=Paper=
{{Paper
|id=None
|storemode=property
|title=An FCA-based Boolean Matrix Factorisation for Collaborative Filtering
|pdfUrl=https://ceur-ws.org/Vol-977/paper8.pdf
|volume=Vol-977
}}
==An FCA-based Boolean Matrix Factorisation for Collaborative Filtering==
https://ceur-ws.org/Vol-977/paper8.pdf
An FCA-based Boolean Matrix Factorisation for
Collaborative Filtering
Elena Nenova2,1 , Dmitry I. Ignatov1 , and Andrey V. Konstantinov1
1
National Research University Higher School of Economics, Moscow
dignatov@hse.ru
http://www.hse.ru
2
Imhonet, Moscow
http://imhonet.ru
Abstract. We propose a new approach for Collaborative filtering which
is based on Boolean Matrix Factorisation (BMF) and Formal Concept
Analysis. In a series of experiments on real data (Movielens dataset)
we compare the approach with the SVD- and NMF-based algorithms
in terms of Mean Average Error (MAE). One of the experimental con-
sequences is that it is enough to have a binary-scaled rating data to
obtain almost the same quality in terms of MAE by BMF than for the
SVD-based algorithm in case of non-scaled data.
Keywords: Boolean Matrix Factorisation, Formal Concept Analysis,
Singular Value Decomposition, Recommender Algorithms
1 Introduction
Recently Recommender Systems is one of the most popular subareas of Machine
Learning. In fact, the recommender algorithms based on matrix factorisation
techniques (MF) has become industry standard.
Among the most frequently used types of Matrix Factorisation we definitely
should mention Singular Value Decomposition (SVD) [7] and its various mod-
ifications like Probabilistic Latent Semantic Analysis (PLSA) [14]. However,
the existing similar techniques, for example, non-negative matrix factorisation
(NMF) [16,13,9] and Boolean matrix factorisation (BMF) [2], seem to be less
studied. An approach similar to matrix factorization is biclustering which was
also successfully applied in recommender system domain [18,11]. For example,
Formal Concept Analysis [8] can also be used as a biclustering technique and
there are some of its applications in recommenders’ algorithms [6,10].
The aim of this paper is to compare recommendation quality of some of the
aforementioned techniques on real datasets and try to investigate the methods’
interrelationship. It is especially interesting to conduct experiments on com-
parison of recommendations quality in case of an input matrix with numeric
values and in case of a Boolean matrix in terms of Precision and Recall as well
as MAE. Moreover, one of the useful properties of matrix factorisation is its
ability to keep reliable recommendation quality even in case of dropping some
58 Elena Nenova, Dmitry I. Ignatov, and Andrey V. Konstantinov
insufficient factors. For BMF this issue is experimentally investigated in section
4.
The novelty of the paper is defined by the fact that it is a first time when
BMF based on Formal Concept Analysis [8] is investigated in the context of
Recommender Systems.
The practical significance of the paper is determined by demands of the rec-
ommender systems’ industry, that is to gain reliable quality in terms of Mean
Average Error (MAE), Precision and Recall as well as time performance of the
investigated method.
The rest of the paper consists of five sections. The second section is an intro-
ductory review of the existing MF-based recommender approaches. In the third
section we describe our recommender algorithm which is based on Boolean ma-
trix factorisation using closed sets of users and items (that is FCA). Section 4
contains methodology of our experiments and results of experimental compari-
son of different MF-based recommender algorithms by means of cross-validation
in terms of MAE, Precision and Recall. The last section concludes the paper.
2 Introductory review of some matrix factorisation
approaches
In this section we briefly describe different approaches to the decomposition of
both real-valued and Boolean matrices. Among the methods of the SVD group
we describe only SVD. We also discuss nonnegative matrix factorization (NMF)
and Boolean matrix factorization (BMF).
2.1 Singular Value Decomposition (SVD)
Singular Value Decomposition (SVD) is a decomposition of a rectangular matrix
A ∈ Rm×n (m > n) into the product of three matrices
Σ
A=U V T, (1)
0
where U ∈ Rm×m and V ∈ Rn×n are orthogonal matrices, and Σ ∈ Rn×n is
a diagonal matrix such that Σ = diag(σ1 , . . . , σn ) and σ1 ≥ σ2 ≥ . . . ≥ σn ≥ 0.
