=Paper=
{{Paper
|id=None
|storemode=property
|title=Mining Concepts from Incomplete Datasets Utilizing Matrix Factorization
|pdfUrl=https://ceur-ws.org/Vol-972/paper3.pdf
|volume=Vol-972
|dblpUrl=https://dblp.org/rec/conf/cla/PiskovaPHK12
}}
==Mining Concepts from Incomplete Datasets Utilizing Matrix Factorization==
https://ceur-ws.org/Vol-972/paper3.pdf
Mining Concepts from Incomplete Datasets
Utilizing Matrix Factorization
Lenka Pisková1 , Štefan Pero1 , Tomáš Horváth2 , Stanislav Krajči1
1
ICS, Pavol Jozef Šafárik University, Košice, Slovakia
{Lenka.Piskova, Stefan.Pero}@student.upjs.sk
Stanislav.Krajci@upjs.sk
2
ISMLL, University of Hildesheim, Germany
horvath@ismll.de
Abstract. Formal Concept Analysis aims at finding clusters (concepts)
with given properties in data. Most techniques of concept analysis require
a dense matrix with no missing values on the input. However, real data
are often incomplete or inaccurate due to the noise or other unforeseen
reasons. This paper focuses on using matrix factorization methods to
complete the missing values in the input data such that it can be used
with arbitrary concept analysis technique. The used matrix factorization
model approximates the sparse object-item data matrix by a product of
two dense factor matrices, thus, mapping objects and items to a common
latent space. The mentioned object-factor and item-factor matrices are
obtained by a simple stochastic gradient optimization method. We also
investigate how the amount of missing values influences the output of the
concept analysis. Two measures, well-known in the information retrieval
community, have been used for the evaluation of the proposed framework.
Real datasets from the UCI Machine Learning Repository were used in
our experiments.
Keywords: Formal Concept Analysis, matrix factorization, missing val-
ues completion, clustering
1 Introduction
Formal Concept Analysis (FCA) deals with data in the form of a table which
rows represent objects and columns represent attributes of objects. A table entry
corresponding to an object x and an attribute y indicates whether or not the
object x has the attribute y. The clusters on the output of FCA are called
concepts, each of which consists of a set of formal objects and a set of formal
attributes, called the extent and the intent, respectively, of the given concept.
The set of all concepts ordered by ≤ forms a concept lattice [1]. The data table
required on the input of FCA has to contain all information, i.e. should not
contain missing values.
However, real datasets are often incomplete or inaccurate due to the damage
or inconsistency in the data collection process. Incomplete context was first
c 2012 by the paper authors. CLA 2012, pp. 33–44. Copying permitted only for private
and academic purposes. Volume published and copyrighted by its editors.
Local Proceedings in ISBN 978–84–695–5252–0,
Universidad de Málaga (Dept. Matemática Aplicada), Spain.
34 Lenka Pisková, Štefan Pero, Tomáš Horváth and Stanislav Krajči
introduced by Burmeister and Holzer [2] and applied to attribute implications
and attribute exploration. The analysis of incomplete data in the field of FCA
can be accomplished in one of the following two ways: The first one is to adapt
concept techniques to cope with missing values. Some research was conducted,
recently, to generate a concept lattice of an incomplete context [7]. The approach
presented in [8] is based on many-valued contexts and conceptual scaling. The
second way of handling incomplete data in FCA is to complete the missing values
in a pre-processing step and use a concept analysis on the pre-processed data.
The contribution of this paper follows the latter case described above, namely,
to predict the missing values using a matrix factorization method (MF) in the
pre-processing step such that FCA can be used on the completed data. MF, one
of the useful techniques in data mining, allows the decomposition of a sparse1
matrix into two dense matrices such that the product of these two dense matrices
results in a dense matrix which is an approximation2 of the original, sparse
matrix. We use stochastic gradient descent matrix factorization method (SGD
MF) in this paper, which is fast and effective. It is important to note that we need
to adjust the predicted values since these should be from the input values. The
main reason of it is that the resulting dense matrix is just an approximation
of the original one with specific values, e.g. 0 and 1 or 1, 2, 3, 4, 5, etc. The
resulting dense matrix is then scaled [1] and feed to FCA.
