# TaBIIC: Taxonomy Building through Iterative and Interactive Clustering

Mathieu d’Aquin0000-0001-7276-4702 Corresponding Author: Mathieu d’Aquin,
<EMAIL_ADDRESS>K Team, LORIA CNRS/INRIA/Université de Lorraine,
Nancy, France

###### Abstract

Building taxonomies is often a significant part of building an ontology, and
many attempts have been made to automate the creation of such taxonomies from
relevant data. The idea in such approaches is either that relevant definitions
of the intension of concepts can be extracted as patterns in the data (e.g. in
formal concept analysis) or that their extension can be built from grouping
data objects based on similarity (clustering). In both cases, the process
leads to an automatically constructed structure, which can either be too
coarse and lacking in definition, or too fined-grained and detailed, therefore
requiring to be refined into the desired taxonomy. In this paper, we explore a
method that takes inspiration from both approaches in an iterative and
interactive process, so that refinement and definition of the concepts in the
taxonomy occur at the time of identifying those concepts in the data. We show
that this method is applicable on a variety of data sources and leads to
taxonomies that can be more directly integrated into ontologies.

###### keywords:

Applications and Methodsbottom-up ontology constructiontaxonomies

,

##  1 Introduction

Taxonomies are an integral part of ontologies, whether they represent an
aspect of highly axiomatized ontologies in OWL [13], or simpler concept
structures such as those constructed using SKOS [14]. While ontology building
methodologies (see, for example, [9, 19, 20]) focus on identifying and
defining core concepts, understanding how to hierarchically divide such
concepts is often not trivial and is not as well covered by those
methodologies. In addition, when the aim is to use such a taxonomy to classify
entities into the corresponding categories from existing data, it is important
to ensure that the definitions of concepts and subconcepts align well with the
attributes and groups that exist in the data, as well as with what is
meaningful in the domain.

For these reasons, some of the tools aiming to support ontology building have
focused on (semi)automatic ways to construct such taxonomies in a bottom-up
manner, that is, from existing data. We describe such works in more detail in
the following section, focusing on approaches taking structured data as input
(i.e., ignoring the many approaches building taxonomies from text). In
summary, however, depending on the specific technique applied, those
approaches suffer from a number of drawbacks that significantly limit their
applicability. Some, for example, are constrained to using only nominal data
as input and come up with very large and complex hierarchies that require to
be pruned. Others rely on grouping data objects together, coming up with flat
structures, or hierarchies and definitions that are difficult to reconnect
with intensional definitions of concepts.

Taking inspiration from these techniques, we explore here the use of an
iterative and interactive method for building taxonomies from data. The idea
is to enable the user to edit and label subconcepts in the taxonomy as they
are being created, therefore, ensuring that what is created is aligned with
both the domain and the data. We describe the principles and implementation of
a tool called TaBIIC (Taxonomy Building through Interactive and Iterative
Clustering) and show how it applies to a variety of data inputs to create
taxonomies that match the user’s view of the domain, are ready (compatible in
format and structure) for integration in ontologies, and are aligned with the
source data.

In the following (Section 3) we consider common approaches to building
taxonomies from structured data. The form of structured data and of the
taxonomies created can vary from one approach to the other, even if
similarities exist. Therefore, we first introduce some notation and elements
of vocabulary in relation to datasets and concept taxonomies (inspired by
formal concept analysis and description logics) that can be used as a shared
framework to describe existing work.

##  2 Preliminaries

As mentioned above, we consider the semiautomatic construction of taxonomies
from structured data. Although structured data can come in a wide variety of
shapes and formats, we restrict ourselves to a common and widely available
one: data tables or data frames.

######  Definition 1 (Dataset)

A dataset $D=(O,A)$ is defined as a set of objects $O$ (lines), described
through a list of attributes $A$ (columns). Each object $o\in O$ is described
as a vector of size $|A|$ of values for each attribute $a\in A$.

At this stage, the definition of datasets is relatively unconstrained. In
particular, there is no restriction on the types of the values used to
describe each object of the dataset. As we will see later, most approaches to
taxonomy construction from structured data however have limitations regarding
the types of values they can take as input. In practice, it is also often
assumed that all the values for a given attribute are of the same type.

The notion of concept is widely used in various areas, including description
logics, formal concept analysis, ontologies, and machine learning, to
represent a set of entities either present in the data or in a wider
interpretation domain. Here, we adopt a definition inspired by formal concept
analysis.

######  Definition 2 (Concept)

A concept in a dataset $D=(O,A)$ is defined as a pair $(I,E)$. $I$, the
intension, is a set of necessary and sufficient conditions for an object $o\in
O$ to be considered a member of the concept. $E$, the extension, is defined as
the set of all objects in $O$ that validate all conditions in $I$ and
therefore belong to the concept.

In other words, a concept is considered a set of objects from the dataset (the
interpretation domain) that is characterized by a set of conditions (the
intension). Those conditions naturally relate to the values of the attributes
that are used to describe each object in the dataset. We remain vague on the
definition of those conditions at this point as it is another of the aspects
that is specific to different approaches to taxonomy building from structured
data.

While concept taxonomies generally include both the notion that a concept
represents a set of objects (extension) and that they are associated with a
given definition (intension), some approaches do not go as far as to make the
intensional part of concepts explicit. We therefore here make use of a
slightly different, weaker notion corresponding to groups of data objects
defined purely extensionally (i.e. by enumeration).

######  Definition 3 (Cluster)

A cluster $C\subseteq O$ in a dataset $D=(O,A)$ is defined as a set of data
objects from $O$ that are explicitly stated as being grouped together.

Clusters can be built based on various criteria, but those criteria are not
expressed in the form of a set of conditions on the values of attributes of
objects.

Finally, a taxonomy is more than a set of concepts: It organizes them in a
hierarchy from more general to more specific.

######  Definition 4 (Concept subsumption)

A concept $A=(I_{A},E_{A})$ is said to subsume ($B\sqsubseteq A$) a concept
$B=(I_{B},E_{B})$ if and only if $E_{B}\subseteq E_{A}$.

