Retracted: E-Learning Recommender Systems Based on Goal-Based Hybrid Filtering

This research work is based on the thesis contribution by proposing the goal-based hybrid filtering approach in e-learning recommender systems (eLearningRecSys). The proposed work has been used to analyze the personalized similarities between learner’s profile preferences collaboratively. The proposed work consists of two hybridizations: the first hybridization has been made with content-based filtering and collaborative features to overcome the new-learners zero-rated profile recommendations issue; the second hybridization has been done with collaborative filtering and k-neighborhood scheme features to improve the average-learner’s low-rated profile recommendations issue. Therefore, the proposed goal-based hybrid filtering approach that hybridized content-based filtering, collaborative filtering and k-neighborhood features simultaneously works on both types of learner’s profiles recommendation issues in e-learning environments. The experiments in the proposed work are done using the famous “MovieLens” dataset, while the evaluation of experimental results has been performed with mean of precision 83.44% and mean of recall 85.22%, respectively. t -test result shows the probability difference value of 0.29 between the proposed hybrid approach and the evaluated literature work. The results demonstrate the effectiveness of the proposed hybrid recommender systems in e-learning scenarios.

Introduction
The intensification of web increased the difficulties to find the relevant learning contents quickly and efficiently, but electronic learning or e-learning scenarios provide many benefits to learners. They have some drawbacks too, for example, big data mismanagement [1] by increasing the number of pages, which have not been considered as an effective and good data management strategy [2] for e-learning. The reason for discouraging this strategy is that the learner's spend a huge time in visiting every page for retrieving their required learning content, which increases the leaning content's retrieving time and cuts the learner's interest. However, recommender systems in e-learning know as e-learning recommender systems [3], offer more flexibility for learner's to decrease the learning content searching time, increase the learner's interest, and provide the recommendations relevant to learner's goals or interests [4,5]. Recommendation or recommender system (RS) is a branch of information retrieval; gradually it filters the learning content in three ways, namely, content-based filtering (CBF), collaborative filtering (CF), and hybrid filtering (HF) [6]. In content-based filtering (CBF), it recommends only relevant learning contents to learners that are similar to the ones they preferred themselves in the past [7], while in the collaborative filtering (CF), the learners recommend relevant learning contents that other learners with similar interest and preferences liked in the past [8]. The hybrid filtering (HF) is a third way to tackle the filtering results [9] that plays a controversial role to tackle the learner's required goals. Gradually, the hybrid filtering hybridized the features of content-based filtering and collaborative filtering. Somehow, the encouraged researchers combine some adopted machine learning techniques or approaches [7] with content-based filtering or collaborative filtering to emerge the artificial intelligent (AI) aspects in it. These adaptations construct 2 International Journal of Distributed Sensor Networks the intelligent hybrid approaches that drive the way of knowledge and management in an automotive intellectual era; perhaps they depend upon the controversial issues and domain requirements.

Problem Background
The major challenge in e-learning that affects the performance of recommender systems is learner-based cold-start problem [8] or new-starter problem that moderated against the learners' profile preferences in all types of recommender systems [15]. This problem endorses the learner-based profile recommendations issue and encompasses in two conditions.

(a) New-Learner Zero-Rated Profile Recommendations Issue.
When the learner is new in the system, the system is unable to extract sufficient information from the learner profile that is required for the starter recommendations [19,20]. In other words, the recommender systems generally work with the learners' own historical or rated profile preferences. Therefore, the new learner in the system does not visit and rate any learning content, so the system does not acclaim the learners required goals and is unable to filter the starter or new recommendations.

(b) Average-Learners Low-Rated Profile Recommendations
Issue. The contemporary literature induced that the learners, which are not new in the system, are also facing the forecasting recommendation issues. In other words, the traditional e-learning recommender systems are unable to forecast the recommendations for low-rated learners [21,22]. Critically, it has occurred when the learner is not regular in the system or learner has not rated and visited much learning contents. In both of the cases, the system is not able to recommend the learners required learning content, which loses the learner's interest.

Research Contribution towards E-Learning Recommender
Systems. Salient aspects of our research work, which would contribute towards the recommender systems in e-learning, are as follows: (i) improve the traditional dataset driven method with -fold cross validation and normalization for eLearn-ingRecSys; (ii) drive a goal-based hybrid filtering approach that hybridized content-based filtering, collaborative filtering, and k-neighborhood scheme features to improve the content filtering feature of eLearningRec-Sys; (iii) overcome the new-learner zero-rated profile recommendation issue with the hybridization of contentbased filtering and collaborative feature; (iv) improve the average-learner low-rated profile recommendation issue with the hybridization of collaborative filtering and k-neighborhood scheme features; (v) the proposed work is easily implemented for all sort of web-based learning recommender systems and implies the simplest way to improve the problem background.

Research Flexibility.
The proposed goal-based hybrid approach interworks with learner's personalized profile preferences such as age, gender, and occupation, instead of metakeywords or historical informational aspects to overcome the learner-based cold-start problem that encompasses in two conditions: (i) overcome the new-learner zerorated profile recommendations issue, and (ii) improve the average-learner low-rated profile recommendations issue. In other words, the proposed hybrid approach decreases the e-learning recommender systems dependency on learner's historical or past preferences. The proposed hybrid approach may affect the learner's profile flexibility to make the e-learning recommender system (eLearningRecSys) more appropriate than the other traditional recommendation scenarios working in e-learning domain. In terms of the flexibility, the proposed approach has worked in all types of elearning scenarios; for example, some scenarios need long profile details and some need short. In that case, the proposed work just needs the three personalized profile preferences such as age, gender, and occupation to filter the recommendations for learners in the eLearningRecSys collaboratively.