We also address the problem of the robustness of formal concepts, i.e. the
ability to remain unaffected by small variations in an input table. As far as we
know, there is no related work investigating the issue that how the concepts
computed from a full table differs from those computed from a pre-processed
table, i.e. what happens when we remove some values from the original table,
complete them with a pre-processing method and compute the concepts on the
pre-processed table? We think that this question is quite important for a real-life
application of FCA.
2 Related Work
One direction of estimating missing values in an input matrix is by the use of
association rule mining techniques consisting of two approaches: The first one
discards instances which contain missing data and generate association rules
from the remaining complete data. However, excluding instances with missing
data can bias the result. The second approach takes into account the presence
of missing values in the rule mining process. Note that two or more association
rules may result to different predicted values, thus the so called conflict problem
have to be tackled here. The large number of association rules and the effort to
reduce the conflict problem have led to the usage of generic bases of association
rules. We refer to [3], [4] for details and further references. The percentage of
correctly completed missing values is affected by the number of missing values.
1
We will call a matrix with missing values sparse.
2
The difference of the values in the non-empty cells of the original matrix and the
predicted values for these cells is minimal.
Mining Concepts from Incomplete Datasets Utilizing Matrix Factorization 35
A “hidden” problem is the percentage of missing values that these approaches
permits to complete.
[5] and [6] show how to use extracted knowledge represented by formal con-
cepts for completing a relation or at least augmenting it in the case when we
do not get each missing value. They generate concepts from the sparse matrix
such that they remove rows (objects) containing missing values. However, these
approaches have two drawbacks, namely, that the large number of concepts (also
for a small relation) makes more difficult to predict missing values, and, more
missing values leads to more biased completion.
Combinations of MF and FCA are introduced in [9], [10] and [11], where sin-
gular value decomposition, semi-discrete decomposition and non-negative matrix
factorization are used for reducing the number of formal concepts. These works,
however, use matrix factorization for a different purpose (reduction of the num-
ber of concepts) and consider full input matrix.
A novel approach to combine matrix decomposition and factor analysis is
presented in [12], [13]. There are two main differences from ordinary factor anal-
ysis. First, a table entry represents a degree to which an object has an attribute
(table entries are from a bounded scale L). Second, the matrix composition
operator is a so-called t-norm-based product. The aim of this approach is not
the completion of missing values but finding a set of factors that correspond to
formal concepts.
3 Preliminaries
We will just briefly describe FCA, and focus instead on a more detailed descrip-
tion of the used MF technique in this section since we think it could be more
interesting to the FCA community.
3.1 Formal Concept Analysis
A formal context is a triple (X, Y, I) consisting of two non-empty sets X and Y
and a binary relation I between them. The elements of X are called the objects
and the elements of Y are called the attributes. (x, y) ∈ I means that the object
x has the attribute y.
For a set A ⊆ X and a set B ⊆ Y , define A′ = {y ∈ Y : (∀x ∈ A)(x, y) ∈ I}
and B ′ = {x ∈ X : (∀y ∈ B)(x, y) ∈ I}. A′ is the set of attributes common to
the objects in A and B ′ is the set of object which have all attributes in B.
A formal concept of the context (X, Y, I) is a pair (A, B) of a set A ⊆ X of
objects and a set B ⊆ Y of attributes such that A′ = B and B ′ = A. A and B
are called the extent and the intent of the concept (A, B), respectively. Denote
the set of all concepts in (X, Y, I) by B(X, Y, I).
Introduce a relation ≤ on B(X, Y, I) by (A1 , B1 ) ≤ (A2 , B2 )) ⇔ A1 ⊆ A2 ⇔
B2 ⊆ B1 , (A1 , B1 ) is called the subconcept of (A2 , B2 ) and (A2 , B2 ) is called the
superconcept of (A1 , B1 ). The set of concepts B(X, Y, I) ordered by ≤ constitutes
the concept lattice (or Galois lattice) of the context (X, Y, I). The so-called main
36 Lenka Pisková, Štefan Pero, Tomáš Horváth and Stanislav Krajči
theorem of concept lattices characterizes the structure of concept lattices. For
further details we refer to [1].
