# A Polarity Capturing Sphere for Word to Vector Representation
## 1 Introduction  
Word vector embeddings seek to model each word as a multi-dimensional vector. There are many distinct benefits to modeling words as vectors. Implementing natural language processing (NLP) algorithms using machine learning models necessitates converting a textual word or sentence to a numeric format. Moreover, this conversion has to be meaningful. For example, words with similar meanings should possess vectors that are in close proximity in the embedded space. This is because such words should produce similar outputs, if applied to a machine learning model. Additionally, a word embedding of a complete dictionary will provide a complete representation of the corpus. In this, each word is assigned to its unique position in the vector space, which reflects its aggregate relations with all other words in one cohesive construct. Word vector spaces have been very useful in many applications; for example, machine translation [1], sentiment analysis [2; 3; 4; 5], question answering [6; 7], information retrieval [8; 9], spelling correction [10], crowdsourcing [11], named entity recognition (NER) [12], text summarization [13; 14; 15], and others. The problem of existing word vector embedding methods is that the polarity of words is not adequately considered. Two antonyms are considered polar opposites, and have to be modeled as such. However, most current methods do not deal with this issue. For example, they do not differentiate in the similarity score between two unrelated words and two opposite words. In this work we propose a new embedding that takes into account the polarity issue. The new approach is based on embedding the words into a sphere, whereby the dot product of the corresponding vectors represents the similarity in meaning between any two words. The polar nature of the sphere isthe main motivation behind such embedding since this way antonymous relations can be captured such that a word and its antonym are placed at opposite poles of the sphere. This polarity capturing feature is essential for some applications such as sentiment analysis. Recently, word embedding methods have started to pervade the sentiment analysis field, at the expense of traditional machine learning algorithms, which rely on sentiment-polarity data that are annotated manually, and expensive feature engineering. Using word embeddings, a sentence can be mapped to a number of features based on the embeddings of its words. However, these embeddings should reflect the polarity of words in order to be able to perform the sentiment analysis task. As mentioned before, this polarity issue is completely addressed in our word embedding approach. Figure 1 illustrates the concept of the proposed approach. As shown in the figure, the two vectors corresponding to the words \"happy\" and \"joyful\" have close similarity because they are synonyms. On the other hand, the vectors corresponding to \"happy\" and \"sad\" are on opposite poles and have a similarity score of \(-1\), because they are antonyms. The algorithm can infer a new relation that \"joyful\" and \"sad\" are antonyms too, as their corresponding vectors are also placed close to opposite poles of the sphere. Two words that are similar in meaning are assigned vectors whose similarity measure is \(1\). Two unrelated words will be assigned a similarity close to zero, and two antonyms are assigned similarity close to \(-1\). In other words, we specifically consider the polarity issue and make a distinction between \"opposite\" and \"unrelated\". The fact that the similarity between opposing words or antonyms is close to \(-1\) is consistent with the logically appealing fact that negation amounts to multiplying by \(-1\). Double negation amounts to multiplying \(-1\) twice, giving a similarity close to one (the antonym of an antonym is a synonym). In our geometry, negation can also be considered as reflection around the origin. Thus, the designed model makes logical sense. Another question to investigate is whether the unrelated words should have similarity close to zero.