Literature Background
The learning goals provide a sense of direction and purpose to guide the learner for next learning step and promote clarity to achieve learners required learning content accurately [23,24]. Arguably, the learning goals are only effective if learners use them to pinpoint and process goal-relevant information.
The abusement of learning goals will lay down the learner's interest and learner could not retrieve the learning contents as per their required goal or need. The importance of learning goals attract e-learning researchers to adopt this term in e-learning environment [25]. E-Learning enhanced the individual's problem-solving skills and enticed learner's to focus full attention on a task through the vividness of the learning concepts, intriguing or fascinating learning behavioral activities [26][27][28][29][30][31]. Some research studies define elearning as an effective way for enhancing the individual's problem-solving skills [6,32,33]. The studies reported that collaborative learning reflects the interaction with other learners in the formal knowledge-retrieving environment, such as recommender technology, to know their learning goals and support them to be involved in practical work placements [34]. Recommender technology is a part of information retrieval [35]. It has been popular in the 1990s when researchers used this technology the first time to overcome the information overloading or big data mismanagement issue [35]. The working criterion of this system is far from page-to-page knowledge explorer scenario, which is also useful to enhance the knowledge management efficiency and increase the learner's interests. It helps the target individuals navigate through a complex information retrieval by making suggestions of which the bit of information that target individual should consume, that is, read, watch, learn, and so forth. This recommendations are based on numeric form of data such as the like, dislike or ratings of learning International Journal of Distributed Sensor Networks 3 content by the learners are operated numerically from 1 to 5 [36]. Such systems efficiently deal with any big length of data, evaluate the learner's interests, and automatically generate relevant materialistic learning content suggestions or recommendations [19].
The emerging term of recommender systems in elearning is known as e-learning recommender systems [3]. In e-learning, the recommender systems work as instructor to acclaim the learner's goals, increase their interests, reduce boredom, and promote clarity to achieve learning requirements accurately [2]. Generally, "a goal in e-learning recommender systems is to specify the objectives that a learner may have when learners consult learning web services [37]. " Goal, in e-learning recommender systems, is an identification of requirements and achievements of relevant learning content that are required by the learner. To do so, recommender systems are implied on three famous filtering approaches that help to tackle the learning content most relevant to learner's goals. These approaches are content-based filtering, collaborative filtering, and hybrid filtering. Every approach has its great benefits and little drawbacks [8], that is why most of researchers like to work with hybrid filtering approaches in which they peer different filtering approach with respect to domain and relevant issue [7][8][9][38][39][40]. Recommendation or recommender system (RS) is a branch of information retrieval; gradually it filters the learning content in three ways, namely, content-based filtering (CBF), collaborative filtering (CF), and hybrid filtering (HF) [6]. In content-based filtering (CBF), it recommends only relevant learning contents to learners who are similar to the ones they preferred themselvs in the past [1,7,13]. Typically, this technique has less effectiveness on zero-rated cold-start problem [8] which is one of the major weaknesses of this approach. In these systems the recognition of learner's interest in a particular learning content has been derived on the base of vector similarities such as cosine similarity measurement [16]. By using this type of filtering approach, the recommender systems fully rely on the learners and learning contents profile past preferences; pendora.com is an example of it.
In the collaborative filtering (CF), the learners recommend relevant learning contents that other learners with similar interest and preferences liked in the past [7,8]. This approach depends on the collaborative learner's profile preferences more than learner's own profile and past preferences and tackles the learner's interest by using weight, distance, or correlation between two learners such as euclidean distance. The filtration process works collaboratively or on multiusers network with similar interests. Progressively, this technique works fine to overcome the zero-rated cold-start problem [16] in some domains, but this technique has less effectiveness on low-rated cold-start problem which is one of the major weaknesses of this approach. Its recommendations are based on numeric data, means the learners likes, dislikes or ratings for learning content which calculated as 1 to 5 and the number of clicks per learning content collaboration, and so forth; newsweeder.com is an example of it.
Gradually, the hybrid filtering hybridized the features of content-based filtering and collaborative filtering [3]. Somehow, the encouraged researchers combine some adopted machine learning techniques or approaches [7] with contentbased filtering or collaborative filtering to emerge the artificial intelligent (AI) aspects in it [38,[41][42][43][44][45]. For example, [11] proposed user-oriented content-based recommender system (UCB-RS) [11] with two stages. In the first stage, the author in [11] used fuzzy theoretic content-based filtering to generate the initial population of users' preferences by interactive genetic algorithm (IGA) using reclusive methods (RMs). In second stage, the author in [11] used k-mean algorithm for clustering the item in order to handle time complexity of interactive genetic algorithm (IGA). Usually, the traditional k-mean is unable to handle time computational complexity with genetic algorithm if "k" is small [46]. Mostly with large set of data, the traditional k-mean does not give sufficient performance.

Literature Gap Analysis.
With the deep analysis on the existing literature, this study's results show that the three approaches that are famous in the study domain are contentbased filtering CBF, collaborative filtering CF, and hybrid filtering HF, which are generally the combination of contentbased filtering and collaborative filtering. The study also sees that researches adopted machine-learning technique KNN to become hybrid. These hybridizations depend upon the domain requirement. The deep and comprehensive eye of this study tackles that most of researches have done their work to solve the issue called cold-start. Table 1 shows the analyzed results of this literature review.
From a critical and a deep analysis, Table 1 shows that the cold-start problem is one of a common problems in recommender systems. The issue cause that is described in Table 1 is divided into four different types of learner-based profile issues, in which causes are elaborated in Table 2. Table 2 defines the four issues and their causes in learnerbased cold-start problem. In the first issue, the new-learner or user cold-start occurs when the learner is new in the system and learner or user has zero past preferences. In the second issue, the new-item or content cold-start occurs when the item or content is new in the system and no learners have browsed it and rated or voted it before [7]. The third issue is low-rated user or learner cold-start; it occurs when the collaborative users or learners could not have high rated learning contents in the past preferences. And the fourth issue is low-rated item or content cold-start; it occurs when the item or content profile is unable to clarify the high rating by collaborative learners [38]. In these four issues, the domain systems are unable to suggest the recommendations [15].
In recommender system approaches, if the content-based systems could not find the required information in the learning contents profile [13] or new or other learners did not have any past learning preferences, so the approach is unable to filter any recommendation for the learners. This cause generated an issue called cold-start in content-based filtering systems [7,14]. The low-rated learner's profiles in cold-start are the main problem in collaborative filtering [38]. In both cases, the recommender system is unable to filter  [11] 2013 Cold-start New learner or user Shinde and Kulkarni [12] 2012 Cold-start Low-rated learner or user Souali et al. [13] 2011 Cold-start New-learner or user Ghani et al. [14] 2010 Cold-start New-learner or user Lécué [15] 2010 Cold-start Low-rated learner or user Ahn [16] 2008 Cold-start New-learner or user Wang et al. [17] 2008 Cold-start New-learner or user Adomavicius and Tuzhilin [7] 2005 Cold-start Low-rated learner or user Schein et al. [18] 2002 Cold-start New-learner or user Table 2: Two issues in learner-based cold-start problem and their causes.
New-learner or user cold-start Zero rating to the item or learning content or no past preferences 2 Low-rated learner or user cold-start No high rated item or learning contents in past preferences the recommendations [18]. Instead of this, hybrid filtering systems delightedly help to improve the learner low-rated profile cold-start issue but it still may have zero-rated profiles cold-start issue. Due to these aspects, the study identified that cold-start is an important limitation of recommender systems that need more intention and integration from future researchers.

A Brief Introduction of Research Methodology
Gradually, the research methodology has been envisioned to meet the research study objectives and serves to provide subsequent processes. Table 3 defines the objectives of each research methodological phase that includes each phase objective, its activities that are involved in this research work, its processes, and the possible outcomes.
The dataset "MovieLens" [21,48] generally comes with validated data, which are defined in Table 4. The density of the learners and learning contents matrix with respect to the ratings to the learning contents by the learners, created from the "MovieLens" dataset, is Density = total ratings ( , ) (total users | ∈ ) × (total items | ∈ ) . (1a) Equation (1a) is formulated as Here, (1b) shows the results of the percentage of 15.86 density in "MovieLens" dataset " " that can be considered as appropriate enough in terms of sparsity for the evaluation of the dataset. Perhaps we see that the general validated dataset is not quite sufficient for the implementation of the proposed work. Alternatively, as per this study requirement, we operate -fold cross-validation [10] method for revalidation of "MovieLens" dataset " ", which is shown in Figure 1.
In Figure 1, the individual dataset sample ( ∈ [ 1 , . . . , ]) was used to obtain the random generation of subdataset samples, where 1 , . . . , ( = ) are equal in size [10,58]. The normalized estimation of each subdataset, 1 , . . . , | ∈ rating of each learner against the learning content is the overall number of correct identification. The identifications have been helpful for the classification of the learners personalized profiles as zerorated, low-rated, and high-rated learners dataset class from individual "MovieLens" " " dataset sample. Each looping has been averaged and produced the actual normalization of the learners personalized profile dataset. Finally, the learners personalized profile classification is used to categorize the number of training and testing sets. The revalidated dataset key points are defined in Table 5.
In Table 5, the revalidated key points of "MovieLens" dataset " " are mentioned in detail. The study found only 10 learner's " ", which have the lowest ratings from 943 learner's " "; therefore, the study sees them as new-learner profiles. In the rest of 843 profiles, only 61 learner's " " profiles have high ratings " " and 872 have average ratings " ". Therefore, the system counts the 872 average ratings " " of learner's " " profiles as average-learner profiles and International Journal of Distributed Sensor Networks 5 Table 3: Overview of research activities involved and its possible outcomes.

Action
Activities Involved Possible Outcome

Phase-I: Dataset validation
Validation of dataset " ", and classifies it into subsequent training " " and testing " " sets for the further experimental process on problem domain.
(i) Categorize the zero-rated new-learners and low-rated average-learners profile.
(ii) Classify training-set training set and testing set for the zero-rated new-learner cold-start recommendation experimental process.
(i) Categorize the low-rated average-learners and high-rated super-learners profile.
(ii) Classify training-set training set and testing set for the low-rated average-learner cold-start recommendation experimental process.