3.2 Stochastic Gradient Descent Matrix Factorization
Stochastic gradient descent matrix factorization (SGD MF) is one of the most
popular factorization techniques [14] because of its scalability and good accuracy.
The goal of this approach is to approximate a sparse matrix X ∈ R|O|×|I|
with a product of two (much smaller) matrices W ∈ R|O|×K and H ∈ R|I|×K
such that
X ≈ X̂ = W H T , (1)
where |O|, |I| are the number of objects and items, respectively, and K is number
of (latent) factors. The ith row of the matrix W is a vector containing K latent
factors describing the object i, and the jth row in matrix H is a vector containing
K latent factors describing the item j [15].
The estimate x̂ij of a missing value at the row i and column j of the sparse
matrix X is computed as
K
X
x̂ij = (W H T )ij = wik hjk (2)
k=1
where wik (hjk ) is the value of W (H) at the ith (jth) row and kth column.
We are interested in such X̂ which estimates the missing values of X well.
Since we do not know the missing values, we use the following trick, well-known
in the machine learning community [18]: We split X into two complementary
parts, called the train set X train and the test set X test . The model X̂ is then
approximated from X train and it’s quality is assessed by the root mean squared
error (RMSE) loss computed on X test , defined as
v P
u
u (xij − x̂ij )2
t xij ∈X test
RM SE = , (3)
|X test |
where xij are the values of the non-empty cells of X belonging to X test . In this
way, we simulate the case when we have a sparse matrix (X train ) which missing
values (X test ) are known, however.
Our goal is to find those parameters W and H of the model X̂ for which the
RMSE is minimal. However, we have to keep in mind that X test is “hidden”, i.e.
it represents the “unknown“ missing values, thus, we have to use3 the data we
know, i.e. the train set X train . Since |X train | is constant,Pminimizing RMSE (3)
on X train is equivalent to minimizing the sum of errors xij ∈X train (xij − x̂ij )2 .
Since RMSE is a (loss) function, we use a stochastic gradient descent (SGD)
3
Here we also expect that X test has similar statistical characteristics as X train .
Mining Concepts from Incomplete Datasets Utilizing Matrix Factorization 37
optimization method [16] for searching those parameters W and H of RMSE
such that the following objective function is minimal:
X
(xij − x̂ij )2 + λ(kW k2 + kHk2 ) (4)
xij ∈X train
where x̂ij is defined as in (2) and λ is a regularization term to prevent the so-
called overfitting (i.e. when a model estimates very well the training data but
poorly the test data [18]). λ controls the magnitudes of the factor vectors such
that W and H would give a good approximation of the whole original input
matrix X (containing both X train and X test ). kW k2 means square of the one
vector in matrix W.
In the training phase we first initialize two matrices W and H with some
random values (for example from the normal distribution N (0, σ 2 ) with mean 0
and standard deviation 0.01) and compute the estimation error
X
err = e2ij (5)
xij ∈X train
where
K
X
e2ij = (xij − x̂ij )2 = (xij − wik hjk )2 (6)
k=1
We minimize this error by updating the values of W and H in the following
way [16]: we randomly choose a value xij from X train and compute the gradient
of the objective function (6) in this value w.r.t. the parameters4 wi and hj , i.e.
δ 2
e = −2eij hjk = −2(xij − x̂ij )2 hjk (7)
δwik ij
δ
e2 = −2eij wik = −2(xij − x̂ij )2 wik (8)
δhjk ij
In the next step we use the computed gradient to update the values of wik
and hjk :
(new) δ 2
wik = wik − β e = wik + 2βeij hjk (9)
δwik ij
(new) δ
hjk = hjk − β e2 = hjk + 2βeij wik (10)
δhjk ij
where β is the learning rate controlling the step sizes.
We update W and H for each value of X train in one iteration. The number
of iterations, i.e. the number of passes over the full X train is a hyper-parameter
of the factorization technique as well as the regularization term λ, the learn rate
β and the number of factors K.
4
Only the vectors wi and hj contribute to e2ij , thus we treat the other vectors (rows
of W and H) as constants (which derivatives are zero).
