Similarity from Multi-Dimensional Scaling: Solving the Accuracy and Diversity Dilemma in Information Filtering

Recommender systems are designed to assist individual users to navigate through the rapidly growing amount of information. One of the most successful recommendation techniques is the collaborative filtering, which has been extensively investigated and has already found wide applications in e-commerce. One of challenges in this algorithm is how to accurately quantify the similarities of user pairs and item pairs. In this paper, we employ the multidimensional scaling (MDS) method to measure the similarities between nodes in user-item bipartite networks. The MDS method can extract the essential similarity information from the networks by smoothing out noise, which provides a graphical display of the structure of the networks. With the similarity measured from MDS, we find that the item-based collaborative filtering algorithm can outperform the diffusion-based recommendation algorithms. Moreover, we show that this method tends to recommend unpopular items and increase the global diversification of the networks in long term.


Data
We present here the recommendation results of different algorithms in three real networks, namely Movie-Lens, Netflix and RYM. MovieLens (http://www.grouplens.org) is a movie recommendation website, which provides personalized recommendations by using users' historical ratings. The data used in this paper consists of 6040 users and 3900 movies. Netflix (http://www.netix.com) is an online DVD and Blu-ray Disc rental service in the US. The data we used is a random sample that consists of 3000 users and 3000 movies. RYM (http://rateyourmusic.com/) is an online community of people who love music and allows individuals to catalog, rate, tag and review music. The data used in this paper is also a random sample consisting of 33762 users and 5267 songs. More details about these datasets are presented in table S1.
The data is randomly divided into two parts: the training set (E T ) and the probe set (E P ). The training set contains 80% of the original data and the recommendation algorithm runs on it. The rest of the data forms the probe set, which will be used to examine the recommendation performance. We apply precision and recall to measure the accuracy of recommendation algorithms (see definition in the main manuscript). Higher precision and recall indicate higher accuracy of recommendations. The hamming distance are chosen to measure the diversity of recommendation list and the novelty are selected to measure the capacity of the recommender system to generate novel and unexpected results (see definition in the main manuscript).
The relationship between the accuracy (precision and recall) and the dimension d of Y computed by the ICF-MDS on Netflix and RYM dataset is presented in figure S1. The accuracy of other recommendation approaches is presented in table S2. The diversity and novelty results on the Netflix and RYM are presented in figure S2.

Simulations in Artificial Data
Our artificial data consists of 500 users and 500 items. The distributions of the user degree (the number of items the user has collected) and the item degree (the number of users who have collected the item) follow the power-law form k −2.5 . To generate connections between users and items, we assume that each user i has a taste vector with U-dimension, p i = (p i,1 , p i,2 , ..., p i,U ) [1]. Similarity, attributes of the item α are represented by the U-dimensional attribute vector q α = (q α,1 , q α,2 , ..., q α,U ). We set U = 9 and randomly let 5 elements of p i and q α to be 1 and the remaining elements to be 0. Each time we randomly select a user i to rate an random selected item α and the rating reads r iα = p i q α . According to the definition, r iα ranges from 1 to 5. In this paper, we let the number of ratings equal to 0.1 × 500 × 500.
Based on the item attribute vectors, one can get the "true" similarity between item α and item β as, . (1) One can get their cosine similarity s cos αβ based on the artificial rating matrix R = {r ia }. We also compute their MDS similarity s mds αβ . All the similarities are normalized: where max(s) and min(s) are the maximum and minimum of all the similarities, respectively. To compare the cosine and M DS method, we compute the mean absolute error between s true and s cos or s mds for all item pairs. That is, where N = 500 is the number of items. s * αβ denotes the s cos αβ or s mds αβ . The results of δ are presented in Fig. S1. We compare these two methods on the unweight (the left figure of Fig. S1) and the weight (the right figure of Fig. S1) adjacency matrix. For the unweight adjacency matrix, we set r iα = 1 if r iα > 0 in R. The dimension of Y is increased from 2 to 50. In Fig. S1, there are three main findings: Firstly, the δ of both the MMDS and NMDS decreases with their dimension H. That's to say, it is more accurate to compute the similarities of item pairs by the MMDS and NMDS when using a higher dimension. Secondly, The MMDS is slightly better than the NMDS in computing the item similarity as the δ of MMDS is smaller than that of NMDS. However, their differences tend to be smaller when the dimension increases. Thirdly, both the MMDS and NMDS is significantly more accurate than the cosine method in both unweighted and weighted networks.

Dimension of the MDS method
The relationship between the accuracy (precision and recall) and the dimension of Y computed by the MDS is presented in Fig. S2. One can observe that both the accuracy of ICF-MMDS and ICF-NMDS increases with the dimension when the dimension H is smaller than 100. The slope of the precision and recall curves is larger when the dimension H < 30 and smaller when H > 30. Moreover, the ICF-MMDS's accuracy is larger than the ICF-NMDS but their differences gradually become smaller as H increases. The MDS technique aims to place each item in H-dimensional space such that the between-item distances are preserved as well as possible. When H is reduced, a lot of unimportant information containing noise is neglected. Therefore, MDS can be regarded as a noise reduction method. This is shown in the recommendation results in Fig. S2 in which one can see the precision cannot be constantly increases by increasing H (when H is large enough, further enlarging H only includes noisy information).

MDS under the user-based CF framework
In this section, we carry out the simulation to compare MDS and cosine similarities under the UCF framework. The results are presented in table S2. Consistent with the manuscript, we here use the precision and recall to measure the recommendation accuracy. UCF-cos and ICF-cos are respectively the user-based CF and item-based CF with nodes' similarity calculated with the cosine method. UCF-MDS and ICF-MDS are respectively the user-based CF and item-based CF with nodes' similarity calculated with the MMDS method. UMCF is short for the modified UCF which is proposed in [2]. From the table, one can see that the user-oriented CF is worse than the item-oriented CF. The accuracy of both UCF and ICF can be improved by the MDS method. The best method here is the ICF-MDS method.

Definition of distances in MDS
Apart from the Euclidean distance, we consider here two other definitions of distances: Euclidean Commute-Time Distance (ECTD). Denote by m(α, β) the average number of steps required by a random walker starting from item α to reach item β, the average commute time between α and β is dist(α, β) = m(α, β) + m(β, α), which can be obtained in the terms of pseudoinverse of the Laplacian matrix, L + (L = D − B), as dist(α, β) = V (l + xx + l + yy − 2l + xy ), where l + xy is the entry in L + . B is the adjacent matrix projected by the user-item network with the element b αβ = |Γ α Γ β |, where Γ α is the user set who have collected item α. D is a diagonal matrix with d αα = [D αα ] =    Figure S4. The diversity and novelty results on the Netflix and RYM data. The recommendation length L is 20.