First-Level: One-Class SVM—Second-Level: Multi-Class SVM
LetU = {u1,u2, . . . ,um}andI = {i1,i2, . . . ,in}be the sets of users and items, respectively of the database from which our RS makes recommendations. Each itam in the database corresponds to a feature vector (e.g., a 30-dimensional MARSYAS feature vector) in a high-dimensional Euclidean vector spaceV. Each user assigns a unique rating value for each item in the database within the range of{0,1,2,3}. Thus, user ratings define four disjoint classes of increasing degree of interest, namelyC0, C1,C2andC3.C0corresponds to the class of non-desirable/negative patterns, while the class of desirable/positive patterns may be defined as the union (C1∪C2∪C3) of C1,C2andC3. In order to indicate the user involvement in defining the four classes of interest, we may write that
∀u∈U, V =C0(u)∪C1(u)∪C2(u)∪C3(u), (6.1) where
C0(u)∩C1(u)∩C2(u)∩C3(u)= ∅. (6.2) More specifically, lettingR(u,i)be the rating value that the useruassigned to itemi, the four classes of interest may be defined via the following equations:
C0(u)= {i∈I:R(u,i)=0}
C1(u)= {i∈I:R(u,i)=1}
C2(u)= {i∈I:R(u,i)=2}
C3(u)= {i∈I:R(u,i)=3}
(6.3)
At this point, we need to mention that ifI(u)denotes the subset of items for which useruprovided a rating, it follows that∀u∈U,I(u)=I. Thus, the positive (desirable) and the negative (non-desirable) classes of patterns for each user may be defined as follows:
∀u∈U, P(u)=C1(u)∪C2(u)∪C3(u)
∀u∈U, N(u)=C0(u) (6.4)
The training/testing procedure for the classifiers at both levels involves partitioning each class of the desirable patterns for each user intoKdisjoint subsets such that:
6.2 Cascade Content-Based Recommendation 103
LettingCj(u,k)be the set of patterns from the positive classjthat is used through-out the testing procedure, the corresponding set of training patterns will be denoted asCj(u,k)and the following equation holds:
∀j∈ {1,2,3},∀k∈ [K], Cj(u,k)∪Cj(u,k)=Cj(u). (6.7) In other words, Eq.6.7defines theK-fold cross validation partitioning that is uti-lized to measure the performance accuracy of our cascade classification scheme. Let P(u,k)andN(u,k)be the sets of positive and negative patterns, respectively, as they are presented to the first-level classifier during the testing stage at foldkfor a particular useru. We have:
P(u,k)=C1(u,k)∪C2(u,k)∪C3(u,k) (6.8) N(u,k)=C0(u,k)=C0(u)=N(u) (6.9) In case theK-fold cross validation partitioning is not taken into consideration, the set of positive patterns for a particular user may be referred to as P(u), so that P(u)=C1(u)∪C2(u)∪C3(u).
The training procedure of the first level of our cascade classification architecture aims at developing one one-class classifier per user. These one-class classifiers are trained to recognize those data instances that have originated from the positive class of patterns. In other words, each one-class classifier realizes a discrimination func-tion denoted byfu(v), where vis a vector inV that is learnt from the fraction of training positive patterns. More specifically, iffu,k(v)is the discrimination function that corresponds to useruat foldk, then this function would be the result of training the one-class classifier onC1(u,k)∪C2(u,k)∪C3(u,k). The purpose of each dis-crimination functionfu(v)is to recognize the testing positive patternsP(u)against the complete set of negative patternsN(u).
On the other hand, the training procedure of the second level of our cascade classification architecture aims at developing one multi-class classifier per user. This is achieved by training the multi-class classifier on the same set of positive data, C1(u,k)∪C2(u,k)∪C3(u,k), but this time with the purpose to discriminate among the various (sub-)classes of the data pertaining to the set P(u,k). In other words, each second-level classifier realizes a discrimination function denoted bygu(v), the purpose of which is to partition the space of testing (positive) dataP(u)into the 3 corresponding subspaces,C1(u),C2(u)andC3(u), of desirable patterns. To explicitly indicate the discrimination function concerning useruat foldk, we usegu,k(v).
Fig. 6.1 Cascade content-based recommender
The recommendation ability of our system is based on its efficiency when predict-ing the ratpredict-ing value that a particular user assigned to an item which was not included in the training set. Having in mind that P(u,k)∪N(u,k)is the full set of testing data presented to the fist level of our cascade classification mechanism, the one-class component operates as a filter that recognizes the items that a user assigned to the class of desirable patterns. Specifically, the first-level discrimination functionfu,k(v) for useruat foldkpartitions the set of testing data into positive and negative pat-terns as illustrated in Fig.6.1. In other words, the testing procedure concerning the first-level of our cascade classifier involves the assignment of a unique value within the set{−1,1}for each input element such that:
∀u∈U, ∀k∈ [K], ∀v∈P(u,k)∪N(u,k), fu,k(v)∈ {−1,+1}. (6.10) The subset of testing instances that are assigned to the class of desirable patterns are subsequently fed into the second-level classifier which assigns to them a particular rating value within the range of{1,2,3}. Specifically,
6.2 Cascade Content-Based Recommendation 105
∀u∈U,∀k∈ [K],∀v∈P(u,k)∪N(u,k):fu,k(v)= +1, gu,k(v)∈ {1,2,3}
(6.11) Let the true rating value concerning an objectv∈Vfor a particular useruat foldk beRu,k(v)so that the following equation holds:
∀u∈U, ∀k∈ [K], ∀j∈ {0,1,2,3}, Ru,k(v)=j⇔v∈Cj(u,k). (6.12) The estimated rating value assigned by our system will be indicated asRu,k(v)and can be computed as in the following equation:
Ru,k(v)=
As stated previously, the problem addressed in this chapter is that of building an efficient RS in the complete absence of negative examples. Since negative examples are, in general, extremely difficult to obtain, we employed classification paradigms that operate exclusively on the basis of positive patterns. This justifies the incorpo-ration of the one-class classification component within the first-level of our cascade classification architecture. Thus, the first classification level serves the purpose of filtering out the majority of the non-desirable patterns. However, as the ultimate purpose of any RS is to provide its users with high quality recommendations, a com-ponent is needed which predicts the true class of unseen patterns with high accuracy.
This is the rationale behind the second classification level which takes as input the set of patterns that were assigned to the class of desirable items by the first classification level. In order to provide high quality recommendations, it is vital that the second-level classifier correctly discriminate among the various (sub-)classes of desirable patterns. Thus, the second-level classifier is a multi-class one.
A natural modification of our cascade classification architecture consists of replac-ing the second (multi-class) classification level with a CF component, as illustrated in Fig.6.2. Having in mind that the first classification level realizes the broader distinc-tion between positive and negative patterns, the subsequent CF component produces specific rating values within the range of{1,2,3}. Specifically, the CF methods that we utilized were:
• Pearson Correlation [1]
• Vector Similarity [1] and
• Personality Diagnosis [2].