In other words, concept subsumption corresponds to the notion of a
generalization relation between concepts. Following this definition, if a
concept subsumes another, its intension will also be less constraining (with a
lower number of or weaker conditions). The base notion of concept subsumption
as defined through the inclusion of extensions naturally also applies to
clusters.

######  Definition 5 (Taxonomy)

In this work, a taxonomy is defined as a set of concepts, each associated with
a label, and organized according to the subsumption relation.

In other words, we consider a taxonomy to be a hierarchy of concepts, with
every concept being given a name.

##  3 Related Work

Based on the definitions above, we consider the task of semiautomatic taxonomy
building from structured data as one that takes as input a dataset and
produces as output a structure as close as possible to a taxonomy. We consider
three prominent families of approaches here, i.e. those based on formal
concept analysis, on clustering, and on decision trees. Although these
approaches have also been used for the bottom-up construction of taxonomies
from text (see, for example, [4, 5]), the corresponding methods do not apply
directly to structured datasets as considered here. We therefore do not
discuss those works here.

###  3.1 Approaches based on formal concept analysis

Formal concept analysis (FCA) [11], broadly summarized, consists in building
concept hierarchies from formal contexts, which are representations of data
objects with binary attributes. In other words, in formal concept analysis,
the input dataset $D=(O,A)$ is such that any $o\in O$ is represented by a
binary vector.

The hierarchy built through formal concept analysis is formally a lattice in
which a concept’s intension is a binary vector, an itemset, indicating which
attribute has to be true for a data object to belong to the extension. Those
itemsets are said to be closed in the sense that all the elements of the
extension of a concept have to contain the itemset in the intension (by
definition), but also that no other attribute (item) should be shared by all
elements of the extension in addition to the ones in the intension of the
concept. In other words, if in a dataset $(O,A)$, we call $I^{\prime}\subseteq
O$ all objects that share all the items of the itemset (intension) $I$, and
$E^{\prime}\subseteq A$ all the attributes (item) shared by all the objects of
$E$ (extension), then the concept $(I,E)$ is closed if $I^{\prime\prime}=I$
and $E^{\prime\prime}=E$.

Since it is building concept hierarchies from structured data, FCA has
naturally been applied multiple times to support the semiautomatic creation of
taxonomies. In [6] for example (applied to the field of data licenses in [7]),
it is used to create an initial hierarchy of concepts from the data, which is
then manually pruned and labeled to form the core of an ontology. Many other
works have made use of FCA to build taxonomies or parts of ontologies [2, 15],
or to align and merge them [12, 18].

From these works, we can see that the principles of FCA represent a strong
foundation for semiautomatically building taxonomies from data. Indeed, apart
from the labels of concepts, which would have to be entered manually, what is
produced perfectly fits the definition of a concepts taxonomy as considered
above. One of the key issues however is that all possible closed concepts are
created, which often leads to very large and very complex hierarchies, from
which it is difficult to figure out what is relevant and what can be
discarded. In other words, editing a concept lattice created through FCA can
potentially take more effort than manually building the taxonomy.

Furthermore, FCA suffers from a number of technical limitations. First, it
requires that objects in the dataset are described through binary attributes.
Although categorical data can always be encoded and numerical data
discretized, a certain level of precision can be lost in the process. In
addition, those steps of encoding and discretization significantly increase
the number of attributes in the data, and FCA is known to have issues in
scaling both in number of objects and in number of attributes. As a result,
FCA is really only applicable to relatively small datasets of low
dimensionality.

###  3.2 Approaches based on clustering

Clustering represents a family of machine learning techniques to
automatically, in an unsupervised way, find groups (clusters) of data objects
in a dataset, based on the values of their attributes [17]. Most clustering
techniques are based on measuring the similarity of data objects and aim to
create homogeneous clusters, that is, clusters of data objects similar to each
other and dissimilar with those outside the clusters.

The majority of clustering techniques (including popular ones such as K-Means
and all its variants) are not suitable for building taxonomies since their
output corresponds to a set of clusters (adhering to the definition given
previously) that are not organized in a hierarchy. Despite this, some works
such as [3] have applied these techniques to support the creation of
taxonomies. In addition, methods such as agglomerative clustering [1] (a.k.a.,
bottom-up hierarchical clustering) are a particular set of techniques that
rely on the creation of a hierarchy of clusters in which every cluster is a
set of data objects that is a subset of its parent cluster. Those hierarchies
are therefore based on a relation that corresponds to the one of subsumption
introduced earlier.

As an example of an application of hierarchical clustering for taxonomy
building, in [10], the authors use this approach on a dataset of articles on
the topic of design research, as a starting point for the creation of a
taxonomy of design research issues. In [21], a simmilar process is used to
build a taxonomy of IT subjects.

An obvious disadvantage of the use of (hierarchical) clustering for building
taxonomies is that those methods produce clusters and not concepts, i.e., they
do not produce an intensional definition for the groups of data objects they
create. The clusters are formed on the basis of a notion of similarity that
can be difficult to reconcile with providing clear definitions (necessary and
sufficient conditions) for the set of objects they represent.

In addition, hierarchical clustering can suffer a similar issue to FCA: It
naturally produces a large number of clusters in a hierarchy that can
therefore be very deep. Sorting relevant from irrelevant clusters, editing
them to reflect useful categories in the domain, and labeling them can,
therefore, require significant effort.

However, on the technical side, those clustering techniques are less
constrained than FCA: They can deal with a wider variety of attribute types
and larger datasets. It is worth mentioning still that most techniques only
work on numerical inputs (so nominal attributes need to be encoded), and
remain slow and memory heavy on large datasets.

###  3.3 Approaches based on decision trees

Decision trees are a popular form of machine learning models and are often
used in classification and regression tasks. The main idea is to progressively
divide the dataset into subsets based on basic conditions on the values of
attributes in the dataset, in order to create subsets that are increasingly
more homogeneous with respect to a given target variable (see, for example,
[16]). Interestingly, nodes in a tree therefore correspond to the given
definition of concepts, since they represent sets of data objects (extension)
with an intensional definition (the condition attached to the node, in
conjunction with all the conditions of the parent nodes). The relation on
which the tree is built also corresponds to the one of subsumption, and adding
labels to those nodes would achieve making it a concept taxonomy.