Phase-II: Propose approach
Content-based filtering hybridized with collaborative features.
Overcome new-learner zero-rated profile recommendations issue. Collaborative filtering inherited with -neighborhood scheme features.
Overcome average-learner low-rated profile recommendations issue. the 61 high ratings " " of learner's " "profiles as superlearner profiles. The classification of zero-rated, low-rated and high-rated learner profiles have been done from database " ".

Phase-II: Proposed Approach.
In second phase, the research work accommodates proposed approach in two eras. First, it overcomes the new-learner zero-rated profile recommendations [19,20] with the hybridization of content-based filtering and collaborative features. Secondly, it improves the average-user low-rated profile recommendations [21,22] with the hybridization of collaborative filtering and kneighborhood scheme features. Figure 2 demonstrates the working flow of proposed approach, its working features and dimensions of learner's profile recommendations to encompass the learner-based cold-start problem in the above two eras simultaneously. The description of Figure 2 parameters is mentioned in Table 6. Figure 2 demonstrates the proposed approach recommendation model, its working features, and dimensions of learner's profile recommendations that encompass the learner-based cold-start problem in two ways simultaneously: the first is new-learner zero-rated profile recommendations and the second is average-learner low-rated profile recommendations. The description of Figure 2 parameters is mentioned in Table 6.
In Table 6, represents the training set and represents the testing sets from "MovieLens" dataset. For fulfilling the objectives of this proposed work, the proposed goalbased hybrid filtering approach works with (i) content-based filtering with collaborative features that helps to overcome the new-user profiling issue and show the usefulness of user-to-user personalized profile similarities; and (ii) collaborative filtering with k-neighborhood scheme features that helps to overcome the new-user profiling issue and show the usefulness of user-to-user personalized profile neighborhood similarities. In both of the above approaches, (a) and (b) performed their work simultaneously in the proposed goal-based hybrid approach to overcome the learnerbased cold-start recommendation challenges. The system attempt the learner's profile " " personalized preferences similarity with learner's profile " " and recommends the relevant learning content " " to the target learner. In the proposed approach, the system generates relevant learning contents to overcome two types of learner's cold-start profile recommendation issues, which are as follows: (a) new-learner zero-rated profile recommendations [19,20] and (b) average-learner low-rated profile recommendations [21,22].

Goal-Based Hybrid Filtering
This research proposed a goal-based hybrid filtering approach in e-learning recommender systems, defined as eLearningRecSys to provide a sense of learning direction and purpose to guide the learner for the next learning step. This research may promote the clarity to achieve required learning contents, enhancing the learning interest and personal satisfaction that help to build the learner interest. Usually, recommender systems in e-learning are based on the learner's profile extra information, such as past history, 6 International Journal of Distributed Sensor Networks f-fold cross validation [10] "MovieLens" dataset Accuracy Accuracy data set Figure 1: Process in -fold cross validation [10]. Table 5: "MovieLens" dataset D revalidated key points.
Goal-based hybrid filtering model The relevant Personalized similarity with The system recommends to Target learner (ii) Average-learner low-rated profile recommendations (i) New-learner zero-rated profile recommendations Learning contents "I" (a) T S1 T R2 T S2 Figure 2: Proposed approach recommendation model, its features, and dimensions. First training set of learner's "U'' profile with ratings ''R'' range (0 ≤ R ∈ r ≤ 60). 1 First testing set of learner's "U'' profile with ratings ''R'' range (60 < R ∈ r ≤ 1000). 2 Second training set of learner's "U'' profile with ratings ''R'' range (60 < R ∈ r ≤ 1000). 2 Second is testing set of learner's "U'' profile with ratings "R'' range (R ∈ r > 1000). (a) Hybridization of content-based filtering with collaborative.
Hybridization of collaborative filtering with -neighborhood features.

Related learning contents
Step 6 To new/low rated users Step 7 Personalized profiles similarities (age, gender, occupation)

Preferences Preferences
Personalized profiles similarities Step 4 Data processing

Categorization
Step 3 Zero-rated profiles

Classification
Step 2 Low-rated profiles High-rated profiles

Initialization
Learners profiles (U-ID, age, gender, occupation) Learning content profiles (I-ID, title, year, URL) Learners and contents ratings (U-ID, I-ID, rating, timestamp) Step 1

Similar profiles
Step 5 k-neighbors similar learners Learners (T R1 and T R2 ) Training sets (T R1 and T R2 ) Testing sets (T S1 and T S2 ) Learners (T S1 and T S2 ) profiles from T S1 and T S2 ratings, or likes and dislikes. Alternatively, the proposed hybrid approach does not require any extra information or historical data of the learner; it only requires the learner's personalized profile preferences, such as age, gender, and occupation that a recommender knows to produce the possible recommendation outcome for end learner in the focused domain of e-learning. The proposed hybrid filtering allows the system to simultaneously assist the learner of zerorated and low-rated profile recommendations and retrieve their required goals. Figure 3 demonstrates the framework of the proposed goal-based hybrid filtering approach.

5.1.
Step 1: Initialization. The first step is a part of data preprocessing; it has initialized the three dataset classes form "MovieLens" as learner and content ratings, learner's profiles, and learning content profiles where learner profiles class " " uses their ratings " " information for each learning content " ". The initialization of data is based on the users personalized profile preferences, for example, age, gender, occupation, and learning content for recommendation of the learner's rating information. The set of users, items, and ratings has been initialized as (2a), (2b), and (2c):

5.2.
Step 2: Classification. The classification of learners' class " " from "MovieLens" dataset has been tackled as per of their total ratings against each learning content. In this system, the recommendations are based on the k-neighbors similar  Figure 4: Classification of learners profiles class " " using their ratings " " information from "MovieLens" dataset " ".
Equation (3) is being used to analyze the new and other learners by their ratings against all learning contents. Here, defines the total learner rating, defines the total number of learner's, and defines total learning contents and range of rating from 1 to 5 to the learning contents . Perhaps, we could not find any zero-rated user profile in "MovieLens" dataset. Alternatively, with the help of (3), this research work classified the learner's profiles class into three types of learner's profiles classes described in Table 7. Table 7 shows the classified classes as zero-rated from 0 to 60, low-rated from 61 to 1000, and super-rated from 1000 to onward rating profiles. Figure 4 demonstrates the classification method of "MovieLens" dataset learner " " profiles class into 3 subsequent learner profiles dataset classes for further operations. The classified classes names are set as new-learner zero-rated profiles "NLP" class, average-learner low-rated profiles "ALP" class, and super-learner's high-rated profiles "SLP" class using the count of their total ratings " " information for learning contents " ". Figure 4 demonstrates the classification of "MovieLens" dataset " ". By referring to Figure 4, the classification of learner's class can be described as follows: (a) The New-Learner's Zero-Rated Profiles. The new-learner's profile classification has been concurred from the "Movie-Lens" dataset learner's " " class. To handle the zero-rated profile situation, this study makes the strategy to classifiy those profiles as new-learner, in which count of ratings is in range of 0 to 60 against all the learning contents. The study found only 10 learner profiles with less than 60 ratings " ∈ " history preferences. Therefore, the new-learner's zero-rated profiles "NLP" are settled in the range from (0 ≤ ∈ ≤ 60) that are shown in Figure 4. By using the new-learner zerorated strategy, which has been settled within the range of 0 to 60, the system automatically classifies the upcoming newlearner's as zero-rated category with the help of (4a) to go through the next process: The Average-Learner Low-Rated Profiles. In this, the classification of learner profile has been done on behalf of ratings of each learner as like new-learner profiles. Although, the scenario slightly changed with average-learner profiles, here the system collects those learners' profiles, which are not new in the system nor visit or rate " ∈ " much learning contents. Therefore, the learners have less or average level of rated profiles and system recorded those profiles as average-learner low-rated profiles "ALP. " The study examined that "MovieLens" dataset learners " " class has only 872 learner's profiles, in which count of rating has been higher than 60 but lower than 1000, as shown in Figure 5. Therefore, the strategy automatically settled the learner profiles, in which range of ratings is from (60 < ∈ ≤ 1000) and classifies them as average-learner low-rated profiles with the help of (4b) to go through the next process: (c) The Super-Learner High-Rated Profiles. The classification of super-learner profiles also has been made on behalf of ratings of each learner as like average-learner and new-learner profiles. The super-learners are so called super because they visit the learning content more than average learner. In the super-learner's scenario, the system examined "MovieLens" dataset and found that in the 61 learners' profiles the ratings range has been above 1000 as shown in Figure 5. In the high-rated learner's profiles, strategy settles the range from ( ∈ > 1000), so the system automatically classifies superlearner's with the help of (4c) as high-rated category and is represented as "SLP": International Journal of Distributed Sensor Networks 9 For new-learner zerorated profile recommendations For average-learner low-rated profile recommendations "NLP" (low-rated) "ALP" "ALP" "SLP" (high-rated) (zero-rated) (low-rated) 60  T S1 Figure 5: The categorization of "MovieLens" dataset in subsequent training and testing sets. First training set of new learner's profiles "NLP" with ratings"R" to item "I. " 0 ≤ R ∈ r ≤ 60 (3) (i) To overcome the new-learner zero-rated profile recommendations 1 First testing set of average learner's profiles "ALP" with ratings "R" to item "I'. ' 60 < R ∈ r ≤ 1000 (5) 2 Second training set of average learner's profiles "ALP" with ratings"R" to item "I'. ' 61 < R ∈ r ≤ 1000 (6) (ii) To improve the average-learner low-rated profile recommendations 2 Second testing set of super learner's profiles "SLP" with ratings "R" to item "I. " R ∈ r > 1000 (6)

