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<article xmlns:xlink="http://www.w3.org/1999/xlink">
  <front>
    <journal-meta />
    <article-meta>
      <title-group>
        <article-title>FrESH - Feedback-reliant Enhancement of Subjective Content Descriptions by Humans</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Magnus Bender</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Kira Schwandt</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Ralf Möller</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Marcel Gehrke</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>University of Lübeck, Institute of Information Systems</institution>
          ,
          <addr-line>Ratzeburger Allee 160, 23562 Lübeck</addr-line>
          ,
          <country country="DE">Germany</country>
        </aff>
      </contrib-group>
      <abstract>
        <p>An agent in pursuit of a task may work with a corpus containing text documents. To perform information retrieval on the corpus, the agent internally maintains a model of the documents in the corpus. This model may contain annotations such as Subjective Content Descriptions (SCD)-additional data associated with diferent sentences of documents. In our scenario, a human interacts with the information retrieval agent: The human sends a query to the agent, the agent uses its internal model to calculate a response and returns this response. However, the response may contain erroneous parts. Such errors, like faulty SCDs, may be send back to the agent by the human as feedback. Then, the agent can incorporate the feedback to improve its internal model. However, removing a faulty association of a sentence with an SCD in a previously trained model is a dificulty task-often the model needs to be retrained from scratch. To circumvent this, this paper presents FrESH an approach for Feedback-reliant Enhancement of Subjective Content Descriptions by Humans. Using FrESH the model keeps fresh and maintained with human feedback.</p>
      </abstract>
      <kwd-group>
        <kwd>eol&gt;Subjective Content Descriptions (SCDs)</kwd>
        <kwd>Text Annotation</kwd>
        <kwd>Information Retrieval Agent</kwd>
        <kwd>Incorporate Human Feedback</kwd>
        <kwd>Incremental Model Adjustment</kwd>
        <kwd>Information System</kwd>
      </kwd-group>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>1. Introduction</title>
      <p>Typically, in machine learning a model is trained to then perform tasks on the model instead
on the data. Training a model is an expensive task. However, if data points later on have to be
removed, we want to discreetly manipulate the model instead of retraining it from scratch. For
example, it may turn out that an item from the data is erroneous or there may be privacy or
copyright related problems with an item. Additionally, the model may produce wrong results
because some item may be associated incorrectly. In all these cases, the item needs to be
removed from the data and the model. However, removing an item from a model is a dificult
task because the model only encodes the data and does not contain the data in such a way that
individual elements can be distinguished and removed. Therefore, models are often retrained
from scratch after items are removed from the data.</p>
      <p>
        For diferent models, there are diferent approaches to avoid retraining from scratch, e.g., for
k-Means [
        <xref ref-type="bibr" rid="ref1 ref2">1, 2</xref>
        ] or linear and logistic regression [
        <xref ref-type="bibr" rid="ref3">3</xref>
        ]. The common idea is to avoid retraining the
model and instead updating the model by applying an inverse operation that removes an item
of the data from the model.
      </p>
      <p>
        In this paper, we consider the scenario of a information retrieval agent [
        <xref ref-type="bibr" rid="ref4">4</xref>
        ], which is an is a
rational and autonomous unit acting in a world fulfilling a defined task. In our case the agent
works with a corpus of text documents and provides an information retrieval service for human
users. Internally, the agent uses Subjective Content Descriptions (SCD), whereas each SCD
represents a concept in the corpus and is associated to similar sentences across the corpus.
When a human sends a query to the agent, the agent determines similar SCDs representing
similar concepts and returns the sentence associated with these SCDs to the human. The SCDs
of a corpus are modeled by an SCD matrix which can be trained by the Supervised Estimator of
SCD Matrices (SEM) [
        <xref ref-type="bibr" rid="ref5">5</xref>
        ] or the UnSupervised Estimator of SCD Matrices (USEM) [
        <xref ref-type="bibr" rid="ref6">6</xref>
        ].
      </p>
      <p>As the name subjective content description already states, it is subjective which sentences are
associated with which description. From the perspective of a human users of the agent, there
may be a faulty association between a sentence and an SCD. If the agent uses such a faulty
association while computing the response for a user, the response will be determined erroneous
by the user. In this situation, the human user may give feedback to the agent and the agent
needs to update its SCD matrix by removing the faulty association between sentence and SCD.