A key issue with decision trees is that they are designed to be used in
supervised machine learning tasks. This means that the construction of the
decision tree, the successive splittings of the dataset, is driven by the goal
of being able to predict the value of a given output variable. This was used,
for example, in [8] for the related task of concept induction. In this case,
however, decision trees were used to automatically derive intensional
definitions (in description logic) of concepts for which the label and
extension were already known. We consider here a broader case where we are
trying to build a taxonomy that is not tailored to a specific task or a
specific objective.

###  3.4 Filling the gap

Based on the review of the three categories of approaches described above, our
goal is to take inspiration from them in order to build a method to create,
with little effort from the human user, taxonomies that can be directly
integrated into ontologies. In other words, the concepts in those taxonomies
should be defined intentionally and divide the dataset hierarchically in a way
that is driven by the data in input.

To achieve that, we take an interactive and iterative approach: Instead of
building an entire structure that needs to be heavily edited by the user in a
second step, we take inspiration from decision trees in the way they
progressively subdivide the dataset. Contrary to decision trees, however,
first, this division is not driven by a specific task but by grouping data
objects in homogeneous clusters. Importantly, the user can (and is encouraged
to) intervene on the emerging taxonomy at any step to label the clusters
created and provide an intensional definition for them, therefore making them
concepts. These definitions, in the form of conditions on the values of
attributes, have a direct effect on the extension of the concepts (which data
objects are being grouped) and, therefore, on subsequent divisions. By
enabling the user to carry out small editing actions at the time when the
taxonomy is being built, we avoid creating a large, complex structure
requiring a much more significant editing effort afterwards.

##  4 Interactive and Iterative Approach

We expect the user of the process being considered here to be someone who is
trying to build a taxonomy of a particular type of entities, so that the
taxonomy is meaningful, useful, and well aligned with available data
describing some of those entities. Also, while the data in question should be
broadly understandable by this user, it might be too broad or complex to
enable them to build such a taxonomy by manually exploring it. Further, while
there might be multiple uses of such a taxonomy, the one we are aiming to
support is its integration into an ontology, i.e. that it will form one
component/one part of an ontology in which the type of entities described by
the data is a core concept.

To achieve this and illustrate the process of iterative and interactive
clustering for taxonomy building from data, we take a simple but realistic
example of a dataset of characters from comic
books.111https://github.com/rfordatascience/tidytuesday/blob/master/data/2018/2018-05-29/week9_comic_characters.csv
This dataset takes the form of a table (a CSV file) where each line represents
a comic book character, and each column a descriptor for those characters.
Therefore, it naturally fits our definition of a dataset $D=(O,A)$ where $O$
is a set of data objects that each represents a character, and $A$ is a set of
attributes. There are more than 20,000 characters described in this dataset
($|O|>20,000$) through 11 attributes ($|A|=11$). Those attributes include the
publisher of the comic books where the characters appear (DC comics or
Marvel), whether they are good or bad, their gender, hair and eye colors, date
of first appearance, etc. Our objective here is therefore to build a taxonomy
of comic book characters that is built from and aligned to those data, while
having control over the process.

The idea of iterative and interactive clustering is that the dataset can be
divided progressively, in a top-down manner, through a clustering method, and
the results of those basic divisions be editable by the user at the time they
are created. We therefore start with one concept, which will include as an
extension the whole set of objects in the dataset (all the comic book
characters), and with an empty intension.

From this step, the user can divide this initial concept into two parts using
an automatic clustering method such as K-Means. What is obtained are two
clusters that are based on grouping objects through their similarity.
Inspecting each of the clusters, the user might notice that one contains most
of the Marvel characters and the other most of the DC characters. Other
characteristics are different between those two clusters, such as that one has
more characters with public identities and the other has more with secret
identities, or that the average year of first appearance of characters in one
is earlier than in the other.

Having considered the characteristics of the created cluster, the user can
decide to give them a proper definition in the form of conditions on the
values of their attributes. In our case, this would mean, for example, fixing
the intention of the first cluster by adding the condition that the value of
the publisher attribute is equal to “Marvel”, therefore transforming the
cluster created automatically into the concept of “Marvel Character” (which
the user might decide to label it at this point).

As the intension of a concept is being fixed, two things happen. First, the
extension of the concept is recalculated so to correspond to the intension,
hence here placing all and only Marvel characters in the newly defined
concept. Second, the cluster present at the same level is automatically given
an intension that corresponds to the complement of the one just created. In
this case, there is only one other value for the attribute publisher: DC
Comics. Therefore, the other cluster is transformed into a concept with one
condition in its intension: $publisher=DC$.

At the next iteration, the user can choose one of the newly created concepts
and repeat the process. Dividing in the same way the concept “Marvel
Character”, we might obtain a cluster where the majority of characters are
“Bad Characters” (attribute $align$) and another where the majority are “Good
Characters”. Hence, the concept “Marvel Villain” can be created by fixing the
value of $align$ to $bad$ from the first cluster. Here, there is another value
for the attribute $align$: $changing$. Therefore, the counterpart to “Marvel
Villain” is created, with a condition in its intension indicating that the
value of the attribute $align$ is either $good$ or $changing$. This concept
might be kept as is and named by the user, or further refined by restricting
it to only one of the values (e.g., $good$, for the concept “Marvel Hero”),
leading to the automatic creation of a third concept, complement of the two
others, with $changing$ as value for the $align$ attribute.

While the two examples above relate to attributes with nominal values, the
same principle is used when the definition of a concept is based on a numeric
attribute (such as the year of first appearance), with conditions indicating
whether the value of the attribute is greater than or lower than a certain
value. In this way it should be possible to distinguish for example “first
generation Marvel heroes” from those of later generations.