The columns of the matrix U and V are called singular vectors, and the numbers
σi are singular values [7].
In the context of recommendation systems rows of U and V can be interpreted
as vectors of the user’s and items’s loyalty (attitude) to a certain topic (factor),
and the corresponding singular values as the importance of the topic among the
others. The main disadvantage is in the fact that the matrix may contain both
positive and negative numbers; the last ones are difficult to interpret.
The advantage of SVD for recommendation systems is that this method
allows to obtain the vector of its loyalty to certain topics for a new user without
SVD decomposition of the whole matrix.
An FCA-based Boolean Matrix Factorisation for Collaborative Filtering 59
The evaluation of computational complexity of SVD according to [15] is
O(mn2 ) floating-point operations if m ≥ n or more precisely 2mn2 + 2n3 .
Consider as an example the following table of movie ratings:
Table 1. Movie rates
The Artist Ghost Casablanca Mamma Mia! Dogma Die Hard Leon
User1 4 4 5 0 0 0 0
User2 5 5 3 4 3 0 0
User3 0 0 0 4 4 0 0
User4 0 0 0 5 4 5 3
User5 0 0 0 0 0 5 5
User6 0 0 0 0 0 4 4
This table corresponds to the following matrix of ratings:
4450000
5 5 3 4 3 0 0
0 0 0 4 4 0 0
A= 0 0 0 5 4 5 3.
0 0 0 0 0 5 5
0000044
From the SVD matrix decomposition we get:
0.31 0.48 −0.49 −0.64 −0.06 0
0.58 0.50 0.03 0.63 0.06 0
0.29 0 0.57 −0.23 −0.72 0
U = 0.57 −0.37 0.31 −0.30 0.57
,
0
0.29 −0.47 −0.43 0.15 −0.28 −0.62
0.23 −0.37 −0.35 0.12 −0.22 0.78
12.62 0 0 0 0 00
0 10.66 0 0 0 0 0
Σ 0 0 7.29 0 0 0 0
= ,
0 0
0 0 1.64 0 0 0
0 0 0 0 0.95 0 0
0 0 0 0 0 00
0.32 0.41 −0.24 0.36 0.07 0.70 0.13
0.32 0.41 −0.24 0.36 0.07 −0.62 −0.35
0.26 0.37 −0.32 −0.79 −0.12 −0.06 0.17
VT =
0.50 0.01 0.55 0.05 0.24 −0.21 0.57 .
0.41 0.01 0.50 −0.14 −0.42 0.21 −0.57
0.42 −0.53 −0.27 −0.15 0.57 0.10 −0.28
0.33 −0.46 −0.36 0.21 −0.63 −0.10 0.28
60 Elena Nenova, Dmitry I. Ignatov, and Andrey V. Konstantinov
It can be seen that the greatest weight have the first three singular values,
which is confirmed by the calculations:
3
σi2
P
i=1
· 100% ≈ 99%.
σi2
P
2.2 Non-negative matrix factorisation (NMF)
Non-negative Matrix Factorization (NMF) is a decomposition of non-negative
matrix V ∈ Rn×m for a given number k into the product of two non-negative
matrices W ∈ Rn×k and H ∈ Rk×m such that
V ≈ W H. (2)
NMF is widely used in such areas as finding the basis vectors for images,
discovering molecular structures, etc. [16].
Consider the following matrix of ratings:
4450000
5 5 3 4 3 0 0
0 0 0 4 4 0 0
V = 0 0 0 5 4 5 3.
0 0 0 0 0 5 5
0000044
Its decomposition into the product of two non-negative matrices for k = 3
can be, for example, like this:
2.34 0 0
2.32 1.11 0
0 1.28 0
1.89 1.89 1.71 0.06 0 0 0
V = 0 1.46 1.23 · 0.13 0.13 0 3.31 2.84 0.27 0
.