We are now back to the point, where the model of the agent needs to be updated. Additionally,
the agent should not need to retrain the SCD matrix from scratch after the sentence has been
removed from the corpus.</p>
      <p>To solve the problem, this paper provides FrESH, an approach for Feedback-reliant
Enhancement of Subjective Content Descriptions. The agent may use FrESH to process feedback by
removing a faulty sentence from the SCD matrix and only needs to update the SCD the faulty
sentence has been associated with. Furthermore, FrESH can be used to remove sentences from
the SCD matrix if their content should be removed entirely, i.e., to keep Fake-News out of the
corpus or to remove copyright or privacy protected content.</p>
      <p>
        An agent as described above may be integrated in an information system working with
SCDs and providing a human friendly way of interacting with SCD based models [
        <xref ref-type="bibr" rid="ref7">7</xref>
        ]. Adding a
feedback mechanism to the agent allows the users of the information system to build their own
subjective model by enhancing an initial SCD matrix learned by USEM on their corpus.
      </p>
      <p>
        In the humanities, corpora of text documents are often rather small and need to be annotated
by scientists. An information system using SCDs helps theses scientists doing their work, e.g.,
SCDs are used to identify comments and actual content in Tamil and Greek language corpora [
        <xref ref-type="bibr" rid="ref8">8</xref>
        ].
Furthermore, SCDs provide techniques for (i) estimating SCDs for a single previously unseen
text document using the Most Probably Suited SCD (MPS2CD) algorithm [
        <xref ref-type="bibr" rid="ref9">9</xref>
        ], (ii) classifying a
text document as related, extended, revised, or unrelated to a corpus [
        <xref ref-type="bibr" rid="ref9">9</xref>
        ], (iii) moving the SCDs
from one corpus to another similar corpus by adapting the SCDs’ domain [10], (iv) enriching
SCDs in a corpus already sparsely associated with SCDs [11], or (v) detecting complementary
documents to a corpus [12].
      </p>
      <p>For all these techniques, FrESH makes it easier for a scientist in the field of humanities to build
a custom SCD based model for their corpora. A corpus can be automatically annotated with
SCDs, and then it is optimized and corrected using FrESH by feedback from the scientist. Without
FrESH, it would not be possible to optimize and correct a corpus automatically annotated with
SCDs, thus requiring more manual work by the scientist.</p>
      <p>The remainder of this paper is structured as follows: First, we recap the basics of SCDs,
the SCD matrix and the estimation of SCD matrices. Second, we formalize the problem of
incorporating feedback by removing false associations of sentences with SCDs in an SCD matrix.
Afterwards, we provide three consecutive methods to solve the problem and evaluate each.
Finally, we conclude with a summary and short outlook.</p>
    </sec>
    <sec id="sec-2">
      <title>2. Preliminaries</title>
      <sec id="sec-2-1">
        <title>2.1. Notations</title>
        <p>This section specifies notations, recaps the basics of SCDs and describes the estimation of SCD
matrices.</p>
        <p>To begin with, we formalize our setting of a corpus.</p>
        <p>• A word  is a basic unit of discrete data from a vocabulary  = {1, . . . , },  ∈ N.
• A sentence  is defined as a sequence of words  = (1, . . . ,  ),  ∈ N, where each
word  ∈  is an element of vocabulary . Commonly, a sentence is terminated by
punctuation symbols like “.”, “!”, or “?”.
• A document  is defined as a sequence of sentences  = (1, ...,  ),  ∈ N.
• A corpus  represents a set of documents {1, . . . , },  ∈ N.
• An SCD  is a tuple of the SCD’s additional data  and the referenced sentences
{1, ...., },  ∈ N. Thus, each SCD references sentences in documents of , while in
the opposite direction a sentence is associated with an SCD.
• A sentence associated with an SCD is called SCD window, inspired by a tumbling window
moving over the words of a document. Generally, an SCD window might not be equal to
a sentence and may be a subsequence of a sentence or the concatenated subsequences of
two sentences, too. Even though, in this paper, an SCD window always equals a sentence.