The pseudocode in Algorithm 1 summarizes the process informally illustrated
above. In this code, the constructed hierarchy is represented by a root node
with four elements: the set of objects that the node represents (the
extension), the conditions defining the node (the intension), the label of the
node, and the list of its descendants. Clusters are created with empty
intensions, labels, and descendants. Creating a condition in line 10
corresponds to the user choosing an attribute $a$ from $A$ and a set of
possible values for it if $a$ takes nominal values, or a treshold in the
values if $a$ takes numeric values.

Algorithm 1 Pseudocode of the process for taxonomy building from data

1:$D=(O,A)$ $\triangleright$ The dataset.

2:$H\leftarrow(O,\emptyset,\\_,\emptyset)$ $\triangleright$ The hierarchy

3:while $not~{}done$ do

4: $C\leftarrow$ a leaf concepts chosen by the user from $H$

5: $Ac\leftarrow$ the user selected action on $C$

6: if $Ac$ is $cut$ then

7: $C1,C2\leftarrow$ results of clustering the extension of $C$

8: Set $(C1,\emptyset,\emptyset)$ and $(C2,\emptyset,\emptyset)$ as
descendants of $C$ in $H$

9: else if $Ac$ is $define$ then

10: $c\leftarrow$ condition created by the user for $C$

11: Add $c$ to the intension of $C$

12: Recalculate the extension of $C$

13: Replace all not explicitly (by the user) defined siblings of $C$ by the
complement of $C$ and its explicitly defined siblings

14: else if $Ac$ is $label$ then

15: $l\leftarrow$ user entered label fo $C$

16: Set the label of $C$ to $l$

17: end if

18:end while

As visible here, this process is designed to enable quickly identifying
candidate concepts from potentially large data, but leaves the user in control
of the definition of those concepts at every step, therefore minimizing the
editing effort required, and ensuring that the resulting taxonomy is aligned
with both the knowledge of the user and with the data from which it was built.

##  5 Implementation

A proof of concept implementation of the process described above is available
online222https://github.com/mdaquin/TaBIIC (see screenshot Figure 1). It takes
the form of a tool called TaBIIC (Taxonomy Building through Iterative and
Interactive Clustering), which allows a user to load a dataset in the form of
a CSV file, select the variables to be considered in the construction of the
taxonomy, divide the dataset and already formed concepts through clustering,
provide intensional definitions for those concepts and label them.

Figure 1: Interface of the TaBIIC tool applied to the comic book character
dataset. The dataset has first been divided by publisher (DC or Marvel). We
then divided DC Characters based on the date of first appearance. Marvel
characters were divided between heroes and villains, with heroes being further
divided based on whether they have a public, secret or mixed public/secret
identity. Finally, Marvel villains were divided according to whether or not
they are alive in the most recent comic book series.

In this prototype, a number of choices have been made to simplify the role of
the user and decrease the amount of effort required. This includes pre-
processing the data so that rows with missing values are removed, numeric data
are processed, and identifiers are automatically excluded from the variables
used in the taxonomy building process. This also includes providing a single
pre-configured approach for dividing concepts into clusters. Here, we use the
K-Means algorithm with the parameter $k$ (number of clusters to create) set to
2. Although this algorithm might not always give perfect results, the fact
that the user always has the option to correct those results by specifying the
intensional definition of concepts makes these imperfections less impactful.

Among other useful features, TaBIIC includes a panel on the left of the
interface to display details of the values of attributes for any selected
cluster/concept. This includes, first, any intensional definition not
inherited from parent concepts (in Figure 1, that the attribute $id$ takes the
values $No~{}Dual~{}Identity$ or $Known~{}To~{}Authorities~{}Identity$). This
is followed by statistics for each attribute, that is, average and standard
deviation for numeric attributes or the most frequent value and its frequency
for nominal attributes. The attributes here are ranked based on the difference
in those statistics with sibling concepts/clusters. In other words, this puts
first attributes that seem to best characterize the objects in the concept or
cluster, as opposed to those outside of it.

Editing the intensional definitions of concepts is done through first
selecting, in the panel described above, the attribute on which the user
wishes to set a condition. If the attribute is nominal, the list of possible
values for it can be entered by separating each value with the keyword $OR$.
The input field is pre-filled with the most frequent value in the
concept/cluster. If the variable is numeric, the user can enter either the
target average value in the concept or the threshold below or above which an
object will be considered part of the concept (with the dependency between
those two values being automatically calculated, see Figure 2 for an example).
The input field for the average is pre-filled with the average value of the
attribute in the cluster/concept currently being edited.

Figure 2: Interface for entering a condition for a numeric variable. In the
example here, we are looking to define the concept of DC characters having an
average year of first appearance of 1965, which corresponds to the DC
characters in the dataset who appeared first before 1988. It is also specified
in the interface that those represent 47% of the DC characters.

From a technical point of view, the backend of the tool is written in Python,
using the libraries scikit-learn for K-Means and pandas for data manipulation,
and flask as a web framework. The interface is written in HTML/CSS/Javascript.

##  6 Evaluation and Discussion

As described previously, we expect the interactive and iterative approach
implemented in the TaBIIC tool to reduce the effort required to come up with a
taxonomy that is valid, aligned with the data and that can easily be
integrated into an ontology. We also expect it to be applicable efficiently to
a variety of different datasets. To provide initial tests of these
assumptions, we applied the tool on three datasets of varied sizes and
richness. We discuss the results in the following.

The first dataset on which we tested TaBIIC is the toy IRIS
dataset333https://archive.ics.uci.edu/ml/datasets/iris often used as an
example to illustrate machine learning methods. This dataset includes the
description of 150 iris flowers according to the length and width of their
petals and sepals. Each flower is associated with one of three species of iris
(50 flowers per species).

The dataset being very simple, building a taxonomy for it with TaBIIC did not
raise much challenge and computation time was in particular negligible at
every step. In applying TaBIIC, we naturally excluded the variable
corresponding to the species. The resulting taxonomy is shown in Figure 3, and
consisted of first separating (based on the most differentiated attribute, and
using the suggested value) the flowers according to the length of their
petals, and then further dividing the “long petal irises” based on the width
of their petal. Interestingly, the leaf nodes of the created taxonomy
correspond almost exactly to the species of iris as classified in the dataset.

