# Tensor Space Model

Pascal Francq

February 17, 2016 (April 20, 2011)

Abstract

In information science related problems (such as document clustering), the objects are generally described with the vector space model. This is especially true for poorly semantically structured documents (for example PDF). This approach doesn’t take other types of information into account (for example metadata or the structure of XML documents). In this article, a tensor space model is presented. It is a simple adaptation of the extended vector model.

## 1 Introduction

Despite the fact that most documents available today are poorly semantically structured, solutions were developed to overcome this. For years, document management systems add metadata to the documents stored, and several document types propose them in some way (headers of emails, <META> tag in HTML or the “properties” of office documents). More recently, the W3C introduces the XML as a set of technologies to structure documents.
Today, more and more documents in intranets are XML ones, and their number increases on the Internet too. Despite the numerous existing XML databases, there is a lack of tools using the structure to solve several classical problems such as keyword-based queries for information retrieval or clustering of XML documents. Moreover, it is still difficult to manage collections (such as the Internet) where XML documents are mixed with non-XML documents (such as Postscript, PDF, etc.).

### 1.1 A First Example: emails

A simple email is a text document divided into two parts (generally separated by a blank line): the header and the content. The following text proposed a simple example of an email:
From: pascal@source.org (Pascal Francq)
Subject: Example of an email
Article-I.D.: src.17194
Date: Thu, 20 Jul 2006 10:08:50 +0200
Distribution: Paul Otlet Institute

Do you buy the remaster version of the In Rock album from Deep Purple?
Pascal
The header contains several fields such as the author (From), the subject or the date when the email was send. These fields are non only structured metadata that enhances the content of the email, but it provides also information that links emails (for example emails sent by the same author or sharing an identical subject [A]  [A] To detect that, it is often necessary to ignore leading “Re:” in the subject field.) .

### 1.2 A Second Example: HTML Documents

HTML are text documents enhanced with presentation and hyperlink related tags and attributes [1]. The hyperlinks between HTML documents form the Web structure. The following text proposed two simple HTML documents:
<html>
<body>
This is a link to the <a href=”mysite.com/doc2.html”>second document</a>.
</body>
</html>

<html>
<body>
This is a link to the <a href=”mysite.com/doc1.html”>first document</a>.
</body>
</html>
Hyperlinks convey some information. In particular, a hyperlink from a document to the document may be seen as if recognizes a given authority to . This assumption is in fact the base of the famous PageRank ranking method of Google [2].

### 1.3 A Third Example: XML Documents

XML documents are characterized by text-oriented content structured with tags and attributes [3]. The following text proposed a simple example of an XML document:
<?xml version="1.0" standalone="no"?>
<discography xmlns="http://music.org/">
<group groupID="Deep Purple">
<groupName> Deep Purple </groupName>
<album>
<albumName> In Rock </albumName>
<published> 1969 </published>
</album>
</group>
</discography>
A XML structure is a tree containing four different types of tokens:
tags Elements structuring the document.
attributes Elements related to tags and influencing the structure.
attributes values Most attributes are associated with a given value (for example groupID=”Deep Purple”).
content The textual content corresponds to the “knowledge” encapsulated into XML documents.
In a XML document, knowledge expressed textually is associated to tags and attributes that are supposed to add structure information to the document. These structure information of XML documents (the tree of tags, attributes and attributes values) are defined through XML schema. Every XML schema can be build upon existing XML schema. In the emerging semantic Web [4], the documents depend from application-specific XML schema (for example technical reports) built on top of generic XML schema (for example to represent a person or the abstract of a document). In this context, XML documents with similar tags and attributes (using elements of common XML schema) are supposed to be related to “something common” (from the content or the functional point of view). For example, in a corpus of XML documents dedicated to clinical tests, similar XML parts (tags and content) may appear in different documents related to the same authors (using tags from the Dublin Core Metadata [5]) or the same chemical molecules (using tags from the Chemical Markup Language [6]).

