Some Thoughts on Using Annotated Su for Natural Language Processing x Trees

Abstract

The paper defines an annotated su x tree (AST) - a data structure used to calculate and store the frequencies of all the fragments of the given string or a collection of strings. The AST is associated with a string to text scoring, which takes all fuzzy matches into account. We show how the AST and the AST scoring can be used for Natural Language Processing tasks.

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Introduction

In this paper we are going to concentrate on the so-called annotated su x tree (AST), introduced in [ 8 ]. We will present the data structure itself and several Natural Language Processing tasks where the AST representation is successfully used. We are not going to make any comparisons to other text representation models, but will show that using the AST approach helps to overcome some exciting problems. The paper is organized as follows: the Section 2 presents the definition of the AST and the algorithm for the AST construction, Sections from 3 to 7 present exciting applications of the AST (almost all developed with author’s contribution), Section 8 lists some future application, Section 9 suggests how to compare the AST scoring to other approaches, Section 10 is devoted to the AST scoring implementation. Section 11 concludes.

The project is being developed by the “Methods of web corpus collection, analysis and visualisation” research and study group under guidance of prof. B. Mirkin (grant 15 - 05 - 0041 of Academic Fund Program). 2

Annotated su

x tree 2.1

Definition The su x tree is a data structure used for storing of and searching for strings of characters and their fragments [ 3 ]. When the su x tree representation is used, the text is considered as a set of strings, where a string may be any significant part of the text, like a word, a word or character n-gram or even a whole sentence. An annotated su x tree (AST) is a su x tree whose nodes (not edges!) are annotated by the frequencies of the strings fragments.

An annotated su x tree (see Figure 1)[ 7 ] is a data structure used for computing and storing all fragments of the text and their frequencies. It is a rooted tree in which: – Every node corresponds to one character – Every node is labeled by the frequency of the text fragment encoded by the path from the root to the node. 2.2

AST construction Our algorithm for constructing an AST is a modification of the well-known algorithm for constructing su x trees [ 3 ]. The algorithm is based on finding su xes and prefixes of a string. Formally, the i-th su x of the sting is the substring, which starts at i-th character of the string. The i-th prefix of the string is the substring, that ends on the i-th character of the string. The AST is built in an iterative way. For each string, its su xes are added to the AST one-by-one starting from an empty set representing the root. To add a su x to the AST, first check, whether there is already a match, that is, a path in the AST that encodes / reads the whole su x or its prefix. If such a match exists, we add 1 to all the frequencies in the match and append new nodes with frequencies 1 to the last node in the match, if it does not cover the whole su x. If there is no match, we create a new chain of nodes in the AST from the root with the frequencies 1. 3 7 1. The string is represented by the set of its su xes; 2. Every su x is matched to the AST starting from the root. To estimate the match we use the average conditional probability of the next symbol: f(node) score(match(suf f ix, ast)) = Pnode2 match ( f(pa|sruenftf(inxo|de)) ) , where f (node) is the frequency of the matching node, f (parent(node)) is it’s parent frequency, and |suf f ix| is the length of the su x; 3. The relevance of the string is evaluated by averaging the scores of all su xes: relevance(string, text) = SCORE(string, ast) =

Psuffix score(match(suf f ix, ast)) = where |string| is the length of the string.

Note, that “score” is found by applying a scaling function to convert a match score into the relevance evaluation. There are three useful scaling functions, according to experiments in [ 8 ] for spam classification: – Identity function: (x) = x – Logit function:

x 1

x (x) = log = log x log(1 x) – Root function (x) = p x The identity scaling stands for the conditional probability of characters averaged over matching fragments (CPAMF).

Consider an example to illustrate the described method. Let us construct an the for the string “mining”. This string has six su xes: “mining”, “ining”, “ning”, “ing”, “ng”, and “g’ . We start with the first su x and add it to the empty AST as a chain of nodes with the frequencies equal to unity. To add the next su x, we need to check whether there is any match, i.e. whether there is such a path in the AST starting at its root that encodes / reads a prefix of “ining”. Since there is no match between existing nodes and the second su x, we add it to the root as a chain of nodes with the frequencies equal to unity. We repeat this step until a match is found: a prefix of the fourth su x “ing” matches the second su x “ining”: two first letters, “in”, coincide. Hence we add 1 to the frequency of each of these nodes and add a new child node “g” to the leaf node “n” (see Figure 1). The next su x “ng” matches the third su x and we repeat the same actions: increase the frequency of the matched nodes and add a new child node that does not match. The last su x does not match any path in the AST, so again we add it to the AST’s root as a single node with its frequency equal to unity. Now let us calculate the relevance score for string “dining” using the AST in Figure 1. There are six su xes of the string “dining”: ‘dining”, “ining”, “ning”, “ing”, “ng”, and “g’ . Each of them is aligned with an AST path starting from the root. The scorings of the su xes are presented in Table 1.

