Title: Tensor-Based Preposition Representation. Abstract: Prepositions are among the most frequent words. Good prepositional representation  is of great syntactic and semantic interest  in computational linguistics. Existing methods on preposition representation either treat prepositions as content words (e.g., word2vec and GloVe) or depend heavily on external linguistic resources including syntactic parsing, training task and dataset-specific representations. In this paper we use word-triple counts (one of the words is a preposition) to  capture the preposition's interaction with its head and children. Prepositional  embeddings are derived via tensor decompositions on a large unlabeled corpus.  We reveal a new geometry involving Hadamard products and empirically demonstrate its utility in paraphrasing of phrasal verbs. Furthermore, our prepositional  embeddings are used as simple features to two challenging downstream tasks: preposition selection and prepositional attachment disambiguation. We achieve comparable to or better results than state of the art on  multiple standardized datasets.  