Considering that this case is far from challenging, we can compare it with
another approach, which could be too cumbersome to apply on datasets with more
dimensions. In particular, we used bottom-up hierarchical clustering
(agglomerative clustering) to create a hierarchy based on the similarity
between the values of the four variables that describe each iris flower444We
do not apply FCA here since, even on such a small scale example, the added
complexity from having to transform attributes into ones with binary values,
therefore multiplying their numbers, would require significant computation and
necessarily lead to results that are cumbersome to exploit.. The result is
shown in Figure 3 next to the TaBIIC-created taxonomy. While it is obvious
that the result from hierarchical clustering is much less interpretable than
the one obtained with TaBIIC, it is not a fair comparison since its
construction would generally be followed by a level selection step, reducing
it to the top levels of the hierarchy at a depth of the user’s choosing.
Depending on the configuration of the algorithm, cutting it at level 2 in this
case could lead to similar results as provided by TaBIIC. However, choosing
this level and understanding the content and possible definition of the
clusters produced would certainly require more effort even in this simple
case.

Figure 3: TaBIIC-created taxonomy (left) and hierarchy from agglomerative
clustering (right) for the iris dataset.

The second dataset on which we tested is the comic book character dataset that
we have already described. While it contains more than 20,000 rows, once rows
with missing values are removed, only about 9,500 remain. This dataset is more
realistic, as it is larger in scale, but also since it includes more
attributes, with a mix of nominal and numeric values. Because of this, there
is more user choice involved in the construction of the taxonomy. As shown
earlier, the first division naturally separates characters from Marvel comics
from characters from DC comics. Following this, whether characters are good or
bad, have a secret identity, or their gender tend to appear as highly divisive
attributes, but the user is free to construct the taxonomy on whichever
criteria they choose based on the information provided.

Since the dataset is larger, the computation time increases significantly,
although K-means clustering in the worst case (the first division) did not
take more than a few seconds. The hierarchy created by agglomerative
clustering cannot be shown here as it is extremely large. It took several
minutes to display on a recent laptop. Also, it is clear, considering the
choices made during the construction of the taxonomy with TaBIIC that, first,
choosing the right level at which to cut this hierarchy would be far than
trivial, and second, that the results might require extensive adjustments to
match the need of the user.

The last dataset555https://www.kaggle.com/datasets/antoinekrajnc/soccer-
players-statistics we tested contains many descriptors (more than 50) of
soccer players (including mostly statistics and ratings for different aspects
of the game). It is large (over 17,000 rows, even once rows with missing
values are removed), and most of the attributes are numeric. It is a much more
challenging dataset in the sense that it can be difficult to understand which
attribute to select when creating a condition. This is not helped by the fact
that many of those attributes are highly correlated, which means that choosing
one over the other is unlikely to make a significant difference.

To start with the technical aspects, despite working on a more challenging
dataset, the tool remains usable with the soccer dataset. Dividing even the
root node takes about half a minute on a recent laptop, and other operations,
while slower, remain within acceptable times. It is worth mentioning that we
did not manage to obtain a hierarchy from agglomerative clustering on this
dataset.

Despite being challenging and therefore requiring more effort, our tests show
that it is possible, sometime after multiple attempts through trial and error,
to structure the dataset with TaBIIC to differentiate, for example, highly
rated from lower rated players, older from younger players, or players
stronger in defense from those stronger in attack, and to combine those
criteria to create a reasonable taxonomy. An interesting aspect here is that
this process is not only helpful in creating the taxonomy, but also provides a
way for the user to explore the dataset and gain an understanding of its
content.

Based on the results of those tests, we believe that the approach presented
here is not only feasible but represents an efficient (in terms of computing
time and user time) approach to building taxonomies from data, which can be
applied on a variety of datasets. As mentioned earlier in this paper, the
objective was also that the taxonomy created could easily be integrated into
an ontology. The definition we relied on for concepts is indeed consistent
with the one used in ontology formalisms such as OWL, especially those based
on description logics. In addition, the types of conditions used for nominal
attributes can naturally be represented in those formalisms through value-
based restrictions and concept unions. The conditions based on numeric values
require the ability to apply restrictions on concrete domains, which is
feasible according to the specification of OWL, and is supported by a number
of description logic reasoners. Finally, it could be remarked that the
taxonomies created by TaBIIC are trees rather than more general hierarchies
and that, by construction, the set of direct descendants of a given concept
represents a partition of that concept. This can be represented through
disjointness axioms in OWL.

##  7 Conclusion

In this paper, we presented an approach to semiautomatically build taxonomies
from data, using an iterative and interactive process where the user can
control and edit the definitions of concepts at every step. Through several
tests, we showed that the approach is conceptually and technically feasible
and, through the proof-of-concept implementation presented, provides an
efficient tool to structure and organize data into a taxonomy. We further
discussed how that taxonomy can be straightforwardly integrated as a component
of an ontology. In addition, it is worth mentioning that besides the practical
objective of building a taxonomy, this tool provides an interesting way to
explore a dataset in a potentially unknown domain and gain an understanding of
its inherent structure.

The tool presented can at this stage only be considered a proof of concept,
and following the observations presented above, a key aspect of future work
relates to strengthening this implementation so as to enable conducting more
extensive user studies on the approach. Some aspects could be
straightforwardly improved, including the ability to select different
clustering methods and the actual export of the created taxonomy in the OWL
format. Related to the former, an aspect that can be interesting to study is
how the clustering technique used, as well as its parameters, could affect the
quality of the end result (the taxonomy created). This kind of question can
also be extended to the quality of the data. Even if the user can control the
construction of the taxonomy at every step, it is expected for noise in the
data to still have an effect, especially considering that we have implicitly
assumed here that a taxonomy can be reliably created from a snapshot of
relevant data at a given time. Maintaining the alignment between the created
taxonomy and evolving data is therefore another potentially interesting
research direction following this work (for example, in the spirit of [22]).