5.3.
Step 3: Categorization. In this step, the proposed work categorizes the learner's profiles into the training set and testing set . These training and testing sets have been used to in the personalized profile similarities computation. Gradually, the categorization of dataset has been done as 50 percent of training and rest of dataset for testing [11] or as 70 percent of training and rest of dataset for testing [18] and vice versa. Alternatively, this research work classifies and uses the training set and testing set in a slighter different way, which are defined in Figure 5.
To handle the training and testing sets strategy, the zero-rated learner's profiles with ratings range from 0 ≤ ∈ ≤ 60 profiles have been set as first training set 1 and average-rated learner's profiles with ratings range from 60 < ∈ ≤ 1000 are as first testing set 1 which were discussed earlier. The first training set 1 and first testing set 1 are being used in the proposed hybrid approach to overcome the new-learner zero-rated profile recommendations issue. Afterwards, for the low-rated profile recommendation issue, 2 and 2 are being used in the proposed hybrid approach to improve the average-learner low-rate profile recommendations issue. With this being settled, the second training set 2 has been set with the average-learner's low-rated profiles with rating range from 60 < ∈ ≤ 1000 and the second testing set 2 is set with the super-learner's high-rated profiles with the ratings range as ∈ > 1000. The detailed description on new-learner, average-learner, and super-learner profiles with respect to their learning content ratings is presented. The definitions of these training and testing sets that are shown in Table 8.
This research study categorized the training and testing sets in a slighter different way from the previous literature [11,18]. This research study system categorizes two training sets, such as 1 and 2 , and two types of testing sets, such as 1 and 2 , for the further experiments. Methodologically, the first training set 1 and first testing set 1 have been used to overcome the new-learner zero-rated profile recommendations issue with the help of content-based filtering and collaborative features in the proposed hybrid filtering approach. Simultaneously, the second training set 2 and second testing set 2 have been used to improve the average-learner low-rated profile recommendations issue with the help of collaborative filtering and k-neighborhood scheme features in hybrid filtering approach.

Step 4: Personalized Profile Similarities.
After the categorization of dataset into training and testing sets in step 3, step 4 collects the personalized profile preferences, such as age, gender, and occupation with respect to the training sets 1 , 2 and testing sets 1 , 2 . These personalized profile preferences are tackled to perform the learner's personalized profile comparisons with each other. The profile comparisons help to compute the similarities between the learner's goals or interests in collaborative network with the hybridization of content-based filtering, collaborative filtering, and k-neighborhood features, which are described as follows.

(a) Content-Based Filtering with Collaborative Features.
In traditional content-based recommender systems, the recommendation is being predicted through the learner's own historical profile activities which means that the system produces recommendations using only those clusters which belong to the learner historical activities. Figure 6 shows the workflow of content-based filtering with collaborative features to overcome the new-learner zero-rated profile recommendations issue, and it shows the usefulness of collaborative personalized profile similarities scheme. Figure 6 shows the operational workflow of contentbased filtering with collaborative filtering features to overcome the new-learner zero-rated profile "NLP" recommendations issue. The conventional, content-based approach [13,43,59] recommends relevant items, if and only if the learners have their own background profile history of rated or visited learning contents. In the other way, the traditional contentbased approach worked on learner's own historical details or rating details . Alternatively, the proposed hybridization of content-based filtering with collaborative features worked in a slightly different way from the traditional content-based filtering approach. In this, foremost the total rating of each learner is being used to classify the new-learner's zero-rated profiles "NLP" and average-learner's low-rated profiles "ALP" for the further process. The personalized profile similarities between collaborative learner's profile preferences such as age, gender, and occupation in content-based filtering with collaborative features method have been measured to overcome the new-learner zero-rated profile recommendation. The following similarities are being measured by using cosine similarity method with the help of (5), as follows: With the help of (5), the personalized similarities have been measured between new-learner's zero-rated profiles "NLP" which are considered as first training set 1 and averagelearner's low-rated profiles "ALP" which are considered as first testing set 1 . Both of training 1 and testing 1 sets have worked as cosine vector and the similarities computational results are represented as Sim( 1 , 1 ). The similarity results are demonstrated further in this paper.

(b) Collaborative Filtering with k-Neighborhood Features.
Generally, in the collaborative filtering approaches [22,50,60], the system recommends those learning contents as relevant learning contents to the target learner's, which are highly rated or sequently visited by the other learner's. For example, if one learner highly rated or visited any learning content of category 1 and the second learner may highly rate or visit other learning contents of the same category 1, so the traditional collaborative system recommends the first learner rated learning content to second learner and the visited or rated learning content of second learner has been recommended to the first learner. This means that the traditional collaborative approaches did not work efficiently if the learners have low rated or less sequently visited learning contents or have minimum rated or visited historical profile. Alternatively, the collaborative filtering with kneighborhood scheme features foremost worked with total rating of per learner in a slightly different way. The proposed work used the total ratings of each learner to classify the average-learner low-rated profiles "ALP" and super-learner's high-rated profiles "SLP" to overcome the average-learner low-rated profile recommendations issue and show the usefulness of user-to-user personalized profile similarities using k-neighborhood scheme features. Figure 7 shows the workflow of famous machine learning k-neighborhood scheme features with collaborative filtering approach. Figure 7 shows the operational workflow of collaborative filtering with k-nearest neighbor's scheme features for the improvement of average-learner's low-rated profiles "ALP" recommendations issue. The personalized similarities have been computed between average-learner low-rated collaborative profiles and super-learner's high-rated collaborative profiles by using their personalized profile preferences such as age, gender, and occupation. The average-learner low-rated profiles "ALP" are considered as second training set 2 and super-learner's high-rated profiles "SLP" are considered as second testing set 2 for the personalized similarities measurement by using the euclidean cosine similarity method as follows: With the help of (6), the personalized similarities havee been measured between the average-learner's low-rated profiles "ALP" as first vector class which is represented as second training set 2 and super-learner's high-rated profiles "SLP" as second vector class which is represented as second testing set 2 by using their personalized profile preferences such as age, gender, and occupation. The similarities results of "ALP" vector class as training set 2 2 and "SLP" vector class as testing set 2 2 are represented as Sim( 2 , 2 ) and are demonstrated further in this paper.