### 1.4 Vector Space Model

Historically, in information science, the documents were represented as bags of words [7]. Several filtering techniques are used to carefully choose the sequences of characters (tokens) indexing the documents in order to limit the amount of memory needed to represent the documents and to speed up the treatment.
The vector space model was introduced to represent documents in search engines, but it can be used to represent any kind of objects. In this model, the objects (for example documents) are represented as a set of features, . For each object, , a vector of features is defined and a weight is associated with each feature ( being the total number of different features). Initially, the model supposes that for documents , but negative weights may be allowed to represent other objects [8]. Several methods exist to compute these weights based on quantity, , related to the feature for the object [9].
For textual documents, the features are selected terms contained in the documents, and is the number of occurrences of these terms. As explained, some filtering techniques can be used, such as replacing the different words by their stems (“connects”, “connected”, etc. are replaced by “connect”) and to remove the stopwords (“and”, “or”, “the”, etc. in English). These techniques are well known [9], and some of them are implemented by the GALILEI platform when it builds a document description.

### 1.5 Our Approach

As the examples shown, new types of information appear in documents. In particular, the metadata, the (hyper)links, and the structure information provided by XML tags should certainly contribute to the representations of documents. Therefore, a major issue in information science is to propose a model to represent structured documents in general, XML documents in particular.
In practice, since most collections do not content structure documents only, a main constraint for the model is to manage documents with different levels of semantic information. Moreover, numerous algorithms, from information retrieval [9] to social browsing [11], were developed for non-structured textual documents. Ideally, the new model representing documents should be “as much compatible as possible” to the previous model in order to take advantage of existing researches in information science.
Previously, extended vector space models were proposed to take new kind of information into account [12, 13]. The GALILEI framework adapts these models for objects (such as XML documents). Specifically, it is suggested that each object is represented by a tensor in a concept space. This adaptation of the classical vector space model has a major advantage. Since there exist several algorithms using the vector space model, the proposed approach makes existing algorithms easily adaptable. In particular, if a new similarity measure taking the richness if the tensors into account exists (as the one proposed in the GALILEI framework), most information science algorithms (such as the documents clustering, users profiling or users clustering) should work without any changes (and use indirectly richest features if available). Moreover, collections mixing XML documents and non-structured documents can be managed.

## 2 Tensor Space Model

As explained in the previous section, taking richer information into account to represent documents (such as XML ones) is useful. We present here a model where each object is represented by a tensor in a concept space.

### 2.1 Concept space

We suppose that the objects are indexed in a concept space, , formed by concepts, . A concept can be a term, a stopword, a metadata, a text block, an Uniform Resource Identifier (URI), a Digital Object Identifier (DOI), an XML tag or attribute, a language-independent meaningful entity (such as names, cities, or organizations) [14], etc. It should be noticed than some concepts (for example the document content) is itself a set of concepts (such as the terms contained in the document).
Each concept, , has a given concept type, . A concept type identifies concepts that share a same “worldview”, for example metadata from the Dublin Core Metadata [5], tags and attributes from a same XML schema, French or English stopwords, an identical interpretation of character strings (such as URI ou DOI), etc. Each concept type acts as a namespace for the corresponding concepts, i.e. the character string representation of a concept must be unique in a given concept type. Moreover, we suppose that each concept type defines one neutral concept, “*”, that represents any set of concepts of that type.
Finally, concept types are regrouped into concept categories that are related to the “nature” of the concept (such as textual content). The GALILEI framework defines five concept categories:
text Textual content (stopwords, “normal” words, groups of words, etc.).
metadata Content representing a kind of descriptive metadata (author, abstract, etc.) or technical one (such as specialized expressions from existing taxonomies).
structure Structure information (such as XML tags and attributes).
predicate First-logic predicates formed by concepts triplets.
Figure 1↓ illustrates the hierarchy of concepts.
The concept types and categories do not influence the concept space, they serve only to organize “semantically” the concepts. The GALILEI framework uses this information, for example, to compute similarity measures. Let us define two functions: that returns the type of a given concept, ; and that returns the category of a given concept, .