0
0 0 0 0.03 0 3.27 2.93
0 1.60
0 0 1.28
2.3 Boolean Matrix Factorisation (BMF) based on Formal Concept
Analysis (FCA)
Basic FCA definitions. Formal Concept Analysis (FCA) is a branch of ap-
plied mathematics and it studies (formal) concepts and their hierarchy. The
adjective “formal” indicates a strict mathematical definition of a pair of sets,
called, the extent and intent. This formalisation is possible because the use of
the algebraic lattice theory.
Definition 1. Formal context K is a triple (G, M, I), where G is the set of
objects, M is the set of attributes , I ⊆ G × M is a binary relation.
An FCA-based Boolean Matrix Factorisation for Collaborative Filtering 61
The binary relation I is interpreted as follows: for g ∈ G, m ∈ M we write
gIm if the object g has the attribute m.
For a formal context K = (G, M, I) and any A ⊆ G and B ⊆ M a pair of
mappings is defined:
A0 = {m ∈ M | gIm for all g ∈ A},
B 0 = {g ∈ G | gIm for all m ∈ B},
these mappings define Galois connection between partially ordered sets (2G , ⊆)
and (2M , ⊆) on disjunctive union of G and M . The set A is called closed set, if
A00 = A [5].
Definition 2. A formal concept of the formal context K = (G, M, I) is a
pair (A, B), where A ⊆ G, B ⊆ M , A0 = B and B 0 = A. Set A is called the
extent, and B is the intent of the formal concept (A, B).
It is evident that extent and intent of any formal concept are closed sets.
The set of formal concepts of a context K is denoted by B(G, M, I).
Description of FCA-based BMF Boolean matrix factorization (BMF) is
a decomposition of the original matrix I ∈ {0, 1}n×m , where Iij ∈ {0, 1},
into a Boolean matrix product P ◦ Q of binary matrices P ∈ {0, 1}n×k and
Q ∈ {0, 1}k×m for the smallest possible number of k. We define boolean matrix
product as follows:
_k
(P ◦ Q)ij = Pil · Qlj ,
l=1
W
where denotes disjunction, and · conjunction.
Matrix I can be considered as a matrix of binary relations between set X
objects (users), and the set Y attributes (items that users have evaluated). We
assume that xIy iff user x estimated object y. The triple (X, Y, I) is clearly
composes a formal context.
Consider the set F ⊆ B(X, Y, I), a subset of all formal concepts of context
(X, Y, I), and introduce the matrices PF and QF :
1, i ∈ Al , 1, j ∈ Bl ,
(PF )il = (QF )lj =
0, i ∈
/ Al , 0, j ∈
/ Bl .
We can consider the decomposition of the matrix I into binary matrix product
PF and QF as described above. The following theorems are proved in [2]:
Theorem 1. (Universality of formal concepts as factors). For every I there is
F ⊆ B(X, Y, I), such that I = PF ◦ QF .
Theorem 2. (Optimality of formal concepts as factors). Let I = P ◦ Q for n × k
and k × m binary matrices P and Q. Then there exists a F ⊆ B(X, Y, I)
of formal concepts of I such that |F| ≤ k and for the n × |F | and |F | × m
binary matrices PF and QF we have I = PF ◦ QF .
62 Elena Nenova, Dmitry I. Ignatov, and Andrey V. Konstantinov
There are several algorithms for finding PF and QF by calculating formal con-
cepts based on these theorems [2].
The algorithm we use (Algoritm 2 from [2]) avoid the computation of all the
possible formal concepts and therefore works much faster [2]. Time estimation
of the calculation algorithm in the worst case yields O(k|G||M |3 ), where k is the
number of found factors, |G| is the number of objects, |M | this the number of
attributes.
Transform the matrix of ratings described above, to a boolean matrix, as
follows:
4450000 1110000
5 5 3 4 3 0 0 1 1 1 1 1 0 0
0 0 0 4 4 0 0
⇒ 0 0 0 1 1 0 0 = I.
0 0 0 5 4 5 3 0 0 0 1 1 1 1
0 0 0 0 0 5 5 0 0 0 0 0 1 1
0000044 0000011
The decomposition of the matrix I into the Boolean product of I = AF ◦ BF is
the following:
1110000 100
1 1 1 1 1 0 0 1 1 0
0 0 0 1 1 0 0 0 1 0
1110000
0 0 0 1 1 1 1 = 0 1 1 ◦ 0 0 0 1 1 0 0 .