• For a corpus  there exists a set  called SCD set containing  associated SCDs () =
︀{  = (︀  , ⋃︀∈{1, ...., }︀)} =1 . Given a document  ∈ , the term () refers to
the set of SCDs associated with sentences from document .
• Each word  ∈  is associated with an influence value (, ) representing the
relevance of  in the sentence . For example, the closer  is positioned to the object
of the sentence , the higher its corresponding influence value (, ). The influence
value is chosen according to the task and might be distributed binomial, linear, or constant.</p>
      </sec>
      <sec id="sec-2-2">
        <title>2.2. Subjective Content Descriptions</title>
        <p>
          SCDs provide additional location-specific data for documents [
          <xref ref-type="bibr" rid="ref5">5</xref>
          ]. The data provided by SCDs
may be of various types, like additional definitions or links to knowledge graphs.
        </p>
        <p>
          Kuhr et al. [
          <xref ref-type="bibr" rid="ref9">9</xref>
          ] use an SCD-word distribution represented by a matrix when working with
SCDs. The SCD-word distribution matrix, in short SCD matrix, can be interpreted as a generative
model. A generative model for SCDs is characterized by the assumption that the SCDs generate
Algorithm 1 Supervised Estimator of SCD Matrices  ()
1: function SEM(, ())
2: Input: Corpus ; Set of SCDs ()
3: Output: SCD-word distribution matrix  ()
4: Initialize an  ×  matrix  () with zeros
5: for each document  ∈  do
6: for each SCD  = (, {1, ..., }) ∈ () do
7: for  = 1, ...,  do
8: for each word  ∈  do
9:  ()[][] += (, )
the words of the documents. We assume that each SCD shows a specific distribution of words
of the referenced sentences in the documents.
        </p>
        <p>The SCD matrix  () models the distributions of words for all SCDs () of a corpus  and
is structured as follows:</p>
        <p>1
1 ⎛ 1,1
2 ⎜⎜ 2,1
 () = ... ⎜⎜⎜ ...</p>
        <p>⎝
2
1,2
2,2
.
.
.</p>
        <p>3</p>
        <p>· · ·
1,3 · · ·
2,3 · · ·
.
.
.</p>
        <p>.
.</p>
        <p>.

,1 ,2 ,3 · · ·</p>
        <p>⎞

1,</p>
        <p>⎟
2, ⎟
... ⎟⎟⎟</p>
        <p>⎠
,</p>
        <p>The SCD matrix consists of  rows, one for each SCD in (). Each row contains the word
probability distribution for an SCD. Therefore, the SCD matrix has  columns, one for each
word in the vocabulary of the corpus .</p>
      </sec>
      <sec id="sec-2-3">
        <title>2.3. Supervised and UnSupervised Estimator for SCD Matrices</title>
        <p>The SCD matrix can be estimated in a supervised manner given the set () for a corpus .
SEM is described in Algorithm 1. Given a corpus , the algorithm iterates over each document
 in the corpus and the document’s SCDs. For each associated SCD , the referenced sentences
, ...,  are used to update the SCD matrix. Thereby, the row of the matrix representing SCD
1
 gets incremented for each word in each sentence by each word’s influence value.</p>
        <p>Finally, the SCD matrix needs to be normalized row-wise to meet the requirements of a
probability distribution. However, the normalization is often skipped because later the cosine
similarity is often used with the rows of the matrix and the cosine similarity does a normalization
by definition.</p>
        <p>Unlike SEM, USEM estimates an SCD matrix  () without needing the SCD set (). USEM
initially starts by associating each sentence to one unique SCD, which leads to an initial SCD
matrix consisting of a row for each sentence in the document’s corpus. Then USEM finds the
sentences in the corpus that represent the same concept and groups them into one SCD.</p>
        <sec id="sec-2-3-1">
          <title>Errors/</title>
        </sec>
        <sec id="sec-2-3-2">
          <title>Feedback</title>
        </sec>
        <sec id="sec-2-3-3">
          <title>Faulty sentences</title>
          <p>and their SCDs</p>
        </sec>
        <sec id="sec-2-3-4">
          <title>Only faulty sentences</title>
        </sec>
        <sec id="sec-2-3-5">
          <title>SCD matrix</title>
        </sec>
        <sec id="sec-2-3-6">
          <title>Frequecies Distribution</title>