### 2.2 Concept Independence

This model shares an hypothesis with the classic vector space model : concepts (such as index terms) are independent. This mean that the presence of a given concept (for example the term “deep”) provides no information on the presence of another concept (for example the term “purple”). This hypothesis is, of course, a simplification: concepts in general, and terms in particular, are dependent. In practice, however, this hypothesis is acceptable.
First of all, taking the dependance into account is a tricky problem: it complicates the model, increases the computational power needed (in particular for a large corpus), and the existing attempts don’t performed better [9].
Secondly, since our tensor space model is statistical (as the classic vector space model), dependencies can be taken into account indirectly through the co-occurrences of the concepts. To illustrate this, let us suppose two domains: genetic algorithms (described with terms such as “genetic”, “algorithm”, “performance”, “chromosome”, “heuristic”, etc.) and genetics (described with terms such as “genetic”, “biology”, “chromosome”, “heredity”, etc.). To describe these domains, the terms “genetic” and “chromosome” are not enough, we must combine them, i.e. they are dependent from other terms. Yet, it is reasonable to suppose that documents about genetic algorithms will also contained “heuristic” or “performance”, while the terms “biology” and “heredity” will appear in documents about genetics. This means that their descriptions don’t include the terms “genetic” and “chromosome” alone, but will also be defined by other terms in order to make clean distinction between the two domains.
Thirdly, it is possible to create concepts that encapsulates some dependencies. Let us take the previous example. A new concept type, “scientific domain” (text), can be added. Two concepts of that type can be created: “genetic algorithms” and “genetics”. These concepts can then be included in the object representations (such as document).

### 2.3 Tensor representation

In his model, Fox suggests to associate to a given object a (sub)vector for different types of concepts called c-types (authors, bibliographical links, etc.). [15]. If Fox’s model is attractive, it has a little drawback: his c-types are specific to a particular document collection (INEX 2005). Therefore, I propose a generalization of it. In fact, for any document corpus, we may suppose that each object has multiple vectors representing it, each vector being associated to a particular meta-concept (what Fox called a c-type).
Le us imagine that an object (for example a document), , has an “author” metadata which is “Pascal Francq”, and contents the single sentence “The connections are optical fibers”. Moreover, we suppose that we remove the stopwords and take only the stems into account. One possible representation of the object can be built with two elements:
1. One metadata meta-concept “author” (of the concept type “DMCI”) defined by the two text terms “Pascal” and “Francq” [B]  [B] Of course, “Pascal Francq” can also be an unique text concept of the type “Person names”..
2. One text meta-concept “*” (of the concept type “Document Content”) defined by the three terms “connect”, “optic” and “fiber”.
Each element may be represented by a vector in the concept space: the first one will have two non-zero elements, the second one having three non-zero elements. As explained above, “author” and “*” are themselves concepts. Therefore, the object can be represented has a tensor, where represents the weight of the concept for a vector associated with the meta-concept . This model makes two assumptions:
1. Multiple elements associated to a common meta-concept (for example two authors) are represented as a single vector.
2. Recurrent descriptions are prohibited, i.e. .
The weights depend on the method used to compute the object descriptions. For example, in the case of a document, the weights of a concept can be the number of its occurrences in the part document represented by the corresponding meta-concept.
Let us suppose that the space contains 8 concepts (types and categories are also given): represents “author” (“DMCI”, “metadata”) , represents the neutral concept “*” (“Document Content”, “structure”), represents the term “Pascal” (“term”, “text”), represents the term “Francq” (“term”, “text”), represents the term “coffee” (“term”, “text”), represents the term “ connect” (“term”, “text”), represents the term “optic” (“term”, “text”) and represents the concept “fiber” (“term”, “text”). The object is represented by the following tensor:
As this example shows, the matrix representing a tensor is highly sparse. This tensor can be seen as a set of vectors , each vector being associated to a particular meta-concept, . As for the concepts, we can define two functions: that returns the type of the meta-concept, , associated to the vector; and that returns the category of that meta-concept.
If we define only one meta-concept (for example ), the tensor space model is identical to the classical vector space model. The tensor space model can therefore be qualified as an extension of the vector space model if the method used to compute the document descriptions can extract multiple meta-concepts.
GALILEI stores the tensor representations associated to a given type of objects in a binary file that allows a fast retrieval of the tensor of a given object.