5.5.
Step 5: Similar Profiles. As discussed earlier, the newlearner zero-rated profiles "NLP" are set to as first training set 1 and average-learner low-rated profiles "ALP" as first testing set ( 1 ) to overcome the new-learner zero-rated profile recommendations. Similarly, to improve the average-learner low-rated profile recommendations, the system categorized average-learner low-rated profiles "ALP" as second training set 2 and super-learner's high-rated profile "SLP" as second testing set 2 .
Step 5 collects the similar learner's profile results as Sim( 1 , 1 ) and Sim( 2 , 2 ) from step 4 for the next process. With respect to the similarities between both training and testing sets, this step chooses only kneighbors similar learner's profile results as Sim( 1 , 1 ) and Sim( 2 , 2 ) in which profiles personalized preferences such as age, gender, and occupation are very similar to other learner's profiles in the collaborative network. Afterwards, this step selects the similar k-neighborhood learner's profiles of 1 and 2 and forwards them to step 6 for further process.

5.6.
Step 6: Related Learning Contents. This step collects the similar k-neighborhood learner's profiles of 1 and 2 from step 5 and selects the learning contents which have been visited or rated " " in the past by the k-neighborhood learner's of 1 and 2 . The selected learning contents " " have been sent to step 7 for recommendations to the target learner's, denoted as ( 1 , 2 ) →̂.

Step 7: Recommendations.
Step 7 is the last step of operational framework of the proposed hybrid approach and plays its part of work between the system and the relevant learners. This step receives the recommendation learningcontents " " from step 6 and recommends them to the newlearner zero-rated profiles "NLP" and average-learner lowrated profiles "ALP. " The recommendations in both cases are denoted as "rm. "

Hybrid Results and Discussion
This section shows the proposed work experiments and results. With respect to the problem background, the experimental work is partitioned into two portions. The first is new-learner zero-rated profile recommendation experimental results and the second is average-learner low-rated profile recommendation experimental results, which are as follows.

New-Learner Zero-Rated Profile Recommendations.
As per reconsideration of (5), the experimental operations are based on the cosine vector-based similarity measurement. Experimental results are structured as per the availability of dataset. For new-learner's zero-rated profile "NLP" recommendations issue, this research work has used 1 as training set 1 and 1 as testing set 1 in the experimental setup of the proposed hybrid approach for this issue. Here, we considered 1 as training set 1 as and 1 as testing set 1 as . The similarity between the and has been used as [ × ] factorization matrix as 0 ≤ ∼ ≤ 1. Therefore, the is set to be as target learner's profile class; perhaps, is set as other learner's profile classes. This study conducts three similarity experiments using different lengths of training sets = ( 1 , 2 , . . . , ) and operates the personalized profile preferences such as age, gender, and occupation similarity measurement with the total range of testing set = (V 1 , V 2 , . . . , V ). The three experiments are discussed as follows.
(a) Experiment 1. In the first experiment of new-learner's profile recommendation issue, the system randomly selects 2 learner's profiles in which count of ratings against learning contents or items is between 0 and 60 as new-learner's = ( 1 , 2 ) as a training set 1 and operates the similarity with the total number of learner's = (V 1 , V 2 , . . . , V 872 ) profiles with ratings 61 to 1000 identifies as other or average learner's and considered as testing set 1 . Table 9(a) shows Table 9: (a) Similarity matrix of training-set 1 and testing-set 1 as = 2 × = 872 . (b) Similarity matrix of training-set 1 and testing-set 1 as = 5 × = 872 . (c) Similarity matrix of training-set 1 and testing-set 1 as = 10 × = 872 .
(a)  and set to be defined as testing set 1 . The results are discussed further in this paper. The similarities in Table 9 have been computed with parameters that work between 0.1 and 1.0 with the help of (7) as follows.
if there is perfect similarity between learner and learner V profiles ≥ 0.1 and < 0.9 if there is mature similarity between learner and learner V profiles 0 if there is no similarity between learner and learner V profiles.
(b) Experiment 2. In the second experiment, the system randomly selected 5 learner's profiles, which total learning contents rating is between 0 to 60 and considered them as a new-learner. The learners have total ratings above then 60 and less than 1000 are considered as average-learners. The range of new-learner class profiles = ( 1 , 2 , 3 , 4 , 5 ) are set to be as training set 1 and the range of average-learner class profiles = (V 1 , V 2 , V 3 , . . . , V 872 ) are set to be as testing set 1 Table 9(b) shows the resulting matrix of experiment 1, which is conducted with = ( 1 , 2 , 3 , 4 , 5 ) training and = (V 1 , V 2 , . . . , V 872 ) testing set. Table 9(b) represents the similarity results of [ 5 × 872 ] learner's profile similarity matrix. The learner's class " " is set to be as training set 1 are helpful to identify the high, average and less similar learner's profiles. The parameters have been defined with the help of (9). Each score of the similarity matrix contains a measure of similarity between the elements of 1 and 1 learner's profiles. The similarities in Table 9(b) have been computed with the parameters that work from higher score 1.0 to lower score 0.1. The higher scores are given to accurate similar learner's profiles; lower scores are given to less similar learner's profiles and in between the score are given to average similar learner's profiles. The parameters have been defined with the help of (9) as follows: (c) Experiment3. In the third experiment, the system select all the new-learners profiles, which total learning contents rating is between 0 to 60 and the range of new-learner's class = ( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ) is set International Journal of Distributed Sensor Networks 13 to be as training set 1 as 1 . The average-learners with total ratings above then 60 and less than 1000 with the range of class = (V 1 , V 2 , . . . , V 872 ) is set to be as testing set 1 as 1 . Table 9(c) represents the similarity results of [ × ] learner's profile similarity matrix. The results of similarity matrix have been computed with the parameters from higher 1.0 to lower 0.1 score. The higher scores are given to accurate the learner's profiles similarity; lower scores shows less similarity of learner's profiles and in between the score are given to average similar learner's profiles of [ × ]. The parameters have been defined with the help of (9).

Average-Learner Low-Rated Profile Recommendations.
In this section, the experimental result has been detailed with the help of (6); the experimental operations are based on the Euclidean distance similarity measurement. The similarity matrix factorization has been adjusted as [ × ]. This study selects all the low-rated learner's profiles "ALP" with ratings between 61 and 1000 as = ( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ,. . ., 872 ) and the total number of learner 61 profiles with high ratings above 1000 are identified as other or super-learner's profiles "SLP. " The low-rated learners have been considered as second training set 2 and highrated learner's profiles are considered as second testing set 2 . Table 10 shows the computational similarity results of [ 872 × 61 ] learner's profile similarity matrix, which is conducted with ( 1 , 2 , . . . , =872 ) training and ( 1 , 2 , . . . , 61 ) testing set. The similarity measurement results between the [ × ] similarity matrix to improve the low-rated average learner's profiles "ALP" recommendations. Here, = ( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ,. . ., 872 ) and is used as training set 2 ; perhaps = (V 1 , V 2 , . . . , V 61 ) and is set to be defined as testing set 2 . The results are discussed further in this paper. The similarities in Table 10 have been computed with parameters that work between 0.1 and 1.0 with the help of (7).

Experimental Results
Analysis. After collecting the experimental data, this section helps to analyze the experimental resutls. Mostly, researchers utilize statistics to determine if the results are statistically significant or not. This research work used the famous statistical methods precision and recall to analyze the results efficiency of proposed filtering approach. The formulation structure of precision Pr and recall Re is as follows: Total size of recommended learning contents Total size of learning contents , Total size of recommended learning contents Total size of relevant learning contents .
The results analysis has been done with the help of precision equation (8) and recall equation (9), respectively. In (8) and (9), rm embodies the total number of recommended contents, represents total contents stored in the system, and ( ⋅V) →̂i s considered as total relevant contents with respect to ratings.