        </sec>
        <sec id="sec-2-3-7">
          <title>Method 1</title>
        </sec>
        <sec id="sec-2-3-8">
          <title>Method 2</title>
        </sec>
        <sec id="sec-2-3-9">
          <title>Method 3</title>
        </sec>
        <sec id="sec-2-3-10">
          <title>Method 4</title>
          <p>Algorithm 2 FrESH Method 1 (M1)
1: function FrESH-M1(SCD Matrix  (), Set of faulty Sentences with SCDs )
2: for each (, ) ∈  do ◁ Iterate over faulty sentences with SCDs
3: for each  ∈  do ◁ Iterate over words
4:  ()[][] − = (, )</p>
          <p>However, the SCDs estimated by USEM and SEM may contain faulty sentences which have
to be removed if a human user gives feedback about such errors.</p>
        </sec>
      </sec>
    </sec>
    <sec id="sec-3">
      <title>3. Incorporate Feedback</title>
      <p>In this section, we present FrESH and thereby how to incorporate human feedback to enhance
an SCD matrix. For our method, we assume that the feedback exactly states which sentences
is falsely associated with its SCD, i.e., the human user may click on a button remove sentence.
Therefore, we consider the problem of removing a faulty sentence from an SCD matrix. This
also removes the sentences from the corpus, making FrESH useful for removing Fake-News,
copyright, and privacy protected content, too.</p>
      <p>We diferentiate between two types of SCD matrices  (): The values contain the frequencies
of the words without normalization or a normalized version containing row-wise distributions of
words. Additionally, we diferentiate between the way the errors are contained in the feedback:
Only a faulty sentence is given as input or a faulty sentence together with it’s SCD is given. In
both cases a set  contains either sentences  or pairs of sentences  and SCDs .</p>
      <p>In total, we have four diferent cases regarding the input data of FrESH, which are also shown
in Figure 1. Next, we will consider each case and develop method 1 to 4 to solve each.
3.1. Method 1
The input consists of an SCD matrix containing the frequencies of the words and a set of faulty
sentences with their associated SCDs . It is then possible to revert the operations of SEM
(Algorithm 1). SEM adds in Line 9 for each word in a sentence the frequency of the word
weighted by an influence value to the row of the matrix representing the SCD. The first method
(M1) reverses this addition by subtracting the same value in Line 4 of Algorithm 2.</p>
      <p>
        M1 inverts the operations of SEM. Therefore, an SCD matrix learned by SEM on the corpus
without  will be identical to an SCD matrix from which  was removed by M1.
Algorithm 3 FrESH Method 2 (M2)
1: function FrESH-M2(SCD Matrix  (), Set of faulty Sentences )
2: for each  ∈  do
3:  = MPS2CD( (), )
4: for each word  ∈  do
5:  ()[][] − = (, )
◁ Each faulty sentence
◁ Get SCD of sentence, use MPS2CD [
        <xref ref-type="bibr" rid="ref9">9</xref>
        ]
◁ Iterate over words
 = min=1,...,;  ′()[][]&gt;0  ′()[][]
for each word  ∈  do
      </p>
      <p>′()[][] − = (, ) · 
return NormalizeRows( ′())
Algorithm 4 FrESH Method 3 (M3)
1: function FrESH-M3(SCD Matrix  ′(), Set of faulty Sentences with SCDs )
2: for each (, ) ∈  do ◁ Iterate over faulty sentences with SCDs
3: ◁ Minimal value in SCD’s row
4: ◁ Iterate over words
5:
3.2. Method 2
The input consists of an SCD matrix containing the frequencies of the words and a set of faulty
sentences . Hence, first the SCD for each faulty sentence needs to be found, afterwards the
sentence can be removed from the matrix as in M1. The MPS2CD algorithm finds a most suitable
SCD for a sentence, and is therefore used with this second method (M2) in Algorithm 3.</p>
      <p>Unlike M1, M2 does not guarantee to produce identical matrices if a matrix is first trained on
a corpus with faulty sentences that are then removed, compared to a matrix trained directly
on a corpus without faulty sentences. This is caused by the fact, that MPS2CD is used to find a
most suitable SCD, which may not be the SCD used initially. However, MPS2CD works quite
accurately, so we expect only a small diference, which should not afect an agent using SCDs.