## 3 Inverse Object Frequency

### 3.1 Feature Space

In the classical vector space model, the inverse frequency factor, , plays an important role in the weighting of the terms in the vector (document representation). The inverse frequency factor, , for a feature is defined by:
where is the total number of documents, and the number of those containing the feature .
For a given number of documents, increases when decreases: the discriminant value of a given feature decreases when it occurs often in the corpus. In fact, its value is null if it appears in each document of the corpus.
This factor can be generalized to any kind objects (documents, profiles, etc.) that are described with terms [8]:
where is the total number of objects, and the number of those containing the feature .

### 3.2 Concept Space

The inverse frequency factor defined by () could be extended to the concept space. After all, “concept” is just another name for “feature”. But a concept “is more” than a feature: it has a type. Let us suppose, for example, a subset, , of all objects of that contains some email addresses (considered as a particular concept type). Moreover, let us suppose that a given email address, , is always present in these objects. Its factor is computed by:
where represents the set of objects which don’t contain any email address.
In practice, since is contained in each object providing some email addresses, its discriminant value should be null. But, due to the quantity , it is not the case. Therefore, a modification is proposed to compute the inverse object frequency factor, , for concept, :
where is the total number of objects described by at least one concept of the same type as , and the number of those containing the concept .
When applying () to compute the inverse object frequency factor in the case of the email address just presented, its value is now null since .

## 4 Concept Weighting

The document descriptions computes tensors which associate to each meta-concept/concept pair,  [C]  [C]  is a meta-concept, while is a “normal” concept., extracted from a document, , a quantity, , that reflects its importance (for example, the number of occurrences of a term in a particular metadata of the document). Before using these tensors in further tasks, such as computing a document similarity or a profile description, it is necessary to adapt these weights in order to take the whole document corpus into account.
Several methods adapt these weights [9]. One of the most used in information retrieval is based on the term frequency () and inverse document frequency () factors [7, 10]:
where represents the weights of an element in the vector, the total number of documents, the number of those containing feature and is the occurrence of the feature for the document .
These factors have an interpretation: the is a sort of document length normalization while the infers the importance of a particular feature regarding its distribution in the corpus (a feature appearing very often in the documents of the corpus has a less importance).
In the previous section, I explained how the inverse object frequency factor has to be adapted to a concept space where the concepts are structured in types. Similar reasoning can be done with the term frequency factor where the normalization of a particular weight is done regarding the corresponding vector (and not the whole tensor). It is therefore possible to propose a concept weighting method for the tensor space model that transform a local weight (such as the occurrence of a term in a given part of a document):
where is also a tensor, and represents the absolute value of and its uses is necessary because, for some objects (such as profiles), the weights may be negative.
This formula can be adapted to any subset of objects, :
where is also a tensor, and is the total number of objects of the subset described by at least one concept of the same type as , and the number of those containing the concept .
This latest formula allows to act as a magnifying glass that focuses on a particular subset of objects (for example all the documents assessed by a user with a given profile).