We have used the identity scaling function to score all 6 su xes of the string “dining”. Now, to get the final CPAMF relevance value we sum and average 5 9 them: relevance(dining, mining) =

In spite of the fact that “dining” di↵ ers from “mining” by just one character, the total score, 0.44, is less than unity. This is not only because the trivial su x “dining” contributes 0 to the sum, but also because conditional probabilities get smaller for the shorter su xes. 3

Spam filtering

The definition of the AST presented above was for first time introduced by Pampapathi, Mirkin and Levene in [ 7 ] for spam filtering. The AST was used as a representation tool for every class (spam and ham). By introducing a procedure for scoring the class AST they developed a classifier that beats the Naive Bayes classifier in a series of experiments on standard datasets. The success of ASTs in domain of email filtering was due to the notion of match permutation normalization, which allowed to take into account some intentional typos developed by spamers to pass over spam filters. Match permutation normalization is in a a sense analogous to the edit distance [ 10 ] that if frequently implemented in spam filters [ 11 ]. 4

Research paper categorization

The problem of text categorization is formulated as follows. Given a collection of documents and a domain taxonomy, annotate a document with relevant taxonomy topics. A taxonomy is a rooted tree, such that every node corresponds to a (taxonomy) topic of the domain. The taxonomy generalizes the relation “is – a” or “is a part of”.

There are two basic approaches to the problem of text categorization: supervised and unsupervised. Supervised approaches give high precision values when applied to web document categorization [ 12 ], but may fail when applied to research paper categorization, since the research taxonomies, such as ACM Computing Classification System [ 13 ], are seldom revised and the supervised techniques may overfit [ 14 ]. The unsupervised approaches to text categorization are based on information retrieval – like idea: given the set of taxonomy topics, let us find those research papers that are relevant to every topic. The question for researcher is the following: what kind of the relevance model and measure to choose? In [ 15 ] we experimentally compared cosine relevance function, which measures the cosine between tf idf vectors in Vector Space Model [ 1 ], BM25, based on the probabilistic relevance framework, and the AST scoring, introduced above. These three relevance measures where applied to a relatively small dataset of 244 articles, published in ACM journals and the current version of ACM Computing Classification System. The AST scoring outperforms cosine and BM25 measures, by being more robust and taking not crisp but fuzzy measures into account. The next step in this research direction would be testing the AST scoring versus w-shingling procedure [ 17 ], which is also a fuzzy matching technique that requires text preprocessing, such stemming or lemmatization. However there is no need in stemming or lemmatization to apply the AST scoring. 5

Taxonomy refinement Taxonomies are widely used to represent, maintain and store domain knowledge, see, for example SNOMED [ 18 ] or ACM CCS [ 13 ]. Domain taxonomy construction is a di cult task and a number of researchers have come out with idea of taxonomy refinement. The idea of taxonomy refinement is the following: having one taxonomy or upper levers of taxonomy refine it with topics extracted from additional sources such as other taxonomies, web search or Wikipedia. We followed this strategy and developed a two-step approach to taxonomy refinement, presented in more details in [ 21 ]. We concentrated on taxonomies of probability theory and mathematical statistics (PTMS) and numerical mathematics (NM), both in Russian. On a first step an expert sets manually the upper layers of taxonomy. On the second step these upper layers are refined by Wikipedia category tree and the articles, belonging to this tree, from the same domain. In this study the AST scoring is used several times: – To clear the Wikipedia data from noise; – To assign the remaining Wikipedia categories to the taxonomy topics; – To form the intermediate layers of the taxonomy by using Wikipedia subcategories; – To use Wikipedia articles in each of the added category nodes as its leaves. The Wikipedia data is rather noisy: there some articles that are stubs or irrelevant to parental categories (the categories, they belong to) and the more so there are subcategories (of a category) that are irrelevant to the parental categories. For example, we found the article “ROC curve” be irrelevant to the category “Regression analysis” and the category “Accidentally killed” to the category “Randomness”. To define what article is irrelevant we exploit the AST scoring twice: – We scored the title of the article to the text of the article to detect stubs; – We scored the title of the parental category to the text of the article to detect irrelevant category.