Hybrid Results
Analysis. This research has been constuct the personalized similarities between new-learner's, averagelearner's and super-learner's profiles. The results analysis has been shown with the help of famous precision and recall methods. Table 11(a) shows the analyzed precision and recall scores for new-learner zero-rated profile "NLP" recommendations and average-learner low-rated profile "ALP" recommendations matrices are shown in Tables 9(a), 9(b), 9(c), and 10 and demonstrated in Figure 8(a) using bar graph diagram.
Table 11(b) shows the mean of precision and mean of recall scores for new-learner zero-rated profile "NLP" and average-learner low-rated profile "ALP" recommendations matrices in Table 11(a). Figure 8(b) shows the resultant values in Table 11(b) using bar graph diagram. With the help of the above parameters of precision Pr and recall Re, Figure 8(a) stated that new-learner zero-rated profiles "NLP" resultant matrices from Tables 9(a), 9(b), 9(c), and average-learner low-rated profiles "ALP" resultant matrices from Table 10 exhibit almost identical efficiency of proposed work that New-learner zero-rated profile recommendations   Table 11(b) and graphically represented as Figure 8(b), respectively.

Results Evaluation and Discussion.
For the evaluation of proposed Goal-based hybrid filtering, the recommendation results are compared with related literature work [11] results scheme. In [11], the author used machine learning to improve content-based filtering and proposed user-oriented contentbased recommender system (UCB-RS) with two stages as like our proposed work. In the first stage, the authors [11] used fuzzy theoretic content-based filtering to generate the initial population of users' preferences by interactive genetic algorithm using reclusive methods to improve the traditional content-based filtering for new cold-start users. Perhaps, in proposed goal-based hybrid filtering approach, the newlearner zero-rated profile recommendations issue has been improved with the hybridization of content-based filtering and collaborative features. In second stage, the authors [11] used k-mean algorithm for clustering the item in order to handle time complexity of interactive genetic algorithm to Proposed goal-based hybrid filtering approach Mean of (8) 66.43% 83.44% Mean of (9) 78.53% 85.22% improve the collaborative recommendations. Perhaps, in proposed goal-based hybrid filtering, the average-learner lowrated profile recommendations have been overcome with the hybridization of collaborative filtering and k-neighborhood features.
The results evaluation helps to indicate that the proposed goal-based approach performance has improved the hybrid filtering recommender systems for new-learner and average-learner recommendations in e-learning scenarios. The evaluation of proposed approach has been compares with related literature work [11] results to see how much the proposed work contributes to increase the recommendations in hybrid recommender systems in e-learning scenarios. The evaluation results of related literature work [11] and the proposed goal-based hybrid filtering are given in Table 12. In the proposed approach, the mean of (8) precision Pr has been used for calculating all the recommendations that were useful to learners, while the mean of (9) has been used to measure the desired learning contents appearing among the recommendations. The results in Table 12 show improvements in proposed goal-based hybrid filtering approach results with the comparison of related literature work [11]. Now this work test the statistical significance to determine the probability difference of improvement between proposed hybrid approach and the literature work [11]. To do this, the -test has been used to compute the comparison differences with the help of (10a), defined as follows: -test = difference between mean of set 1 and set 2 √variance of set 1 and set 2 /size of set 1 and set 2 = mean (Pr) − mean (Re) Equation (10a) is used to compute the difference between evaluated literature work and proposed research work. The difference has been tackled between the mean of set 1 and set 2 .
In the case of this research work, the set 1 of -test has been set as the value of mean of precision (Pr) and the set 2 of -test has been set as the value of mean of recall (Re). The 2 1 shows the population variance standard deviation of precision set and 2 2 shows the population variance standard deviation of recall set. The variance of precision and recall sets is calculated by (10b) and (10c) as follows: (10c) 1 represents the total subjects values in precision set, while 2 shows the subjects values in recall set. By calculating the -test between the precision and recall sets of proposed approach and evaluation literature work [11], the process gets the precision and recall value sets from Table 12 and computes the -test by using famous statistical software [61]. The results of -test are shown in Table 13. Table 13 shows the -test result that shows the probability difference between the proposed approach and evaluation literature work [11]. The -test result shows the difference value 0.29, which comes in the medium effect category [62]. Figure 9 shows the results of Tables 12 and 13 by using bar graph diagram.
In Figure 9, the results show clear improvement of our proposed goal-based hybrid filtering for e-learning recommender systems, defined as eLearningRecSys. The proposed hybrid approach enhanced the learning content recommendations as revealed in the result mean of precision Pr, 82.97%, mean of recall Re, 84.68%. The results demonstrate the controversial combination of content-based filtering, collaborative filtering, and k-neighborhood features that helps to overcome the new-learner zero-rated profile "NLP" recommendations and improve the average-learner low-rated profile "ALP" recommendations simultaneously.
In the evaluation stage compared with the research work of [11], the authors work is based on reclusive traditional content-based recommendation methods aimed at dealing with content-based recommendations to the new-users. This approach contains precision mean 66.43% and recall mean 78.53% as mentioned in [11]. In the comparison evaluation of proposed goal-based hybrid filtering with [11], it is examined that one machine learning technique is enough to enhance the performance of recommender system if the hybridization sequence is controversially accurate. However, the traditional k-mean in [11] is unable to handle time computational complexity with genetic algorithm. Mostly with large set of content, the traditional k-mean does not give sufficient performance if k is small [46].

Conclusion
The e-learning applies to all the web-based styles and webbased fields of learning scenarios. Generally, these days, all the e-learning styles and fields work with the information retrieval crawlers. The information retrievals are gradually being used for retrieving the web-based learning content electronically over the Internet. The information retrieval systems take some initial keyword from the end user to retrieve the learning content. Due to the large size of elearning content on web [63], the end user or learner failed to find their required content in sort time. For this, the recommender systems are proposed in e-learning webbased scenarios. Recommender system is the branch of information retrieval that suggests the recommendations to end user that help to find the related learning content in a less time as compared to traditional e-learning web-based content retrieval systems. Recommender system is a very active research field. A number of works have already been mentioned in the literature background. This work proposed the goal-based hybrid filtering for personalized similarities between collaborative users profile preferences in e-learning recommender systems, defined as eLearningRecSys. This paper discussed the research challenge learnerbased cold-start problem; it occurs in eLearningRecSys when the system tackles the recommendation based on learner's past profile preferences, such as voting or like and dislike, to the specific learning contents by the learner. Previous literature taught that the learner-based cold-start problem is well known in recommendation systems. Bundle of literature has found the improvement of learner-based coldstart problem with the help of collaborative filtering and other approaches. However, none of them considered the learner's own personalized preferences such as age, gender, and occupation similarities and none of them overcome the both zero-rated and low-rated learner's recommendation issues simultaneously in one system. Alternatively, the goalbased hybrid filtering approach divides the learner-based cold-start problem into two issues, called new-learner zerorated profile "NLP" recommendations and average-learner low-rated profile "ALP" recommendations with aim to help the learner's on both recommendation issues simultaneously in one system. This research has used the famous "MovieLens" dataset " " for experiments and proposed a work to overcome the new-learner zero-rated profile "NLP" recommendations and overcome the average-learner low-rated profile "ALP" recommendations using the learner's personalized profile preferences similarity with super-learner's high-rated profiles "SLP" collaboratively. The proposed research work has intended to introduce a goal-based hybrid filtering approach for elearning recommender systems that hybridized the contentbased filtering, collaborative filtering, and k-neighborhood scheme features. The results of proposed work demonstrate that the goal-based hybrid filtering plays its role to improve the new-learner zero-rated profile recommendation issue and overcome the average-learner low-rated profile recommendations issue simultaneously at the mean time. Nevertheless, the average mean of precision Pr scored 83.44% and mean of recall Re scored 85.22% and have been evaluated with related literature work [11]. The -test probability difference of mean(Pr) and mean(Re) with evaluated literature work is 0.29 that seems the proposed hybrid approach have been made the medium level of improvement in the problem background of the relevant research domain.
The contribution of this research work has intend to propose a goal-based hybrid filtering approach that work with the learners personalized profile preferences such as age, gender and occupation to compute the similarity with other learner or group of learners profile in the collaborative network. These personalized similarities in collaborative learner's network are helpful to get the similar goals of different learner's to overcome the zero-rated learner's recommendations and improve the low-rated learner's profile recommendations simultaneously in one proposed system. The results show that the proposed goal-based hybrid filtering has a flexibility to generate good similarities with minimum learner's profile preferences such as age, gender, and occupation without using any extra profile information. Perhaps, it may work with the maximum learner's profile personalized preferences too, if it is applicable in the future.