## References

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Holographic Anomalous Current at a Finite Temperature

Jian-Guo Liu, Rong-Xin Miao 111Email<EMAIL_ADDRESS>
School of Physics and Astronomy, Sun Yat-Sen University, 2 Daxue Road, Zhuhai
519082, China

###### Abstract

Weyl anomaly leads to novel anomalous currents in a spacetime with boundaries.
Recently it is found that the anomalous current can be significantly enhanced
by the high temperature for free theories, which could make the experimental
measurement easier. In this paper, we investigate holographic anomalous
currents at a finite temperature. It is found that the holographic current is
still enhanced by the high temperature in dimensions higher than three.
However, the temperature dependence is quite different from that of free
theories. This may be due to the fact that the holographic CFT is strongly
coupled and there is non-zero resistance in the holographic model. Remarkably,
the temperature dependence of holographic anomalous currents is universal in
the high temperature limit, which is independent of the choices of background
magnetic fields.

###### Contents

  1. 1 Introduction
  2. 2 Holographic current I: exact result
  3. 3 Holographic current II: perturbative result
  4. 4 Holographic current III: numerical result
  5. 5 Conclusions and Discussions

##  1 Introduction

Due to Weyl anomaly [1], an external electromagnetic field can induce novel
anomalous currents in a conformally flat space [2, 3] and a spacetime with
boundaries [4, 5]. Similar to Casimir effect [6, 7, 8], the anomalous currents
arise from the effect of the background gravitational field and the boundary
on the quantum fluctuations of the vacuum. Other anomaly-induced transports
[9] include chiral magnetic effect (CME) [10, 11, 12, 13, 14] and chiral
vortical effect (CVE) [15, 16, 17, 18, 19, 20, 21]. See also [22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] for related
works.

In this paper, we focus on the anomalous current in the spacetime with a
boundary [4, 5]. In four dimensions, it takes a universal form

$\displaystyle<J^{\mu}>=\frac{-2\beta F^{\mu\nu}n_{\nu}}{x}+...,\ x\sim 0,$
(1)

near the boundary. Here $\beta$ is the beta function, $F^{\mu\nu}$ are the
field strength, $x$ is the proper distance to the boundary, $n_{\mu}$ are the
normal vectors and $...$ denotes higher order terms in $O(x)$. Note that (1)
applies to not only the conformal field theory (CFT) but also the general
quantum field theory (QFT). That is because it is derived from Weyl anomaly
[4], which is well-defined for the general QFT [1, 42, 43].

Unfortunately, the anomalous current is heavily suppressed by the mass and the
distance to the boundary [26, 33], which makes it difficult to be measured in
laboratory. Remarkably, recently it is found that the anomalous current can be
greatly enhanced by the high temperature for free theories [44], which could
make the experimental measurement easier. For free theories, the anomalous
current is proportional to the temperature in the high temperature limit

$\displaystyle\lim_{T\to\infty}<J>\ \sim\ T.$ (2)

This means, for any given charge carrier with fixed mass, one can always
produce a detectable anomalous current by increasing the temperature.

In this paper, we investigate the holographic anomalous current at a finite
temperature. We find that the holographic current is still enhanced by the
high temperature in dimensions higher than three. However, the temperature
dependence is quite different from that of free theories. See (3,16,22) for
example. The holographic current decreases with the temperature in three
dimensions, while it increases with the temperature in dimensions higher than
four. In four dimensions, the absolute value of current first decreases and
then increases with the temperature. In the high temperature limit, the
holographic anomalous current takes a universal form,

$\displaystyle\lim_{T\to\infty}<J>\ \sim\begin{cases}\ \ T^{d-4},\ \ \ \ \ \ \
\ \ \ \ d\neq 4,\\\ \ \ \log(T),\ \ \ \ \ \ \ \ \ d=4,\end{cases}$ (3)

which is independent of the choices of external magnetic fields.

The paper is organized as follows. In section 2, by choosing a suitable
external magnetic field, we derive an exact expression of the holographic
anomalous current. We find that the holographic anomalous current is enhanced
by the high temperature in dimensions higher than three. In section 3, we
discuss the general background magnetic fields and derive the holographic
anomalous current in the high temperature limit. In section 4, we present some
numerical results. Finally, we conclude with some open questions in section 5.

##  2 Holographic current I: exact result

The main purpose of this paper is to explore the temperature dependence of
holographic anomalous currents [5, 30]. To warm up, let us first study a
special case that the background magnetic field is given by $B=2b_{a}x$, where
$b_{a}$ are a constant and $x$ is the distance to the boundary. As it is shown
below, we can derive exact expressions of anomalous currents for this case.

For simplicity, we focus on the probe limit, where the background spacetime is
given by AdS-Schwarzschild black hole,

$\displaystyle
ds^{2}=\frac{\frac{dz^{2}}{1-\frac{z^{d}}{z_{h}^{d}}}-(1-\frac{z^{d}}{z_{h}^{d}})dt^{2}+dx^{2}+dy_{a}^{2}}{z^{2}}.$
(4)

Here the bulk boundary $Q$, the AdS boundary $M$ and the horizon $H$ are
located at $x=0$, $z=0$ and $z=z_{h}$, respectively. See Fig. 1 for the
geometry of our holographic model. See also [45] for more explanations of the
geometry in AdS/BCFT.

Figure 1: Geometry of the holographic model. The bulk is an AdS-Schwarzschild
black hole, the bulk boundary $Q$, the AdS boundary $M$ and the horizon $H$
are located at $x=0$, $z=0$ and $z=z_{h}$, respectively.