If the value of the scoring function is less than a threshold we decided that the article is irrelevant. Usually we set the threshold at 0.2. To assign the remaining Wikipedia categories to the taxonomy topics we score the taxonomy topics to all the articles in the category merged into one text. Next we found the maximum value of the scoring function and assigned the category to the corresponding 7 11 taxonomy topic. Finally, we score the title of parental categories to the articles of the subcategories, merged into one. If the subcategory to category scoring is higher than the subcategory to taxonomy topic, the subcategory remains on the intermediate layer of the refined taxonomy tree under its parental category. Finally, the articles left after clearing from noise became leaves in the refined taxonomy tree. The quality of achieved PTMS and NM taxonomies is di cult to evaluate computationally, so the design of the user study is an open question. 6

Text summarization Automatic text summarisation is one of the key tasks in natural language processing. There are two main approaches to text summarisation, called abstractive and extractive approaches [ 22 ].

According to the abstractive approach, the summary of a text is another text, but much shorter, generated automatically to make the semantic representation of the text. According to extractive approach, the summary of a text is nothing else, but some important parts of the given text, such as a set of important sentences.

The extractive summarisation problem can be formulated in the following way. Given a text T that is a sequence of sentences S that consists of words V , select a subset of the sentences S⇤ that are important in T . Therefore we need to define: – what importance of a sentence is; – how to measure importance of the sentence; Hence we need to introduce a function, importance(s), which measures the importance of a sentence. The higher importance is, the better. Next step is to build the summary. Let us rank all the sentences according the values of importance. Suppose we look for the summary that consists of five sentence. Hence we take the five sentences with the highest values of importance and call them top-5 sentences according to importance. Generally, the summary of the text are the top-N sentences according to importance and N is set manually.

The best results for this statement of the problem are achieved by Mihalcea and Tarau [ 23 ], where importance(s) is introduced as PageRank type function [ 24 ] without any kind of additional grammar, syntax or semantic information. The main idea of the suggested TextRank algorithm is to represent a text as a directed graph, where nodes stand for sentences and edges connect sequential sentences. The edges are weighted with sentence similarity. When PageRank is applied to this graph, every node receives its rank that is to be interpreted as the importance of the sentence, so that importance(s) = P ageRank(snode), where snode is the node corresponding to sentence s.

To measure similarity of the sentences the authors of TextRank algorithm suggest to use the basic VSM (Vector Space Model) scheme. First every sentence is represented as a vector in space of words or stems. Next cosine similarity between those vectors is computed. We can use the AST scoring as well for scoring the similarity between two sentences. To do this we have to introduce the common tree technique. 6.1

Constructing common subtree for two ASTs

To estimate the similarity between two sentences we find the common subtree of the corresponding ASTs. We do the depth-first search for the common chains of nodes that start from the root of the both ASTs. After the common subtree is constructed we need to annotate and score it. We annotate every node of the common subtree with the averaged frequency of the corresponding nodes in initial ASTs. Consider for example two ASTs for strings “mining” and “dinner” (see Fig. 1 and Fig. 2, correspondingly). There are two common chains: “I N” and “N”, the first one consists of two nodes, the second one consists of a single node. Both this chains form the common subtree. Let us annotate it. The frequency of the node “I” is equal to 2 in the first AST and to 1 in the second. Hence, the frequency of this node in the common subtree equals to 2+1 = 1.5. In the same 2 way we annotate the node “N” that follows after the node “I” with 2+1 = 1.5 2 and the node “N” on the first level with 2+2 = 2. The root is annotated with 2 the sum of the frequencies of the first level nodes that is 1.5 + 2 = 3.5. 6.2

Scoring common subtree

The score of the subtree is the sum of scores of every chain of nodes. The score of the path is the averaged sum of the conditional probabilities of the nodes, where conditional probability of the node is the frequency of the node divided by the frequency of its parent. For example, the conditional probability of the node “G:1” on the third level of the AST on Fig. 1 is 1/2. Let us continue with the example of “mining” and “dinner”. There are two chains in their common subtree: “I N” and “N”. The score of “I N” chain is (1.5/1.5 + 1.5/3.5)/2 = 0.71, since there are 2 nodes in the chain. The score of one node chain “N” is 1.5/3.5 = 0.42. The score of the whole subtree is (0.71 + 0.42) = 1.13.

The collection for experiments was made of 400 articles from Russian news portal called Gazeta.ru. The articles were marked up in a special way, so that some of sentences were highlighted because of being more important. This highlighting was done either by the author of the article or by the editor on the basis of their own ideas. In our experiments we considered those sentences as the summary of the article. We tried to reproduce these summaries using TextRank with cosine similarity measure and AST scoring.