38 Lenka Pisková, Štefan Pero, Tomáš Horváth and Stanislav Krajči
As we see, the quality of the model X̂ depends on the hyper-parameters
used in the learning phase. Also, it can happen that the split of the data into
the train and the test part was done in a way that these two parts have no
similar characteristics, e.g. one part contains large values while the other the
low ones. To prevent this phenomenon to happen we usually use n-fold cross
validation [18] in which we split X into n equal parts X1 , . . . , Xn and for each
n and hyper-parameter combination (K, λ, β, iterations) do the following: train
a model X̂ n with the given hyper-parameter combination on ∪i6=n Xi and com-
pute its RMSE on Xn . The resulting RMSE of X̂ with the hyper-parameters
(K, λ, β, iterations) is then the average RMSE over all X̂ n . After trying some
hyper-parameter combinations with n-fold cross validation, we choose the best
combination (K, λ, β, iterations)best which has the smallest average RMSE over
all folds. Finally, we train the model X̂ again using the whole input matrix X.
It can happen, that the input matrix X contains an empty line or column
with no values. For those objects (rows) or items (columns) we will not update
the parameters W and H (it follows from the algorithm of SGD MF). In such
cases, estimated missing values could be the global average of non-missing values.
More advanced and sophisticated methods that can improve the estimation in
such cases are listed in [17].
4 Illustrative example
To illustrate our approach we present a small toy example. Our purpose is to
cluster students into groups with respect to their success in solving the tasks.
The key problem is that not every student have solved all of the tasks. We will
proceed as follows. We predict the success of students in tasks that they have
not solved yet (empty values in the input table). After the process of conceptual
scaling is completed, we find formal concepts.
The table in the left side of the figure 1 contains five students who have per-
formed several tasks. The total marks that can be scored are 9; 3 marks for each
task. Table values represent students’ marks for tasks. The matrix represents a
relation between objects and items (which are students and tasks in our case).
Figure 1 shows an example of how we can factorize the students and tasks.
After the training phase with K = 2 latent factors (F1 and F2), we get the
student-factor matrix W and the factor-task matrix H. Suppose we would like to
predict Tom’s (3rd row) performance for the task 2 (2nd column). The prediction
is performed using equation 2
K
X
x̂32 = w3k h2k = 1.80 ∗ 1.20 + 0.13 ∗ 1.08 = 2.30.
k=1
Since the predicted value should be included in the input matrix, we round
this value to one of the values 1, 2 and 3. We have predicted Tom will achieve
average results in task 2. Similarly, we can predict the performance of other
Mining Concepts from Incomplete Datasets Utilizing Matrix Factorization 39
X T1 T2 T3 W F1 F2
Katie 3 3 Katie 1.18 1.52 H T1 T2 T3
Dean 2 2 1 Dean 1.24 0.31
F1 0.98 1.20 0.85
Tom 2 2 Tom 1.80 0.13
F2 1.35 1.08 0.54
Sam 2 Sam 0.91 1.57
Adam 2 3 2 Adam 1.68 0.49
Fig. 1. Factorization of the input matrix.
students in tasks which they have not done yet. The full matrix is depicted in
the left side of the figure 2.
Now, we would like to cluster students according to their results in solving the
tasks. The table in the left side of the figure 2 represents many-valued context.
Since values are ordered and each value implies the weaker one, we use the ordinal
scale in the right side of the figure 2. Performing a task at the highest level implies
performing the task at the average level, too. The table in the figure 3 is the
result of the conceptual scaling process. For details on many-valued contexts and
conceptual scaling we refer to [1].
T1 T2 T3
Katie 3 3 2 3 2 1
Dean 2 2 1 3×××
Tom 2 2 2 2 ××
Sam 3 2 2 1 ×
Adam 2 3 2
Fig. 2. The many-valued context and the ordinal scale S for all attributes of the many-
valued context.
The obtained concept lattice contains 6 concepts (clusters) and is depicted in
the figure 3. A labeling is simplified by putting down each object and attribute
only once. The extent of each concept is formed by collecting all objects which
can be reached by descending line paths from the concept. The intent of each
concept consists of all attributes located along ascending line paths.
We interpret some of the clusters, here: Each student solved the task 1 and
task 2 at least at average level (the top element of the lattice). All students
except for Dean achieved at least average results in solving all tasks. Nobody
performed good results in every task (the bottom element of the lattice).