The temperature of the black hole (4) is

$\displaystyle T=\frac{d}{4\pi z_{h}}.$ (5)

For simplicity, we choose the following gauges for the bulk Maxwell’s fields
[5]

$\displaystyle A_{z}=A_{t}=A_{x}=0,\ \ A_{a}=A_{a}(z,x).$ (6)

Now the bulk Maxwell’s equations become

$\displaystyle\left(3Z^{d}+d-3\right)A_{a}{}^{(1,0)}(Z,x)+Z\left(Z^{d}-1\right)A_{a}{}^{(2,0)}(Z,x)-Zz_{h}^{2}A_{a}{}^{(0,2)}(Z,x)=0,$
(7)

where we have made the coordinate transformation $Z=z/z_{h}$. Following [45],
we impose Neumann boundary condition (NBC) on the bulk boundary $Q$

$\displaystyle\partial_{x}A_{a}(Z,x)|_{x=0}=0,$ (8)

which is equivalent to the absolute BC, i.e., $F_{xa}|_{x=0}=0$.

We take the following ansatz of $A_{a}$,

$\displaystyle A_{a}(Z,x)=b_{a}x^{2}+d_{a}+\bar{A}_{a}(Z),$ (9)

which automatically obeys the NBC (8). Here $b_{a},d_{a}$ are constants and
the background magnetic field on the AdS boundary is

$\displaystyle B=F_{xa}|_{Z\to 0}=2b_{a}x.$ (10)

Substituting (9) into (7), we get

$\displaystyle
Z\left(Z^{d}-1\right)\bar{A}_{a}^{\prime\prime}(Z)+\left(3Z^{d}+d-3\right)\bar{A}_{a}^{\prime}(Z)-2b_{a}Zz_{h}^{2}=0,$
(11)

which can be solved as

$\displaystyle\bar{A}_{a}=\begin{cases}c_{2a}+\frac{c_{1a}Z^{d-2}\,_{2}F_{1}\left(1,\frac{d-2}{d};2-\frac{2}{d};Z^{d}\right)}{d-2}+\frac{b_{a}Z^{2}\text{zh}^{2}\,_{2}F_{1}\left(1,\frac{2}{d};\frac{d+2}{d};Z^{d}\right)}{d-4},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d\neq 4\\\
c_{2a}+\frac{c_{1a}}{4}\log\left(\frac{Z^{2}+1}{1-Z^{2}}\right)-\frac{b_{a}z_{h}^{2}}{8}\left(\text{Li}_{2}\left(Z^{4}\right)-4\text{Li}_{2}\left(Z^{2}\right)+4\log(Z)\log\left(\frac{Z^{2}+1}{1-Z^{2}}\right)\right),\
d=4\end{cases}$ (12)

where $\,{}_{2}F_{1}$ is the hypergeometric function, $\text{Li}_{2}$ is the
polylogarithm function and $c_{1a},c_{2a}$ are integral constants. Note that
$c_{2a}$ can be absorbed into the definition of $d_{a}$ (9). Thus we can set
$c_{2a}=0$ without loss of generality. We impose the natural boundary
condition on the horizon

$\displaystyle A_{a}(Z,x)|_{Z=1}\ \text{is finite},$ (13)

which fixes the left integral constant as

$\displaystyle c_{1a}=\begin{cases}\frac{2b_{a}z_{h}^{2}}{4-d},\ \ \ d\neq
4,\\\ 0,\ \ \ \ \ \ \ \ d=4.\end{cases}$ (14)

The holographic current can be read off from the asymptotic solutions of bulk
vectors near the AdS boundary $z=0$

$\displaystyle A_{a}=A_{a}(z=0)+...+z^{d-2}\left(\frac{J_{a}}{d-2}+f_{a}\ln
z\right)+...,$ (15)

where $J_{a}$ is the holographic current and $f_{a}$ is an irrelevant function
which appears only in even dimensions. Expanding the solutions (12) near $z=0$
and comparing with (15), we finally obtain the holographic anomalous current

$\displaystyle J_{a}=\begin{cases}\frac{2^{2d-7}\pi^{d-4}}{(4-d)d^{d-4}}b_{a}\
T^{d-4},\ \ \ \ \ \ \ \ d\neq 4,\\\ b_{a}\big{(}1-2\log(\pi T)\big{)},\ \ \ \
\ \ \ \ d=4.\end{cases}$ (16)

In dimensions higher than four, the holographic current increases with the
temperature, while in dimensions lower than four, the anomalous current
decreases with the temperature. In four dimensions, the absolute value of
current first decreases and then increases with the temperature.

##  3 Holographic current II: perturbative result

In the above section, we focus on a special kind of external magnetic field
(10) and derive an exact expression of the holographic anomalous current. In
this section, we discuss the currents induced by general background magnetic
fields. In the high temperature limit, we find that the temperature dependence
of holographic currents is universal and is still given by (16).

The general bulk Maxwell’s fields obeying NBC (8) take the following form

$\displaystyle A_{a}(Z,x)=\int dkF(k)\cos(kx)\hat{A}_{a}(Z),$ (17)

where $F(k)$ is an arbitrary function as long as it defines a convergent
integral. From (17), we read off the background magnetic fields on the AdS
boundary

$\displaystyle B=-\int dkkF(k)\sin(kx)\hat{A}_{a}(0).$ (18)

Substituting (17) into (7), we obtain

$\displaystyle
Z\left(Z^{d}-1\right)\hat{A}_{a}^{\prime\prime}(Z)+\left(3Z^{d}+d-3\right)\hat{A}_{a}^{\prime}(Z)+k^{2}Zz_{h}^{2}\hat{A}_{a}(Z)=0.$
(19)

Note that the above equation is quite similar to (11), only the last term is
different. Unfortunately, unlike (11), the above equation cannot be solved
analytically. For simplicity, we investigate the perturbation solutions in
this section, and leave the discussions of numerical solutions to next
section.

