Representation of knowledge in manifold space. Sphere as a manifold embedding is shown in figure.


San Jose State University

The space where entities and relations are represented after analysis. Representation space is also called embedding space. The embedding space should satisfy three conditions differentiability, calculation possibility, and definability of a scoring function. The representation space can be categorized into

 - Point-Wise Space
 - Complex Vector Space
 - Gaussian Distribution
 - Manifold and Group:


Representation Space 

A Survey on Knowledge Graphs: Representation, Acquisition and Applications

Entities and relations are represented in a real valued point -wise euclidean space . The various space for projection are shown in the figure

Point-Wise Space 

Entities and relations are represented in a complex space, $$h, t, r \in \mathcal{C}^d$$.

Complex Vector Space

Manifold and Group

Uncertainties in entities and relations can be modeled using a Gaussian distribution space. In this approach, entities and relations are embedded into a multi-dimensional Gaussian distribution function, such that $$\mathcal{H} \sim \mathcal{N}(\mu_h,\Sigma_h)$$ and $$\mathcal{T} \sim \mathcal{N}(\mu_t,\Sigma_t)$$. The mean vector $$\mu$$ represents the position of the entities and relations, while the covariance matrix $$\Sigma$$ represents their uncertainties.

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