5 Experiments
Experiments were conducted on computing cluster with 7 computing nodes, each
of which has 16 cores, running Red Hat Linux.
40 Lenka Pisková, Štefan Pero, Tomáš Horváth and Stanislav Krajči
Dean ✉ T10.0 , T10.5
T20.0 , T20.5 , T30.0
T1 T2 T3
1 2 3 1 2 3 1 2 3 Tom ✉ T30.5
Katie × × × × × × × ×
Dean × × ×× × Sam ✉ T11.0 Adam ✉ T21.0
Tom × × ×× ××
Sam × × × × × ××
Adam × × ××××× Katie ✉
✉ T31.0
Fig. 3. The one-valued context as the derived context of the many-valued context in
figure 2 and the corresponding concept lattice.
5.1 Data
For these experiments, we consider a complete database to act as a reference
database, and we randomly introduce missing values with the following different
rates : 10%, 20% and 30%. Benchmark datasets used for this experiments are
from the UCI Machine Learning Repository5 . Characteristics of these datasets
are depicted in the table 1.
Table 1. Datasets characteristics
Dataset Objects Attributes Attribute types FCA attributes
Wine 178 13 Integer, Real 68
SPECT Heart 267 22 Categorical 46
Tic-tac-toe 958 9 Categorical 29
The data conversion into FCA format was done as follows. Categorical (many-
valued) attributes were converted by creating a formal context attribute for each
of the values. Attributes with integer or real types were scaled to create context
attributes with ranges of values.
5.2 Matrix factorization model settings
We implemented matrix factorization model in Java described in the section 3.2
on our own. We have used 3-fold cross-validation for testing the proposed model:
In each of the 3 iterations, one fold was used for testing. The remaining two folds
were used for tuning hyper-parameters of the model. Hyper-parameters (number
5
http://www.ics.uci.edu/mlearn/MLRepository.html.
Mining Concepts from Incomplete Datasets Utilizing Matrix Factorization 41
of factors, number of iterations, learn rate and regularization term) were tuned
using grid search [19] (a systematic search over different combinations). The
best hyper-parameter combinations for our model and the root mean squared
error (RMSE) was measured on the test fold for each dataset. The best hyper-
parameters are reported in the table 2.
Table 2. The best hyper-parameters found by grid search
Dataset Missing values (%) Factors Iterations Learn rate Reg. term
10 12 30 0.055 0.1
Wine 20 12 40 0.055 0.05
30 12 50 0.055 0.05
10 22 50 0.055 0.05
SPECT Heart 20 42 40 0.055 0.05
30 72 40 0.055 0.05
10 42 70 0.005 0.05
Tic-tac-toe 20 12 60 0.005 0.05
30 32 90 0.005 0.1
5.3 Results
Concept lattices from complete data sets were computed, the number of concepts
is shown in the table 3.
Table 3. Datasets and the number of concepts computed from the full matrix.
number of concepts
Wine 24423
SPECT Heart 2135549
Tic-tac-toe 59505
Table 4 presents the comparison of the RMSE measured over 3 folds and
average of success rates of the prediction for all datasets. Using matrix factor-
ization model we correctly predicted from 30% to 80% missing values. If we did
not predict a missing value correctly then the predicted value was close to the
original value.
In order to evaluate methods for completing missing values to mine formal
concepts, one have to compare concepts generated from the original data table
(initial context) with concepts generated from the data table containing esti-
mated values (predicted context).
42 Lenka Pisková, Štefan Pero, Tomáš Horváth and Stanislav Krajči
Table 4. RMSE and the average of success rates of the prediction
Dataset Missing values (%) RMSE Average of the success rates (%)
10 0.6944168 53.18
Wine 20 0.7345221 66,46
30 0.6972556 48,84
10 0.37255815 78,61
SPECT Heart 20 0.3785878 78,09
30 0.38591003 77,37
10 0.88052654 31,72
Tic-tac-toe 20 0.9031108 31,56
30 0.9076978 33,71
We propose to use two measures for evaluating the performance of the pro-
posed approach for the missing values completion with respect to mined con-
cepts. Both are well-known in the information retrieval (IR) community. The
first one indicates how many concepts of the initial context occur in the set of
concepts of the estimated context. The second one determines how many con-
cepts of the estimated context are in the set of concepts of the initial context.