## 5 Tensor Operators

The classical vector space model has two nice properties:
1. The sum of the vectors describing two documents, and , equals the vector build from a document obtained by concatenating and ;
2. The multiplication of a vector describing a document, , by a number, , equals the vector build from a document obtained by concatenating times .
These properties are respected by the tensor space model proposed. To show that, let us suppose the following two tensors :
where and represent metadata concepts (such as “author”) and the other concepts correspond to terms.
The sum of the two tensors respects the first property:
The multiplication of the tensor by 3 respects the second property:
Therefore, every method developed for the vector space model that combines vectors with the sum and scalar multiplication operators works unchanged for the tensor space model proposed.

## 6 Implementation

As explained earlier, the matrices representing the tensors are sparse. Moreover, each row, , of a tensor, , represents a (sub)vector, associated to a given concept, .
In the GALILEI library, the class GDescription provides a representation for a tensor describing an object [D]  [D] In practice, each object having a description inherits from the class GDescriptionObject (document, profile, etc.). In practice, it is a container of vectors implemented trough the GVector class. Each vector is associated to a meta-concept, , of the corresponding tensor. The classes GDescription and GVector reimplement the sum and scalar multiplication operators.
The GALILEI library provides the class GConcept that represents a given concept. It has the method GetIf that computes the inverse frequency for a particular type of object. In practice, it is dynamically updated each time an object description is added (method GDescription::AddRefs) or removed (method GDescription::DelRefs) from the session. To compute the inverse frequency for a subset of descriptions given by (), the GALILEI library provides the class GDescriptionSet and its method GetIf.

## References

[1] World Wide Web Consortium, HTML 4.01 Specification, 1999.

[2] Sergey Brin & Lawrence Page, “The Anatomy of a Large-Scale Hypertextual Web Search Engine”, Computer Networks and ISDN Systems, 30(1), pp. 107—135, 1998.

[3] World Wide Web Consortium, eXtensible Stylesheet Language (XSL) Version 1.0, 2000.

[4] Tim Berners-Lee, James Hendler & Ora Lassila, “The Semantic Web”, Scientific American, 284(5), pp. 28—37, 2001.

[5] Dublin Core Metadata Initiative, Dublin Core Metadata Element Set, Version 1.1: Reference Description, 1999.

[6] Peter Murray-Rust & Henry S. Rzepa, “CML: Evolution and Design”, Journal of Cheminformatics, 3(1), pp. 1—15, 2011.

[7] Gerard Salton, Automatic Information Organization and Retrieval, McGraw-Hill, 1968.

[8] Pascal Francq, Collaborative and Structured Search: An Integrated Approach for Sharing Documents Among Users, PhD Thesis, Université libre de Bruxelles, 2003.

[9] Ricardo Baeza-Yates & Berthier Ribeiro-Neto, Modern Information Retrieval: The Concepts and Technology behind Search, Addison-Wesley, 2011.

[10] Gerard Salton, The SMART Retrieval System — Experiments in Automatic Document Processing, Prentice Hall, 1971.

[11] Pascal Francq, “The GALILEI Platform: Social Browsing to Build Communities of Interests and Share Relevant Information and Expertise” in Open Source for Knowledge and Learning Management: Strategies Beyond Tools, Miltiadis D. Lytras & Ambjorn Naeve (ed.), Idea Group Publishing, 2007.

[12] Gerald Salton, Edward A. Fox & Harry Wu, “Extended Boolean Information Retrieval”, Communications of the ACM, 26(11), pp. 1022—1036, 1983.

[13] Carolyn J. Crouch, Samer Apte, Harsh Bapat, “Using the Extended Vector Model for XML Retrieval”, In Proceedings of the 1st Workshop of the Initiative for the Evaluation of XML Retrieval (INEX), pp. 95—98, 2002.

[14] Nancy A. Chinchor, “Overview of MUC-7/MET-2”, In Proceedings of the Seventh Message Understanding Conference (MUC-7), Vol. 1, 1998.

[15] Edward A. Fox, Extending the Boolean and Vector Space Models of Information Retrieval with P-norm Queries and Multiple Concept Types, PhD Thesis, Cornell University, 1983.