0 0 0 0 0 1 1 0 0 1
0 0 0 0 0 1 1
0000011 001
This example shows that the algorithm has identified three factors that sig-
nificantly reduces the dimensionality of the data.
2.4 General scheme of user-based recommendations
Once the matrix of rates is factorized we need to learn how to compute recom-
mendations for users and to evaluate whether a particular method handles well
with this task.
For factorized matrices the already well-known algorithm based on the simi-
larity of users can be applied, where for finding K nearest neighbors we use not
the original matrix of ratings A ∈ Rm×n , but the matrix U ∈ Rm×f , where m
is a number of users, and f is a number of factors. After selection of K users,
which are the most similar to a given user, based on the factors that are peculiar
to them, it is possible, based on collaborative filtering formulas to calculate the
projected rates for a given user.
After formation of the recommendations the performance of the recommenda-
tion system can be estimated by measures such as Mean Absolute Error (MAE),
Precision and Recall.
An FCA-based Boolean Matrix Factorisation for Collaborative Filtering 63
3 A recommender algorithm using FCA-based BMF
3.1 kNN-based algorithm
Collaborative recommender systems try to predict the utility of items for a
particular user based on the items previously rated by other users.
Denote u(c, s) the utility of item s for user c. u(c, s) is estimated based on the
utilities u(ci , s) assigned to item s by those users ci ∈ C who are “similar” to user
c. For example, in a movie recommendation application, in order to recommend
movies to user c, the collaborative recommender system finds the users that have
similar tastes in movies with c (rate the same movies similarly). Then, only the
movies that are most liked by those similar users would be recommended.
Memory-based recommendation system, which are based on the previous
history of the ratings, are one of the key classes of collaborative recommendation
systems.
Memory-based algorithms make rating predictions based on the entire col-
lection of previously rated items by the users. That is, the value of the unknown
rating rc,s for user c and item s is usually computed as an aggregate of the
ratings of some other (usually, the K most similar) users for the same item s:
rc,s = aggrc0 ∈Cb rc,s ,
where Cb denotes the set of K users that are the most similar to user c , who
have rated item s. For example, function aggr may has the following form [1]
X
rc,s = k sim(c0 , c) × rc0 ,s ,
c 0 ∈C
b
sim(c, c0 ).
P
where k serves as a normalizing factor and selected as k = 1/
c0 ∈C
b
The similarity measure between users c and c0 , sim(c, c0 ), is essentially a
distance measure and is used as a weight, i.e., the more similar users c and c0
are, the more weight rating rc0 ,s will carry in the prediction of rc,s .
The similarity between two users is based on their ratings of items that
both users have rated. The two most popular approaches are correlation and
cosine-based. One common strategy is to calculate all user similarities sim(x, y)
in advance and recalculate them only once in a while (since the network of
peers usually does not change dramatically in a short time). Then, whenever the
user asks for a recommendation, the ratings can be calculated on demand using
precomputed similarities.
To apply this approach in case of FCA-based BMF recommender algorithm
we simply consider as an input the user-factor matrices obtained after factori-
sation of the initial data.
3.2 Scaling
In order to move from a matrix of ratings to a Boolean matrix, and use the
results of Boolean matrix factorization, scaling is required. It is well known that
64 Elena Nenova, Dmitry I. Ignatov, and Andrey V. Konstantinov
scaling is a matter of expert interpretation of original data. In this paper, we
use several variants of scaling and compare the results in terms of MAE.
1. Iij = 1 if Rij > 0, else Iij = 0 (user i rates item j).
2. Iij = 1 if Rij > 1, else Iij = 0.
3. Iij = 1 if Rij > 2, else Iij = 0.
4. Iij = 1 if Rij > 3, else Iij = 0.
4 Experiments
To test our hypotheses and study the behavior of recommendations based on the
factorization of a ratings matrix by different methods we used MovieLens data.
We used the part of data, containing 100,000 ratings, while considered only users
who have given over 20 ratings.