Research Limitation and Future Work
The limitation of proposed hybrid approach is intended to facilitate and respond to the recommendations to registered learners only who have their personalized profiles in the system. The nonregistered learners or guest is unable to receive the recommendations by the system.
The learner's requirements are increasing with the passage of time; for this, more research is required to be applicable in real-world situations in the field of e-learning recommender systems [59]. The future work will enhance the learner-profile based similarity results for the significance improvement of proposed goal-based hybrid filtering for learner-based coldstart problem using multiagent-based personalized similarities in different formal and informal e-learning scenarios with other machine learning approaches to check the contentfiltering accuracy, time deficiency, and stability of other normalized datasets too. It may help in the multilearner's personalized similarities, recommendation timeline, and recommendation contents filtration accuracy.


extfieldsofstudy:
Computer Science
Mathematics

text:
Complex Decomposition of the Negative Distance kernel

A Support Vector Machine (SVM) has become a very popular machine learning method for text classification. One reason for this relates to the range of existing kernels which allow for classifying data that is not linearly separable. The linear, polynomial and RBF (Gaussian Radial Basis Function) kernel are commonly used and serve as a basis of comparison in our study. We show how to derive the primal form of the quadratic Power Kernel (PK) -- also called the Negative Euclidean Distance Kernel (NDK) -- by means of complex numbers. We exemplify the NDK in the framework of text categorization using the Dewey Document Classification (DDC) as the target scheme. Our evaluation shows that the power kernel produces F-scores that are comparable to the reference kernels, but is -- except for the linear kernel -- faster to compute. Finally, we show how to extend the NDK-approach by including the Mahalanobis distance.

INTRODUCTION
An SVM has become a very popular machine learning method for text classification [5]. One reason for its popularity relates to the availability of a wide range of kernels including the linear, polynomial and RBF (Gaussian radial basis function) kernel. This paper derives a decomposition of the quadratic Power Kernel (PK) using complex numbers and applies it in the area of text classification. Our evaluation shows that the NDK produces F-scores which are comparable to those produced by the latter reference kernels while being faster to compute -except for the linear kernel. This evaluation refers to the Dewey Document Classification DDC [11] as the target scheme and compares our NDK-based classifier with two competitors described in [6], [10] and [18], respecstively.
An SVM is a method for supervised classification introduced by [17]. It determines a hyperplane that allows for the best possible separation of the input data. (Training) data on the same side of the hyperplane is required to be mapped to the same class label. Furthermore, the margin of the hyperplane that is defined by the vectors located closest to the hyperplane is maximized. The vectors on the margin are called Support Vectors (SV). The decision function dc : R n → {1, 0, −1}, which maps a vector to its predicted class, is given by: dc(x) := sgn( w, x + b) where sgn : R → {1, 0, −1} is the Signum function; the vector w and the constant b are determined by means of SV optimization. If dc(x) = 0 then the vector is located directly on the hyperplane and no decision regarding both classes is possible.
The decision function is given in the primal form. It can be converted into the corresponding dual form: dc(x) = sgn( m j=1 y j α j x, x j + b) where α j and b are constants determined by the SV optimization and m is the number of SVs x j (the input vector has to be compared only with these SVs). The vectors that are located on the wrong side of the hyperplane, that is, the vectors which prevent a perfect fit of SV optimization, are also considered as SVs. Thus, the number of SVs can be quite large. The scalar product, which is used to estimate the similarity of feature vectors, can be generalized to a kernel function. A kernel function K is a similarity function of two vectors such that the matrix of kernel values is symmetric and positive semidefinite. The kernel function only appears in the dual form. The decision function is given as follows: Let Φ : R n → R l be a function that transforms a vector into another vector mostly of higher dimensionality with l ∈ N ∪ {∞}. It is chosen in such a way that the kernel function can be represented by: . Thus, the decision function in the primal form is given by: Note that Φ is not necessarily uniquely defined. Furthermore, Φ might convert the data into a very high dimensional space. In such cases, the dual form should be used for the optimization process as well as for the classification of previously unseen data. One may think that the primal form is not needed. However, it has one advantage: if the normal vector of the hyperplane is known, a previously unseen vector can be classified just by applying the Signum function, Φ and a scalar multiplication. This is often much faster than computing the kernel function for previously unseen vectors and each SV as required when using the dual form.
The most popular kernel is the scalar product, also called the linear kernel. In this case, the transformation function Φ is the identity function: K lin (x 1 , x 2 ) := x 1 , x 2 . Another popular kernel function is the RBF (Gaussian Radial Basis Function), given by: K r (x 1 , x 2 ) := e −γ||x1−x2|| 2 where γ ∈ R, γ > 0 is a constant that has to be manually specified. This kernel function can be represented by a function Φ r that transforms the vector into infinite dimensional space [19]. For reasons of simplicity, assume that x is a vector with only one component. Then the transformation function Φ r is given by: Another often used kernel is the polynomial kernel: For d := 2, a = 1, c = 1, and two vector components, the function Φ p is given by [9]: Φ p : In general, a vector of dimension n is transformed by Φ p into a vector of dimension n+d d . Thus, the polynomial kernel leads to a large number of dimensions in the target space. However, the number of dimensions is not infinite as in the case of the RBF kernel.