3.3. Method 3
The input consists of an SCD matrix containing row-wise distributions of words  ()′ and a
set of faulty sentences with their associated SCDs . The dificulty is to revert the operations of
SEM on the distributions of each SCD. During normalization of the SCD matrix, all frequencies
in each matrix’s row are divided by a divisor. To approximate this divisor with M3, we assume
that in each row at least one value had the frequency one, i.e., there was a word that occurred
only once in the referenced sentences of the SCD. This word will have the minimal value in
the distribution and we assume it had frequency one. So the normalized value is equal to one
divided by the divisor, which in turn is a factor that we can use to decrease the values that are
subtracted from the matrix by the same amount as during the normalization.</p>
      <p>Again, M3 is based on an approximation and may not produce identical matrices.
3.4. Method 4
The input consists of an SCD matrix containing row-wise distributions of words and a set of
faulty sentences . For this input, the ideas from M2 and M3 can be combined. First, the SCD
for each faulty sentences needs to be determined, which can be done via MPS2CD. Second, the
minimal value of this SCD’s distribution can be determined and used as factor while changing
the SCD matrix.</p>
      <p>We will not consider M4 in detail, as it is only a combination of M2 and M3. In most use-cases
there is no benefit in normalizing the SCD matrix, because mostly the cosine similarity is used
and it does a normalization by definition. Therefore, especially M2 is beneficial for our agent
using SCDs and getting feedback from its users, e.g., to remove faulty sentences.</p>
      <p>Next, we present an evaluation of M1, M2, and M3.</p>
    </sec>
    <sec id="sec-4">
      <title>4. Evaluation</title>
      <sec id="sec-4-1">
        <title>4.1. Dataset</title>
        <p>After we have introduced FrESH to remove sentences from an SCD matrix, we present an
evaluation of the methods M1, M2, and M3. First, we describe the used corpus and evaluation
workflow. Afterwards, we present the results of the evaluation.</p>
        <p>In this evaluation we use the 20 newsgroups1 dataset. 20 newsgroups is a well-known corpus
consisting of e-mails from 20 e-mail newsgroups. Thematically, the 20 newsgroups can be
divided into six topics, computer, sport, science, politics, religion and for sale. The entire corpus
consists of 18 828 text documents. The documents have between 1 and 39 682 words with a
median of 160 words.</p>
      </sec>
      <sec id="sec-4-2">
        <title>4.2. Workflow and Metrics</title>
        <p>FrESH is implemented in Python and runs in a Docker container on a machine featuring 8 Intel
6248 cores at 2.50GHz (up to 3.90GHz) and 16GB RAM.</p>
        <p>To evaluate each method, we first create the full corpus  containing all documents from
20 newsgroups. We annotate the sentences with OpenIE [13] and use the extracted triples as
data for the SCDs. Then, we split  into to subsets,  contains the sentences to subtract
that we consider faulty and  contains the non-faulty sentences to keep. For each corpus we
learn an SCD matrix using SEM, i.e.,  ( ) and  (). Afterwards, we apply one of the three
methods to  ( ) and remove the sentences in . This step yields  ′( ), which should be
identical to  ().</p>
        <p>As a metric, we then calculate the diference of the matrices  ′( ) and  () for each SCD 
using the Hellinger distance [14]:</p>
        <p>1 ⎸⎸⎯∑︁ (︂ √︁
( ′( ),  ()) = √2 ⎷
=1</p>
        <p>︂) 2
 ′( )[][] − √︀ ()[][]
1http://qwone.com/~jason/20Newsgroups/</p>
        <p>Method
M1</p>
        <p>M2</p>
        <p>M3</p>
      </sec>
      <sec id="sec-4-3">
        <title>4.3. Results</title>
        <p>In this section, we present the results of the evaluation. In the left part of Figure 2, the duration
of removing one sentence with the three diferent methods is shown. M1 is very fast, while M2
and M3 need more time. M2 needs to determine the SCD via MPS2CD which takes time and
M3 needs to calculate the factor needed to maintain the distribution in the matrix. However,
all methods are reasonably fast, as it takes at most 4.5 ms to remove a sentence which will be
clearly faster than retraining the matrix from scratch.</p>
        <p>In the right part of Figure 2, the diference between the matrices  ′( ) and  () is shown.