User ratings are split into two sets, a training set consisting of 80 000 ratings,
and test set consisting of 20 000 ratings. Original data matrix is 943× 1682, where
the number of rows is the number of users and the number of columns is the
number of rated movies (each film has at least one vote).
4.1 The number of factors that cover p% of evaluations in an input
data for SVD and BMF
The main purpose of matrix factorization is a reduction of matrices dimension-
ality. Therefore we examine how the number of factors varies depending on the
method of factorization, and depending on p % of the data that is covered by
factorization. For BMF the coverage of a matrix is calculated as the ratio of
the number of ratings covered by Boolean factorization to the total number of
ratings.
|covered ratings|
· 100% ≈ pBM F %, (3)
|all ratings|
For SVD we use the following formula:
K
σi2
P
i=1
· 100% ≈ pSV D %, (4)
σi2
P
where K is the number of factors selected.
Table 2. Number of factors for SVD and BMF at different coverage level
p% 100% 80% 60%
SVD 943 175 67
BMF 1302 402 223
An FCA-based Boolean Matrix Factorisation for Collaborative Filtering 65
4.2 MAE-based recommender quality comparison of SVD and
BMF for various levels of evaluations coverage
The main purpose of matrix factorisation is a reduction of matrices dimension-
ality. As a result some part of the original data remains uncovered, so it was
interesting to explore how the quality of recommendations changes based on dif-
ferent factorisations, depending on the proportion of the data covered by factors.
Two methods of matrix factorisation were considered: BMF and SVD. The
fraction of data covered by factors for SVD was calculated as
K
σi2
P
i=1
p% = P · 100%,
σi2
and for BMF as
|covered ratings|
p% = · 100%.
|all ratings|
To quality assessment we chose M AE.
Fig. 1. MAE dependence on the percentage of the data covered by SVD-decomposition,
and the number of nearest neighbors.
Fig. 1 shows that M AESV D60 , calculated for the model based on 60% of
factors, is not very different from M AESV D80 , calculated for the model built for
80% factors. At the same time, for the recommendations based on a Boolean
factorization covering 60% and 80% of the data respectively, it is clear that
increasing the number of factors improves MAE, as shown in Fig. 2.
Table 3 shows that the MAE for recommendations built on a Boolean fac-
torisation covering 80 % of the data for the number of neighbors less than 50 is
better than the MAE for recommendations built on SVD factorization. It is also
66 Elena Nenova, Dmitry I. Ignatov, and Andrey V. Konstantinov
Fig. 2. MAE dependence on the percentage of the data covered by BMF-
decomposition, and the number of nearest neighbors.
Table 3. MAE for SVD and BMF at 80% coverage level
Number of neighbors 1 5 10 20 30 50 60
M AESV D80 2,4604 1.4355 1.1479 0.9750 0.9148 0.8652 0.8534
M AEBM F 80 2.4813 1.3960 1.1215 0.9624 0.9093 0.8650 0.8552
M AEall 2.3091 1.3185 1.0744 0.9350 0.8864 0.8509 0.8410
easy to see that M AESV D80 and M AEBM F 80 are different from M AEall in no
more than 1 − 7%.
4.3 Comparison of kNN-based approach and BMF by Precision and
Recall
Besides comparison of algorithms using MAE other evaluation metrics can also
be exploited, for example
|objects in recommendation ∩ objects in test|
Recall = ,
|objects in test|
|objects in recommendation ∩ objects in test|
P recision =
|objects in recommendation|
and
2 · Recall · P recision
F1 = .
Recall + P recision
An FCA-based Boolean Matrix Factorisation for Collaborative Filtering 67
It is widely spread belief that the larger Recall, Precision and F1 are, the
better is recommendation algorithm.
Figures 3, 4 and 5 show the dependence of relevant evaluation metrics on
the percentage of the data covered by BMF-decomposition, and the number of
nearest neighbors. The number of objects to recommend was chosen to be 20.
The figures show that the recommendation based on the Boolean decomposition,
is worse than recommendations built on the full matrix of ratings.
Fig. 3. Recall dependence on the percentage of data covered by BMF-decomposition,
and the number of nearest neighbors.
