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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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

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


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