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For information processing, the data should be given a representation that is easy to manipulate. Often, geometric objects are used to give a representation of a discrete data type to endow the data with rich structures such as differentiability and metric. For instance, graph embedding techniques associate vertices of a graph with points in a Riemannian manifold so that the adjacency relation is modelled by vicinity. I will talk about the following two representations:- - a directed graph by subspaces of the Euclidean space
- - a probability distribution on a permutation group by hyperplanes in the Euclidean space
The latter data type is relevant to recommendation. I will discuss both mathematical backgrounds and their applicability to real-world data. |