The transformation matrix M k ∈ R d×d that projects the first space to the latter is learned via solving the following constraint optimization problem: The induction of multiple topic embeddings in the unified vector space is achieved via applying Equation 3 to each TDSM.
given a word and its k-th topic distributed representation x k ∈ R d , we compute its projected representation x k ∈ R d as follows: Starting from the set of aligned topic embeddings {x k } K k=1 for each word, we learn a Gaussian Mixture Model with N components, where closely positioned topic embeddings are assigned to the same component.