic training set selection algorithm (Nalepa and Kawulok 2016a).

Image retrieval Tong and Chang (2001) showed that their SVM active learning training set selection algorithm can be successfully applied for image retrieval. It selects the most informative images to effectively query a user and quickly learn the decision hyperplane which should separate unlabeled \(\varvec{T}\) images to satisfy the user’s query. With the use of reallife datasets (encompassing up to 2000 images collected from the Internet), the authors proved their technique to be outperforming other stateoftheart image retrieval approaches. Such image retrieval techniques are commonly applied in the textiles industry, nuditydetection filtering engines, picture and art archives, and even medical diagnosis (Trojacanec et al. 2009).
 Biomedical applications Selecting appropriate training sets is an important problem in biomedical applications since the data quality and volume are big issues in this field. The following points summarize the most interesting biomedical applications in which SVM training set selection algorithms have been tested and utilized.

RNA classification SVMs have been successfully applied to detect noncoding RNAs (ncRNAs) in sequenced genomes (Uzilov et al. 2006). However, RNA datasets are very large which affects the SVM training. Cervantes et al. (2008) exploited their clusteringbased training set selection algorithm for two RNA datasets (the first one included almost \(5\times 10^5\) vectors with 8 features, and the second—\(2\times 10^3\) vectors with 84 features) and showed that is it quite competitive with the state of the art for such largescale data. Wang et al. (2013a) tested their training set selection algorithm on an interesting problem of deciding whether the incoming vector represents RNA of cod fish (the entire training set encompassed more than \(3\times 10^5\) vectors with 8 features).

Diseases classification (e.g., leukemia, diabetes, Parkinson’s disease, hepatitis) There are a bunch of approaches that have been tested on various disease classification tasks. In a standard medical image analysis scenario, the cardinality of a training set is not very large, but such datasets are highly imbalanced (usually, there are much more healthy examples compared with the pathological ones). Therefore, applying an appropriate approach for selecting desired training sets is inevitable. Oh et al. (2011) investigated their SVM training set selection using such imbalanced sets for various diseases (leukemia, diabetes, Parkinson’s disease, hepatitis, breast cancer and cardiac diseases). These datasets included up to 800 vectors (Diabetes dataset), and the number of features was up to almost 7200 in the Leukemia dataset.


Network intrusion detection Yu et al. (2003) focused on an important network intrusion detection problem. Its aim is to build a classifier which is able to distinguish between “bad” connections (intrusions and/or attacks) and normal connections. To test their approaches, the authors exploited a dataset containing a variety of intrusions simulated in a military network environment (42 features, \(4\times 10^6\) vectors). This dataset is available at http://kdd.ics.uci.edu/databases/kddcup99/kddcup99.html, and it was used as a benchmark at the Third International Knowledge Discovery and Data Mining Tools Competition, which was held in conjunction with The Fifth International Conference on Knowledge Discovery and Data Mining (KDD99). Interestingly, the test data does not follow the same probability distribution as the training data (it includes 14 specific attack types that are not present in \(\varvec{T}\) in which 24 training attack types are given). Yu et al. (2003) showed that their clusteringbased training set selection technique can easily outperform random sampling in this scenario.

Text classification Text classification, being the problem of determining to which topic a given text document belongs (it may be in one, multiple or no category because of the overlaps across these categories), is an important research topic which has been accelerated by the rapid growth of online information. Its applications include spam filtering, language identification, email routing, readability assessment and more. Schohn and Cohn (2000) and Tong and Koller (2002) tackled this problem to verify the capabilities of their activelearning SVM approaches. They exploited the Reuters21578 dataset (http://www.daviddlewis.com/resources/testcollections/reuters21578/) in the ModApte data split configuration (there are several predefined trainingtest splits provided by the authors of this dataset) with almost \(1.3\times 10^4\) articles (about \(10^4\) features each) and considered 10 most frequently occurring categories. Another commonly used textclassification dataset is Newsweeder (Lang 1995), also investigated in these papers.

Credit screening Lyhyaoui et al. (1999) tested their SVM training set selection using a dataset of 690 examples (15 features, 2 classes) reflecting customer creditworthiness. Although this dataset is known to be noisy (Quinlan 1999), the authors were able to surpass \(90\%\) of the classification accuracy with the use of their clusteringbased technique.

Summary of the most frequently used benchmark datasets
Dataset  \(t\)  #features  Types of features  #classes  Repository 