Let (X, Y, I) be the initial formal context and (X, Z, J) be the estimated
formal context. Let O be the set of formal concepts of the initial context and
E be the set of formal concepts of the estimated one (O = B(X, Y, I) and
E = B(X, Z, J)). We propose to use the following two measures:
|O ∩ E| |O ∩ E|
precision = and recall = .
|E| |O|
Precision is the fraction of concepts of the estimated context that are in
the set of concepts of the initial context. Recall is the fraction of concepts of
the initial context that are in the set of concepts of the estimated context. The
higher the precision, the more concepts of the estimated context are in the set
of concepts of the initial context. The higher the recall, the more concepts of the
initial context are in the set of concepts of the estimated context.
Two formal concepts are equal when any formal object in the extent of one is
in the extent of the other and vice versa, the same for formal attributes. In this
case, precision and recall are computed. From the extent-ional point of view:
two concepts are equal when they have the same extent. Analogously, from the
point of view of intent, two concepts are equal when they have the same intent. In
these cases, we calculate precisione , recalle and precisioni , recalli , respectively.
The results of our experiments are shown in the table 4 and 5.
It is noteworthy that there is a huge difference between precisione and
precisioni . We assume that the gap between precisione and precisioni is caused
by the fact that the number of objects is greater than the number of attributes.
Moreover, the precision and recall are quite small. The reason is how we have
measured the equality of the sets: the sets are not equal if they differ even only
in one element. Nevertheless, the proposed measures are useful for experimental
Mining Concepts from Incomplete Datasets Utilizing Matrix Factorization 43
Table 5. Precision and recall.
Dataset Missing values (%) prec prece preci rec rece reci
10 0,0005 0,0277 0,2603 0,0005 0,0311 0,2920
Wine 20 0,0004 0,0246 0,2603 0,0004 0,0272 0,2883
30 0,0002 0,0194 0,2187 0,0002 0,0198 0,2238
10 6 × 10−5 0,0107 0,3775 4 × 10−6 0,0007 0,0257
SPECT Heart 20 4 × 10−5 0,0063 0,3667 3 × 10−6 0,0005 0,0279
30 2 × 10−5 0,0039 0,3346 2 × 10−6 0,0003 0,0267
10 0,0011 0,0239 0,3969 0,0002 0,0032 0,0524
Tic-tac-toe 20 0,0011 0,0132 0,3899 0,0001 0,0009 0,0258
30 0,0008 0,0139 0,3477 0,0001 0,0011 0,0282
comparisons of various techniques used in the pre-processing step. We will focus
on finding other measures. We suppose that if we modify measures to use the
similarity (of extents, intents and concepts) better results could be achieved.
6 Conclusion and future research directions
In this paper we have introduced framework for mining concepts from incom-
plete datasets. The proposed framework uses stochastic gradient descent matrix
factorization (SGD MF) in the pre-processing step and after completing the
missing values a formal concept analysis (FCA) is deployed. Using SGD MF
we are able to quite precisely predict values from a sparse data matrix without
having any background knowledge about the data. Two measures (precision and
recall) have been used for evaluating the presented framework.
The experiments on three datasets (with 10, 20 and 30 percent of missing
values) showed that the percentage of missing values does not influence the
prediction rate, and, that there is a large difference in the precision and recall
measures when computed on the extents and the intents of the resulting concepts,
respectively.
Our further research will focus on modifying the (precision and recall) mea-
sures with using the similarity of extents, intents and concepts instead of their
equality. The generation of all concepts from the huge incidence matrix is time
consuming and the number of concepts is very large. Using our matrix factoriza-
tion model we transform input matrix into more smaller latent factor matrices
W and H. These matrices describe objects and items according to latent factors.
We want to explore these matrices and to use them for reducing the number of
concepts.
Even if there are some remaining issues to investigate, experimental results
show that the proposed approach is promising and worth of further research.
Acknowledgements: This work was supported by the grants VEGA 1/0832/12
and VVGS-PF-2012-22 at the Pavol Jozef Šafárik University in Košice, Slovakia.