The diference is shown as percentage of the reduction of the Hellinger distance averaged
for all SCDs: First, the Hellinger distance between  ( ) (full corpus) and  () (non-faulty
part) is calculated. When removing the faulty sentences from  ( ) to get  ′( ) the distance
should become smaller, which is shown as reduction. A reduction of 100 % implies that  ()
equals  ′( ). The reductions are shown for all three methods and diferent numbers of faulty
sentences in . The x-axis represents the size of  as percentage of the entire corpus  .</p>
        <p>M1 inverts the operations of SEM, therefore, after removing all faulty sentences, both SCD
matrices are equal and the diference is reduced completely. M2 reaches very high reductions
around 95% and thus provides a reliable technique to remove faulty sentences. The results of
M3 are significant below M1 and M2. Especially, if only a few faulty sentences are deleted, the
diference keeps high. This implies that our assumption used in M3, to approximate the factor
used to normalize the matrix, does not hold.</p>
        <p>Therefore an agent maintaining an SCD matrix should store it using frequencies instead of
distributions. Doing so, the agent may use M1 and M2 to incorporate feedback. In particular,
storing frequencies s is advantageous because FrESH is more accurate, frequencies can be stored
as integers instead of possibly inaccurate floats, and for most use cases of the SCD matrix,
normalization is done by definition, e.g., as part of the cosine similarity.</p>
        <p>Summarized, an information system using SCDs can automatically train an SCD matrix
storing frequencies using USEM on a corpus. This SCD matrix is automatically trained and
may contain some faulty sentences and SCDs. When a user uses the information system, a
button to remove faulty sentences is displayed, which then triggers FrESH to remove a sentence.
Depending on the specific situation, the SCD of a sentence to be removed may or may not
be known, and M1 or M2 may be used. Thus, the SCD matrix is updated incrementally and
optimized for the user. A limitation of FrESH is that it removes a sentence completely from the
SCD matrix, whereas sometimes it would be better to change only the SCD associated with the
sentence.</p>
      </sec>
    </sec>
    <sec id="sec-5">
      <title>5. Conclusion</title>
      <p>This paper presents FrESH, a technique to enhance SCD matrices using human feedback.
Humans indicate faulty sentences, which then can be removed from the matrix without retraining
the matrix. This allows human users of an agent working with SCDs to build their own
subjective model for the corpus the agent works with. Additionally, it allows the operators of the
agent to remove erroneous, copyright, or privacy protected data quickly and eficiently. Our
evaluation shows, that FrESH works quite reliable when the SCD matrix contains the word
frequencies.</p>
      <p>Currently, a faulty sentence is removed entirely from the SCD matrix and corpus. Furthermore,
dependencies and relations to other SCDs and other sentences are not maintained, if a sentences
is removed. Future work will focus on maintaining these dependencies.
[10] F. Kuhr, M. Bender, T. Braun, R. Möller, Context-specific adaptation of subjective content
descriptions, 15th IEEE International Conference on Semantic Computing, (ICSC 2021),
Laguna Hills, CA, USA, January 27-29 (2021) 134–139.
[11] F. Kuhr, B. Witten, R. Möller, Corpus-Driven Annotation Enrichment, 13th IEEE
International Conference on Semantic Computing, (ICSC 2019), Newport Beach, CA, USA,
January 30 - February 1, (2019) 138–141.
[12] M. Bender, F. Kuhr, T. Braun, To extend or not to extend? complementary documents,
16th IEEE International Conference on Semantic Computing, (ICSC 2022), Virtual, January
26-28 (2022) 17–24.
[13] G. Angeli, M. J. Johnson Premkumar, C. D. Manning, Leveraging linguistic structure for
open domain information extraction, Proceedings of the Association of Computational
Linguistics (ACL) (2015) 344–354.
[14] E. Hellinger, Neue Begründung der Theorie quadratischer Formen von unendlichvielen
Veränderlichen, Journal für die reine und angewandte Mathematik (1909) 210–271.</p>
    </sec>
  </body>
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