Document Simplicial Complex

A k-simplex is de�ned as k-dimensional geometric structure which is the convex hull \nof k+1 points. Given k+1 points x0; :::; xk 2 Rk which are a�nely independent, the \nset \nC = \n( \na0x0 + ::: + akxk \n���� \nXk \ni=0 \nai = 1 and ai � 0 for all i \n) \n; \nis de�ned as the k-simplex determined by them. Simplex is a very basic building \nstructure in abstract topology. Collection of simplexes (or simplices) under certain \ncondition is called geometrical simplicial complex, which further helps to analyze a \ngeometrical structure on bigger scale. An abstract simplicial complex is a purely \ncombinatorial description of the geometric notion of a simplicial complex, consisting \nof a family of non-empty �nite sets closed under the operation of taking non-empty \nsubsets. \nA text document can be visualized as a geometric structure in topology. A docu- \nment is de�ned as a collection of words, where each word is considered to be a part of \nvocabulary having a certain meaning. And an n-gram is a contiguous sequence of n \nitems from a given sample of text. Using the n-gram concept to de�ne a simplex we \ncan construct an abstract simplicial complex out of every text document. Thus from \nthis model, every simplex catches the local structure or behavior while a document \nsimplicial complex, which is the collection of all n-1 simplex, captures the global be- \nhavior of the document. We will study this considering we have a bag of documents \ni.e. the universal set of documents. \nThe aim of this thesis is to understand abstract structure admitted by text doc- \numents to �nd more accurately the similar documents from the given family if text \ndocuments. In our discussion, we will visualize a document as a geometrical entity \nand will make use of such representation of a text document to fast the process of \nquerying, where given a query document one can �nd the semantically similar doc- \numents more e�ciently in the sense of time and similarity. For example, given a \nset of documents as f1.\\after clearing high school one joins college", 2.\\College can \nbe joined only after passing high school" and 3.\\High school and college must be \nattended by everyone"g the document 1 and 2 are more semantically similar that 1 \nand 3 or 2 and 3. \nAfter a brief glance at abstract topology, we study the topological structure and \nbehavior of text documents. A novel representation of documents is given in this \nthesis. Using this new structure of a text document we represent each document as a \ngeometrical entity which further can be analyzed using topological tools. Using Earth \n \nMover's distance and Hausdor� distance we give a new formulation to fetch semantic \ndocuments for a given query. To represent documents as a mathematical structure \nin some Rk, we use Word2Vec model to �nd vector representation of each word in a \ntext document.