Sparse Linear Methods (SLIM) are state-of-the-art recommendation approaches based on matrix factorization, which rely on a regularized ℓ1-norm and ℓ2-norm optimization -an alternative optimization problem to the traditional Frobenious norm. Although they have shown outstanding performance in Top-N recommendation, existent works have not yet analyzed some inherent assumptions that can have an important effect on the performance of these algorithms. In this paper, we attempt to improve the performance of SLIM by proposing a generalized formulation of the aforementioned assumptions. Instead of directly learning a sparse representation of the user-item matrix, we (i) learn the latent factors' matrix of the users and the items via a traditional matrix factorization approach, and then (ii) reconstruct the latent user or item matrix via prototypes which are learned using sparse coding, an alternative SLIM commonly used in the image processing domain. The results show that by tuning the parameters of our generalized model we are able to outperform SLIM in several Top-N recommendation experiments conducted on two different datasets, using both nDCG and nDCG@10 as evaluation metrics. These preliminary results, although not conclusive, indicate a promising line of research to improve the performance of SLIM recommendation.
Categories and Subject Descriptors H.4 [Information Systems Applications]: Miscellaneous; H.3.3 [Information Search and Retrieval]: Information filtering
The main goal of a recommender system is helping users dealing
with information overload by providing personalized suggestions.
This so-called “recommendation problem” has been addressed in
different ways, such as predicting unobserved user ratings or as
Top-N recommendation [
If we compare SLIM with other similar applications that use
ℓ1norm and ℓ2-norm optimization, such as sparse coding used for
dictionary learning in image processing, we can identify two
assumptions. First, SLIM aims at reconstructing directly the
lowdensity user-item matrix A, but low-density datasets decrease the
effectiveness of Top-N recommendations [
In this document, we explain our gSLIM approach, preliminary experiments on two datasets showing promising results, and finally we discuss limitations and some interesting ideas for future work. 2.
GSLIM RECOMMENDATION MODEL
In order to model each user we use a traditional regularized
matrix factorization [
mU;iVn jjA UT Vjj2F + 1(jjUjj2F + jjVjj2F ) (1)
After learning the latent users’ matrix U, we proceed to compute
the latent social prototypes. Similar as Karypis et. al. [
Finally to make a recommendation r^i;j for the user i on the item j we multiply vectors r^i;j = u⃗′iT ⃗vj , where user’s reconstruction defined by equation 3 is solved by an Orthogonal Matching Pursuit (OMP) algorithm with a maximum of non-negative coefficients for coefficients w⃗ .
⃗u′i = P argmin (⃗ui Pw⃗ ) (3)
w⃗
gSLIM compared to SLIM. Unlike our method, SLIM [
1 mWin 2 jjA
AWjj2F + 2 jjWjj2F + jjWjj1; s.t.W
Using the Movielens datasets with 100K (ML-100K, juj = 943,
jij = 1682) and 1M (ML-1M, juj = 6040, jij = 3952) ratings1,
we performed top-N recommendation experiments by using the
LensKit framework2, evaluating the performance with nDCG and
nDCG@10 metrics [
Figure 1: Results in ML-100K. Line styles: dashed=SLIM, dash&dot=MF, solid=gSLIM.
Results. Table 1 presents the results, showing that in the ML100K dataset SLIM is outperformed by both gSLIM and regularized MF, indicating that this smaller and denser dataset (6.3%) might be better served by a traditional MF (hypothesis 1). However, in the larger and sparser ML-1M dataset (4.19%) SLIM out0.24 0.22 0.20 0.18 0.24 0.22 0.20 0.18 0.24 0.22 0.20 0.18 0.940 0.935 0.930 0.925 0.940 0.935 0.930 0.925 0.940 0.935 0.930 0.925
1 2 3 4 5 γ 6 7 8 9 10 (a) nDCG in ML-1M (b) nDCG@10 in ML-1M Figure 2: Results in ML-1M dataset. Line styles: dashed=SLIM, dash&dot=MF, solid=gSLIM. performs both MF and gSLIM in nDCG@10, but gSLIM outperforms both in terms of nDCG, indicating that reconstructing the denser latent user matrix with prototypes can support a recommendation task beyond a small top-N (hypothesis 2). Finally, figures 1a and 1b show the behavior in the ML-100K dataset of the parameters k1, k2 and in gSLIM performance compared to SLIM and MF baselines, and a similar behavior is seen in ML-1M dataset (Figures 2a and 2b). Analysis shows that larger k2 and give better performance but k1 has its peak between 20-30 latent factors for users’ matrix U. 4.
CONCLUSIONS AND FUTURE WORK In this work we introduced a general formulation for SLIM, gSLIM. Although we haven’t addressed the limitation of our model in terms of computational complexity, we think that our results open an opportunity for studying unexplored ideas on Sparse Linear Methods. In the first step, researchers can try different methods to learn a denser representation of the A, U or V matrices. In the step where we perform sparse coding, we could try reconstructing the items’ latent factor matrix V rather than U, or try directly A. We can also try alternative algorithms to OMP for prototype learning. Even more, our intuition is that learning these sparse prototypes can help us finding actual “stereotypes” of users and items, which can be used, e.g., for clustering users and items.