In the high temperature limit, $z_{h}=d/(4\pi T)$ is a small parameter. Thus
we can expand the bulk vectors $\hat{A}_{a}(Z)$ in powers of $z_{h}^{2}$,

$\displaystyle\hat{A}_{a}(Z)=\hat{A}_{a}^{(0)}(Z)+z_{h}^{2}\hat{A}_{a}^{(1)}(Z)+O(z_{h}^{4}).$
(20)

Let us solve (19) order by order in $O(z_{h}^{2})$. One can check that
$\hat{A}_{a}^{(0)}(Z)$ must be a constant

$\displaystyle\hat{A}_{a}^{(0)}(Z)=c_{a}$ (21)

in order to satisfy the EOM (19) and the natural boundary condition (13) at
the same time. Substituting (20,21) into (19), we get the EOM of
$\hat{A}_{a}^{(1)}(Z)$, which is exactly the same as (11) provided that we
identify $-2b_{a}z_{h}^{2}$ with $c_{a}k^{2}$. Now following the approach of
sect. 2, we can derive the holographic current as

$\displaystyle J_{a}=-\frac{c_{a}+O(1/T^{2})}{2}\int
dkF(k)k^{2}\cos(kx)\begin{cases}\frac{2^{2d-7}\pi^{d-4}}{(4-d)d^{d-4}}\
T^{d-4},\ \ \ \ \ \ \ \ d\neq 4,\\\ \big{(}1-2\log(\pi T)\big{)},\ \ \ \ \ \ \
\ d=4.\end{cases}$ (22)

It is remarkable that the temperature dependence of the holographic current is
universal, which is independent of the choices of background vector fields in
the high temperature limit.

##  4 Holographic current III: numerical result

In this section, we discuss holographic anomalous currents at general
temperatures. The main task is to solve (19) numerically. To do so, we need to
specify the boundary condition near the horizon $Z=1$. Assume that
$\hat{A}_{a}(Z)$ takes the following form near the horizon

$\displaystyle\hat{A}_{a}=a_{0a}+a_{1a}(1-Z)+a_{2a}(1-Z)^{2}+a_{3a}(1-Z)^{3}+O(1-Z)^{4},$
(23)

where $a_{ia}$ are constants to be determined. Imposing the natural boundary
condition (13), we solve

$\displaystyle a_{1a}=\frac{k^{2}z_{h}^{2}}{d}a_{0a},$ $\displaystyle
a_{2a}=\frac{\left(2dk^{2}z_{h}^{2}+k^{4}z_{h}^{4}\right)}{4d^{2}}a_{0a},$
$\displaystyle
a_{3a}=\frac{k^{2}z_{h}^{2}\left(-4(d-5)d^{2}+d(d+5)k^{2}z_{h}^{2}+k^{4}z_{h}^{4}\right)}{36d^{3}}a_{0a},$
(24)

where $a_{0a}$ is a free parameter, which can be set to be $a_{0a}=1$. Now the
boundary conditions of the differential equation (19) become

$\displaystyle\hat{A}_{a}(1-\epsilon)=a_{0a}+a_{1a}\epsilon+a_{2a}\epsilon^{2}+a_{3a}\epsilon^{3}+O(\epsilon^{4}),$
$\displaystyle\hat{A}^{\prime}_{a}(1-\epsilon)=-a_{1a}-2a_{2a}\epsilon-3a_{3a}\epsilon^{2}+O(\epsilon^{3}),$
(25)

where $\epsilon$ is a small constant chosen for the numerical calculations,
and $a_{ia}$ are given by (4) with $a_{0a}=1$.

Now it is straightforward to numerically solve the differential equation (19)
together with the BCs (4). Once we obtain the solutions, we can derive the
holographic anomalous current from the asymptotic solutions (15) near the AdS
boundary. Let us draw some figures to show the temperature dependence of the
holographic currents. Without loss of generality, we set $k=a_{0a}=1$ for all
the figures. See Fig. 2, Fig. 3, Fig. 4 for examples. It is found that the
holographic anomalous currents in these figures match exactly the results (22)
of sect.3 in the high temperature limit. This can be regarded as a double
check of our results. Besides, the current decreases with the temperature in
three dimensions, while it increases with the temperature in dimensions higher
than four. In four dimensions, the absolute value of current first decreases
and then increases with the temperature.

Figure 2: Holographic anomalous current at finite temperature in three
dimensions (Left); Compare the numerical result with that of high temperature
limit $-\frac{3k^{2}}{4\pi T}$ (Right). The holographic current decreases with
temperature in three dimensions.

Figure 3: Holographic anomalous current at finite temperature in four
dimensions (Left); Compare the numerical result with that of high temperature
limit $k^{2}\left(\log(\pi T)-\frac{1}{2}\right)$ (Right). The absolute value
of current first decreases and then increases with the temperature in four
dimensions.

Figure 4: Holographic anomalous current at finite temperature in five
dimensions (Left); Compare the numerical result with that of high temperature
limit $\frac{4}{5}\pi k^{2}T$ (Right). The holographic current increases with
temperature in five dimensions.

##  5 Conclusions and Discussions

In this paper, we investigate the holographic anomalous current at a finite
temperature. For the external magnetic field $B\sim x$, we derive an exact
expression of the holographic current. As for general background magnetic
fields, we obtain perturbative and numerical results. It is remarkable that
the temperature dependence of the holographic anomalous current is universal
in the high temperature limit, which is independent of the choices of
background magnetic fields. Similar to the case of free theories, the
holographic anomalous current is still enhanced by the high temperature in
dimensions higher than three. However, the temperature dependence is quite
different from that of free theories. The reasons may lie in the fact that the
holographic CFT is strongly coupled and there is non-zero resistance in the
holographic model [46]. In this paper, we focus on the probe limit, where the
background spacetime is given by the AdS-Schwarzschild black hole. It is
interesting to study the back-reactions and the charged black holes to see if
the universal temperature dependence of holographic currents still holds or
not. It is also interesting to investigate the low temperature regions
carefully. Finally, the temperature dependence of the anomalous Fermi
condensation [31, 32] is also a problem worth exploring. We leave these
problems to future work.

## Acknowledgements

We thank C. S. Chu and M. X. Liu for helpful discussions and comments. R. X.
Miao acknowledges the supports from National Natural Science Foundation of
China (No. 11905297) and Guangdong Basic and Applied Basic Research Foundation
(No.2020A1515010900).

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