THE POWER KERNEL
The PK is a conditionally positive definite kernel given by for some p ∈ R [16], [2], [12]. A kernel is called conditionally positive-definite if it is symmetric and satisfies the conditions [14] n j,k=1 where c j is the complex-conjugate of c j . We consider here a generalized form of the PK for p := 2 (also called NDK): with a, c ∈ R and a > 0. The expression −a||x 1 − x 2 || 2 + c can also be written as: For deciding which class a previously unseen vector belongs to we can use the decision function in the dual form: The decision function shown in formula (9) has the drawback that the previously unseen vector has to be compared with each SV, which can be quite time consuming. This can be avoided, if we reformulate formula (9) to: With z := a m j=1 y j α j x j , u := a m j=1 y j α j x j , x j and c ′ = c m i=1 y j α j , formula (10) can be rewritten as: The expressions u, z, ( m i=1 y j α j ), and c ′ are identical for every vector x and can be precomputed. Note that there exists no primal form for the NDK based on real number vectors which is stated by the following proposition. Proposition 1 Let a,c ∈ R with a > 0 and n, l ∈ N. Then there is no function Φ re : R n → R l (re indicates that Φ re operates on real numbers) with Proof: If such a function existed, then, for all x ∈ R n , which requires that c ≥ 0 since the square of a real number cannot be negative. Now, consider x, y ∈ R n with ||x−y|| 2 > 2c a ≥ 0. On the one hand, we have, by the Cauchy-Schwarz inequality, On the other hand, it holds that Although no primal form and therefore no function Φ re exists for real number vectors, such a function can be given if complex number vectors are used instead. In this case, the function Φ c is defined with a real domain and a complex codomain: Φ c : R n → C 4n+1 and Note that no scalar product can be defined for complex numbers that fulfills the usual conditions of bilinearity and positive-definiteness simultaneously 1 . Thus, the bilinearity condition is dropped for the official definition and only sesquilinearity is required. The standard scalar product is defined as the sum of the products of the vector components with the associated complex conjugated vector components of the other vector. Let x 1 ,x 2 ∈ C n , then the scalar product is given by [1]: x 1k x 2k . In contrast, we use a modified scalar product (marked by a " * ") where, analogously to the real vector definition, the products of the vector components are summated: * x 1 , x 2 := n k=1 x 1k x 2k . This product (not a scalar product in strict mathematical sense) is a bilinear form but no longer positive definite. For real number vectors this modified scalar product is identical to the usual definition. With this modified scalar product we get which is just the result of the NDK. The optimization can be done with the dual form. Thus, no complex number optimization is necessary. For the decision on the class to which a data vector should be assigned we switch to the primal form. The vector w ∈ C 4n+1 is calculated by: Note that the modified scalar product * w, Φ c (x) must be a real number. This is stated in the following proposition. (12) and x ∈ R n . Then * w, Φ c (x) is a real number.
which is clearly a real number.
The NDK is related to the polynomial kernel since it can also be represented by a polynomial. However, it has the advantage over the polynomial kernel that it is faster to compute, since the number of dimensions in the target space grows only linearly and not exponentially with the number of dimensions in the original space [16], [2]. It remains to show that the decision functions following the primal and dual form are also equivalent for the modified form of the scalar product. This is stated in the follow proposition: Normally, feature vectors for document classification represent the weighted occurrences of lemma or word forms [15]. Thus, such a vector contains a large number of zeros and is therefore usually considered sparse. In this case, the computational complexity of the scalar product can be reduced from O(n) (where n is the number of vector components) to some constant runtime, which is the average number of nonzero vector components. Let I 1 be the set of indices of nonzero entries of vector x 1 (I 1 = {k ∈ {1, . . . , n} : x 1k = 0}) and I 2 be analogously defined for vector x 2 . The scalar product of both vectors can then be computed by k∈(I1∩I2) x 1k ·x 2k . Let us now consider the case that both vectors are transformed to complex numbers before the scalar multiplication. In this case, the modified scalar product is considered where φ(x k ) denotes the transformation of a single real vector component to a complex number vector and is defined as: Note that the partial modified scalar product * φ(x 1k ), φ(x 2k ) can be non-zero, if at least one of the two vector components x 1k and x 2k is non-zero, which is easy to see: Only if both vector components are zero, one can be sure that the result is also zero: Thus, the sparse data handling of two transformed complex vectors has to be modified in such a way that vector components associated with the union and not the intersection of non-zero indices are considered for multiplication: A further advantage of the NDK is that the Mahalanobis distance [8] can easily be integrated, which is shown in the remainder of this section. Each symmetric matrix A can be represented by the product V −1 DV where D is a diagonal matrix with the eigenvalues of A on its diagonals. The square root of a matrix A is then defined as √ A := V −1 D 0.5 V where D 0.5 is the matrix with the square root of the eigenvalues of A on its diagonal. It is obvious that √ A · √ A = A. Proposition 4: Let x, z ∈ R n , a := 1, c := 0, then and Cov denotes the covariance matrix between the feature values of the entire training data set.
(since the covariance matrix is symmetric and the inverse and square root of a symmetric matrix is also symmetric) This proposition shows that the NDK is just the negative square of the Mahalanobis distance for some vectorsx andẑ ifx is set to √ Cov −1 x (analogously forẑ), a is set to 1 and c is set to 0. Furthermore, this proposition shows that we can easily extend our primal form to integrate the Mahalanobis distance. For that, we define a function τ : R n → R n as follows: which shows that we have indeed derived a primal form.
We use the NDK for text classification where a text is automatically labeled regarding its main topics (i.e., categories), employing the DDC as one of the leading classification schemes in digital libraries [11]. Documents are classified regarding the 10 top-level categories of the DDC. To this end, we use training sets of documents for which the DDC categories have already been assigned. Lexical features are extracted from this training set and made input to an SVM library (i.e., libsvm [3]) in order to assign one or more DDC categories to previously unseen texts. Features of instance documents are computed by means of the geometric mean of the tfidf values and the GSS coefficients of their lexical constituents [4], [13].

EVALUATION
The evaluation and training was done using 4 000 German documents, containing in total 114 887 606 words and 9 643 022 sentences requiring a storage space of 823.51 MB. There are 400 documents of each top-level DDC category in the corpus. 2 000 of the texts were used for training, 2 000 for evaluation. The corpus consists of texts annotated according to the OAI (Open Archive Initiative) and is presented in [7]. We tested the correctness of category assignment for the 10 toplevel DDC categories. Each document is mapped at least to one DDC category. Multiple assignments are possible. Precision, recall and F-scores were computed for each DDC category using the NDK, the square (polynomial kernel of degree 2), the cubic (polynomial kernel of degree 3), the RBF kernel and the linear kernel (see Tables I and II). The free parameters of the square, the cubic and the RBF kernel are determined by means of a grid search on an independent data set. On the same held-out dataset, we adjusted the SVM-threshold parameter b to optimize the F-scores. The time (on an Intel Core i7) required to obtain the classifications was determined (see Table III for the real / complex NDK, the RBF, square, and linear kernel). The time needed for building the model was not measured because of being required only once and therefore being irrelevant for online text classification.
The F-score for the NDK is higher than the F-scores of all other kernels and faster to compute except for the linear kernel -obviously, the classification using the primal form of the linear kernel is faster than the one using the NDK. Furthermore, the complex decomposition of the NDK leads to a considerable acceleration of the classification process compared to its dual form and should therefore be preferred.
We conducted a second experiment where we compared the linear kernel with the NDK now using text snippets (i.e., abstracts instead of full texts) taken once more from our OAI corpus. This application scenario is more realistic for digital libraries that mainly harvest documents by means of their meta data and, therefore, should also be addressed by text classifiers. The kernels were trained on a set of 19 000 abstracts 2 . This time, the categories are not treated independently of each other. Instead, the classifier selected all categories with signed distance values from the SVM hyperplane that are greater or equal to zero. If all distance values are negative, the category with the largest (smallest absolute) distance was chosen. In this experiment, the linear kernel performed better than the NDK. Using three samples of 500 texts, the F-scores of the linear kernel are (macro-averaged over all main 10 DDC categories): 0.753, 0.735, 0.743, and of the NDK: 0.730, 0.731, 0.737. This result indicates that the heuristic of preferring the categories of highest signed distance to the hyperplane is not optimal for the NDK. We compared our classifier with two other systems, the DDC classifier of [18] and of [6]. [18] reaches an F-score of 0.502, 0.457 and 0.450 on these three samples. The Fscores of the DDC-classifier of [6] are 0.615, 0.604, and 0.574. Obviously, our classifier outperforms these competitors.
Finally, we evaluated the NDK on the Reuters 21578 corpus 3 (Lewis split). This test set is very challenging since 35 of all 93 categories occurring in this split have 10 or less training examples. Furthermore, several texts are intentionally assigned to none of the Reuters categories. We created an artificial category for all texts that are not assigned in this sense. The histogram of category assignments is displayed in Figure 1. The F-scores for the Reuters corpus are given in Table IV. We modified the training data by filling up instances for every category by randomly selecting positive instances such that the ratio of positive examples to all instances are the same for all categories. Hence, in most category samples some training examples were used more than once. This approach prevents from preferring categories due to their multitude of training examples. The F-score of the NDK is lower than the   highest macro-averaging F-score obtained by the linear kernel, but outperforms the square and the cubic kernel. Again, the parameters of the NDK, the square, the cubic and the RBF kernels were determined by means of a grid search.

CONCLUSION AND FUTURE WORK
We derived a primal form of the NDK by means of complex numbers. In this way, we obtained a much simpler representation compared to the modified dual form. We showed that the primal form (and in principle also the modified dual form) can speed up text classification considerably. The reason is that it does not require to compare input vectors with all support vectors. Our evaluation showed that the F-scores of the NDK are competitive regarding all other kernels tested here while the NDK consumes less time than the polynomial and the RBF kernel. We have also shown that the NDK performs better than the linear kernel when using full texts rather than text snippets. Whether this is due to problems of feature selection/expansion or a general characteristic of this kernel (in the sense of being negatively affected by ultra-sparse features spaces), will be examined in future work. Additionally, we plan to examine the PK with exponents larger than two, to investigate under which prerequisites the PK performs well and to evaluate the extension of the NDK that includes the Mahalanobis distance.