Using this algorithm allowed us to gain around 0.05 points of precision according to cosine baseline on our own collection of Russian newspaper texts. This is a great figure for Natural Language Processing task, taking into account that the baseline precision of the cosine measure was very low. The fact that the precision is so low can be explained by some lack of consistency in the constructed collection: the authors of the articles use di↵ erent strategies to highlight the important sentences. The text collection is heterogeneous: in some articles 9 13 there are 10 or more sentences highlighted, in some only the first one. More details of this experiment are presented in [ 25 ]. 7

Association rule extraction Several research group develop di↵ erent approaches to extraction and visualization of association rules from text collections [ 26, 27 ]. Association rule is a rule X =) Y , where both X and Y are sets of concepts, possibly a singleton, and the implication means some sort of co-occurrence relation. An association rule has two important features, called support and confidence. When the rule is extracted from the text collection, the support of the set X support(X) usually stands for the proportion of the documents where concepts X occur and the confidence of the association rule conf idence(X =) Y ) stands for conditional probability of Y given X. The majority of approaches to association rule extraction share the following idea in common: the concepts should be extracted from the text collection. Using the fuzzy AST scoring we can diminish this limitation and produce the rules on the set of concepts provided by a user. In [ 28 ] we presented a so-called “conceptual map”, which is a graph of association rules X =) Y . To make the visualization easy we restricted ourselves only to single item sets, so that |X| = |Y | = 1. We analyzed a collection of Russian language newspaper articles on business and the concepts were provided by a domain expert. We used the AST scoring to score every concept ki to every text from the collection. Next we formed F (ki) the set of articles, to which the concept ki relevant (i.e. the scoring is higher than a threshold, usually 0.2). Finally, there was a rule ki =) kj if the ratio F (kFi)(\kFi)(kj) was higher than the predefined confidence threshold. An example of conceptual map (translated into English) can be found on Fig. 4.

This conceptual map may serve as a tool for text analysis: it reveals some hidden relations between concepts and it can be easy visualized as a graph. Of course, to estimate the power of conceptual maps we have to conduct an user study. 8

Future work In the following sections we will briefly present some Natural Language Processing tasks, where AST scoring might be used. 8.1

Plagiarism detection Ordinary su x trees are widely used for plagiarism detection [ 29 ]. The common subtree technique can also be used in this case. Suppose we have two texts, 11 15 construct two individual ASTs and the common AST. The size of the common AST will show how much these texts share in come. Scoring the common AST allows to measure how significant coinciding parts are. With no doubts, the common AST can be used for indexing of coinciding parts of the texts. Hence, it inherits advantages of ordinary su x trees with some additional functionality. Splitting compounds, such as German compounds, is necessary for machine translation and information retrieval. The splitting is usually conducted according to some morphological or probabilistic models [ 30 ]. We have a hypothesis that scoring prefixes of compound words to the AST, constructed from the collection of simple words, will allow to split compounds without using additional morphological knowledge. The main research in this direction is the design of the collection of simple words. The Russian profanity language is rich and complex and has a complex derivation, usually based on adding prefixes (such as “za”, “pro”, “vy”, etc). New words appear almost every month, so it is di cult to maintain a profanity dictionary. Profanity filtering is an important part of Russian Text or Web mining, specially since some special limitations on using profanity were introduced. The task is to find words in a text that are profane and, for example, to replace them with star symbols “***”. Note, that Russian derivative includes also a variety of endings, so lematization or stemming should be used. Since Porter stemmer [ 31 ] does not cope with prefixes, it can be easily replaced by some sort of the AST-scoring. 9

Comparison to other approaches

Cosine measure on tf idf vectors is a traditional baseline in majority of Natural Language Processing tasks and is easily overcame by any sort of more robust and fuzzy similarity or relevance measure, such as w-shingling [ 17 ], super shingles [ 32 ], mega shingles [ 33 ] and character n-grams [ 34 ]. The main future research concentrates on drawing comparison between these fuzzy measure and AST scoring. 10

Implementation

Mikhail Dubov’s implementation of AST construction and scoring is based on su x arrays, which makes it space and time e cient. It is available at https: //github.com/msdubov/AST-text-analysis. It can be used as console utility or as a Python library.

Conclusion In this paper the notion of annotated su x tree is defined. The annotated su x trees are used by several research groups and in the paper several finished, running or future projects are presented. The annotated su x tree is a simple but powerful tool for scoring di↵ erent types of relevance or similarity. This paper may sound light weighted and to make it more theoretical, we will conclude by provided some insights on probabilistic or morphological origins of ASTs. From one point of view, we have a strong feeling that it can proved that the AST or the common AST is a string kernel, thus it can be used to generate features for text classification / categorization or to measure similarity. From another point of view, the AST is a sort of supervised stemmer, that can be used to generate terms more e ciently than model-based stemmers. 12

Acknowledgments I am deeply grateful to my supervisor Dr. Boris Mirkin for an oppurtunity to learn from him and to work under his guidance for so long and to my colleagues Mikhail Dubov, Maxim Yakovlev, Vera Provotorova and Dmitry Ilvovsky for collaboration. 13 17

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