Adult  15, 082  14  Categorical, integer  2  UCI 
Breast  277  9  Categorical, integer, real  2  KEEL 
Bupa  345  6  Categorical, integer, real  2  KEEL 
German  1000  20  Categorical, integer  2  UCI 
Ionosphere  351  34  Integer, real  2  UCI 
Iris  150  4  Real  2  UCI 
Mushroom  8124  22  Categorical  2  UCI 
Sonar  208  60  Categorical, integer, real  2  KEEL 
Wisconsin  683  9  Integer  2  KEEL 
Yeast  1484  8  Real  10  UCI 
4 Conclusions and outlook
The amount of data produced every day grows tremendously in most reallife domains, including medical imaging, genomics, text categorization, computational biology, and many others. Although it appears beneficial at the first glance (more data could mean more possibilities of extracting and revealing useful underlying knowledge), handling massively large datasets became a challenging issue and attracts attention of researchers from multiple fields, especially in the era of big data. This big data revolution affected many research fields, including statistics, machine learning, parallel computing, and computer systems in general (Haykin et al. 2016). Albeit SVMs have proved extremely effective in solving a variety of pattern recognition tasks, their main drawback lies in huge time and memory complexities, depending on the training set size cardinality. This is a severe shortcoming (it may be even impossible to train the classifier using a dataset encompassing a very large number of vectors), and it may prevent users from using SVMs in reallife scenarios which often require processing massively large datasets. Finally, the classification time grows linearly with the number of SVs, which indirectly depends on the training set size (as already mentioned, the number of SVs is notably smaller for reduced sets, hence the classification is much faster).
In this review, we analyzed the current advances in selecting the SVM training data from large datasets. We divided all the methods into several classes, comprising algorithms utilizing similar approaches (e.g., for extracting information about SV candidates) in their core, as well as exposing similar characteristics. We believe that this taxonomy can be effectively used for emerging techniques, and will help highlight and understand their potential strengths and weaknesses. We presented the main sources of information concerning training set vectors, which are commonly used to assess the importance of these vectors (only important vectors should be assembled into refined sets, because they are likely to be selected as SVs). As presented in Sect. 3.6, the number of algorithms for selection of refined sets is quite large, but their underpinning strategies for extracting such information can be classified into just five categories. Although some methods combine different information sources (see Table 1), they are in the minority, and this approach has not been intensively investigated in the literature so far.
Training SVMs from large datasets remains an open research problem. A plethora of methods for tackling the SVM training from such datasets are an excellent point of departure for further research. We believe that emerging metaheuristics (especially populationbased ones), combined with refinement procedures should be intensively investigated towards parameterless SVMs. Such engines would be extremely useful, since determining appropriate parameter values of an algorithm at hand is very timeconsuming for massive sets, especially if the trialanderror approach is exploited. It will be beneficial to construct hybrid algorithms, which couple methods for selecting refined training sets, and for enhancing the SVM training. It has not been explored in the literature—we believe that it could become an immediate answer to some of the big data problems, where the data veracity, velocity, volume, and variety play the pivotal role and should be treated comprehensively.
An important research direction encompasses creating algorithms, which utilize various information sources in search of important training set vectors. We believe that such techniques (ideally independent from the cardinality of \(\varvec{T}\)) will be the main stream of development soon, since they allow for extracting various bits of information about the dataset, and for combining them into the solid knowledge about the \(\varvec{T}\) vectors. On the other hand, incorporating those methods which benefit from the same source of information into hybrid approaches will most likely not result in boosting the quality of the refined sets. Due to the wide availability of a variety of parallel architectures, it will be beneficial to develop algorithms which analyze datasets in the complementary ways in parallel. Then, the results could be merged in the final decision engine, used for assessing the \(\varvec{T}\) vectors. Finally, algorithms which target learning SVMs from imbalanced and weaklylabeled datasets are becoming crucial due to the nature of the available data (Sáez et al. 2016).
Finally, the research summarized in this survey needs to be confronted with deep learning—a very powerful classification tool for a variety of pattern recognition tasks (LeCun et al. 2016). However, it has also been criticized for being difficult to tune and easy to fool, domainagnostic, and hard to interpret (Nguyen et al. 2015). A very interesting research direction includes coupling deep convolutional neural networks (CNNs) with SVMs (alongside training set selection algorithms) in a comprehensive classification engine. Convolutional layers of CNNs are in fact feature extractors—features automatically elaborated in such layers could be classified using SVMs. This would allow for omitting a tedious process of preparing handengineered features (which is particularly important in the case of image and video data).
Footnotes
 1.
In most applications, a bag is labeled positive if it contains at least one positiveclass vector—it is negative otherwise. Therefore, the implicit labels of all negativeclass vectors belonging to a positiveclass bag are in fact incorrect (Li et al. 2013).
 2.
It was shown that the choice of the clustering technique does not influence the next \(\varvec{T'}\) selection steps significantly (Lyhyaoui et al. 1999).
 3.
This process interactively extracts the desired content for a user based on the user feedback—the user decides whether the presented data are relevant or not.
 4.
All \(\varvec{T}\) vectors have the same weight at the beginning of the algorithm execution.
 5.
All repositories and datasets discussed in this section were accessed on July 7\(\mathrm{th}\), 2017.
 6.
Numerous benchmark datasets are derived for practical problems, however their cardinalities are often much smaller compared with reallife scenarios.
Notes
Acknowledgements
JN and MK were supported by the National Science Centre, Poland, under Research Grant No. DEC2017/25/B/ST6/00474, and JN was supported by the Silesian University of Technology under the Grant for young researchers (BKM509/RAu2/2017). The authors are grateful to the anonymous Reviewers for their constructive and valuable comments that helped improve the paper.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
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