44 Lenka Pisková, Štefan Pero, Tomáš Horváth and Stanislav Krajči
References
1. B. Ganter, R. Wille: Formal concept analysis: Mathematical foundations. Springer,
1999, ISBN 3-540-62771-5.
2. P. Burmeister, R. Holzer: Treating Incomplete Knowledge in Formal Concept Anal-
ysis. In: B. Ganter, G. Stumme, R. Wille (Eds.): Formal Concept Analysis: State of
the Art, LNAI 3626, Springer, Heidelberg, 2005.
3. L. Ben Othman, S. Ben Yahia: Yet Another Approach for Completing Missing Val-
ues. CLA 2006, pages 155-169.
4. L. Ben Othman, S. Ben Yahia: GBARM V C : Generic Basis of Association Rules
Based Approach for Missing Values Completion. The International Journal of Com-
puting and Information Sciences (IJCIS), 2008.
5. S. Ben Yahia, Kh. Arour, A. Jaoua: Completing Missing Values Using Discovered
Formal Concepts. DEXA 2000, pages 741-751.
6. S. Ben Yahia, Kh. Arour.: Mining fuzzy formal concepts for completing missing
values. In Proceedings of the Intl. Symposium on Innovation in Information and
Communication Technology, Amman, Jordan, 2002.
7. M. Krupka, J. Lastovicka: Concept lattices of incomplete data. In: F. Domenach,
D.I. Ignatov, and J. Poelmans (Eds.): ICFCA 2012, LNAI 7278. Springer, Heidelberg,
pages 180-194, 2012.
8. J. Liu, X. Yao: Formal Concept Analysis of Incomplete Information System. In:
FSKDIEEE (2010), pages 2016-2020.
9. P. Gajdoš, P. Moravec, V. Snášel: Concept Lattice Generation by Singular Value
Decomposition, In: V. Snášel, R. Bělohlávek (Eds.): CLA 2004, pages 102-110.
10. V. Snášel, P. Gajdoš, H. Abdulla, M. Polovinčák: Using Matrix Decompositions in
Formal Concept Analysis. ISIM 2007, pages 121-128.
11. V. Snášel, H. Abdulla, M. Polovinčák: Using Nonnegative Matrix Factorization
and Concept Lattice Reduction to visualizing data. SYRCoDIS, 2007.
12. R. Bělohlávek, V. Vychodil: Factor Analysis of incidence data via Novel Decom-
position of Matrices, In: S. Ferré and S. Rudolph (Eds.): ICFCA 2009, LNAI 5548,
pages 83-97, 2009.
13. R. Bělohlávek, V. Vychodil: Discovery of Optimal Factors in binary data via
novel method of matrix decomposition. Journal of Computer and System Sciences
76(1)(2010), pages 3-20.
14. Y. Koren, R. Bell and C. Volinsky. Matrix Factorization Techniques for Recom-
mender Systems. COMPUTER (Vol. 42, 8): IEEE Computer Society Press, Los
Alamitos, USA, 2009.
15. G. Takács, I. Pilászy, B. Németh. Investigation of Various Matrix Factorization
Methods for Large Recommender Systems. KDD Netflix workshop, 2008.
16. L. Bottou. Stochastic learning. In Bousquet, O. and von Luxburg, U., editors,
Advanced Lectures on Machine Learning, Lecture Notes in Artificial Intelligence,
LNAI 3176, pages 146-168. Springer Verlag, Berlin.
17. Z. Gantner, L. Drumond, C. Freudenthaler, S. Rendle, L. Schmidt-Thieme. Learn-
ing attribute-to-feature mappings for cold-start recommendations. In Proceedings of
the 10th IEEE International Conference on Data Mining (ICDM 2010). IEEE Com-
puter Society, 2010.
18. T. Hastie, R. Tibshirani, J. Friedman: The Elements of Statistical Learning: Data
Mining, Inference, and Prediction. Second Edition, Springer Series in Statistics, 2009.
19. J. Bergstra, Y. Bengio: Random Search for Hyper-Parameter Optimization. J.
Machine Learning Research 13, pages 281-305, 2012.