4.4 Scaling influence on the recommendations quality for BMF in
terms of MAE
Another thing that was interesting to examine was the impact of scaling de-
scribed in 3.2 on the quality of recommendations. Four options of scaling were
considered:
1. I0,ij = 1 if Aij > 0, else Iij = 0 (user rates an item).
2. I1,ij = 1 if Aij > 1, else Iij = 0.
3. I2,ij = 1 if Aij > 2, else Iij = 0.
4. I3,ij = 1 if Aij > 3, else Iij = 0.
The distribution of ratings in data is on Figure 6
For each of the boolean matrices we calculate its Boolean factorisation, cov-
ering 60 % and 80 % of the data. Then recommendations are calculated just like
in 4.2. It can be seen that for both types of data coverage M AE1 is almost the
same as M AE0 , and M AE2,3 is better than M AE0 .
68 Elena Nenova, Dmitry I. Ignatov, and Andrey V. Konstantinov
Fig. 4. Precision dependence on the percentage of data covered by BMF-decomposition,
and the number of nearest neighbors.
4.5 Influence of data filtering on MAE for BMF kNN-based
approach
Besides the ability to search for K nearest neighbors not in the full matrix of
ratings A ∈ Rn×m , but in the matrix U ∈ Rm×f , where m is a number of users,
and f is a number of factors, Boolean matrix factorization can be used to data
filtering. Because the algorithm returns as an output not only matrices users-
factors and factors-objects, but also the ratings that were not used for factoring,
we can try to search for users, similar to the user, on the matrix consisting only
of ratings used for the factorization.
Just as before to find the nearest neighbors cosine measure is used, and the
predicted ratings are calculated as the weighted sum of the ratings of nearest
users. Figure 9 shows that the smaller the data we use for filtering the bigger is
MAE. Figure 10 shows that the recommendations built on user-factor matrix,
are better then recommendations, constructed on matrix of ratings filtered with
boolean factorization.
An FCA-based Boolean Matrix Factorisation for Collaborative Filtering 69
Fig. 5. F1 dependence on the percentage of data covered by BMF-decomposition, and
the number of nearest neighbors.
Fig. 6. Ratings distribution in data.
70 Elena Nenova, Dmitry I. Ignatov, and Andrey V. Konstantinov
Fig. 7. MAE dependance on scaling and number of nearest neighbors for 60% coverage.
Fig. 8. MAE depandance on scaling and number of nearest neighbors for 80% coverage.
An FCA-based Boolean Matrix Factorisation for Collaborative Filtering 71
Fig. 9. MAE dependance on percentage of covered with filtration data and the number
of nearest neighbors.
Fig. 10. MAE dependance on data filtration algorithm and the number of nearest neigh-
bors.
72 Elena Nenova, Dmitry I. Ignatov, and Andrey V. Konstantinov
5 Conclusion
In the paper we considered main methods of Matrix Factorisation which are suit-
able for Recommender Systems. Some of these methods were compared on real
datasets. We investigated BMF behaviour as part of recommender algorithm. We
also conducted several experiments on recommender quality comparison with nu-
meric matrices, user-factor and factor-item matrices in terms of Recall, Precision
and MAE. We showed that MAE of our BMF-based approach is not greater than
MAE of SVD-based approach for the same number of factors on the real data.
For methods that require the number of factors as an initial parameter in the
user or item profile (e.g., NMF), we proposed the way of finding this number
with FCA-based BMF. We also have investigated how data filtering, namely
scaling, influences on recommendations’ quality.
As a further research direction we would like to investigate the proposed
approaches in case of graded and triadic data [3,4] and reveal whether there are
some benefits for the algorithm’s quality in usage least-squares data imputation
techniques [19]. In the context of matrix factorisation we also would like to test
our approach on the quality assessment of recommender algorithms that we
performed on some basic algorithms (see bimodal cross-validation in [12]).
Acknowledgments. We would like to thank Radim Belohlavek, Vilem Vy-
chodil and Sergei Kuznetsov for their comments, remarks and explicit and im-
plicit help during the paper preparations. We also express our gratitude to Gul-
naz Bagautdinova; she did her bachelor studies under the second author super-
vision on a similar topic and therefore contributed somehow to this paper.
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