Commit 5bf89da

Basic Usage

DHG provides an interface to visualize the distribution of feaures:

1. Load data of features and lable by specifying the corresponding path;

2. Specify parameters (*i.e.*, `the dimensionality of the visualisation`, `the storage path` and `the method of dimensionality reduction`);

3. Call our funtion to show or save the figure .

Visualization of Features in Euclidean Space

.. image:: ../_static/img/vis_ft_euclidean.png

:align: center

:alt: Visualization of Features in Euclidean Space

:height: 400px

.. code-block:: python

Visualization of Features in Poincare Space

.. image:: ../_static/img/vis_ft_poincare.png

:align: center

:alt: Visualization of Features in Poincare Space

:height: 400px

.. code-block:: python

Make Animation

Furthermore, we also offer interfaces for the visual presentation of

rotation on three-dimensional visualisations with the option of `Rotation` in Euclidean and Poincare Space.

Rotating Visualization of Features in Euclidean Space

.. code-block:: python

Rotating Visualization of Features in Poincare Space

.. code-block:: python

Mathamatical Principles of Hyperbolic Space

The hyperbolic space is a manifold with constant Gaussian constant negative curvature everywhere,

which has several models. We base our work on the Poincaré ball model for its well-suited for gradient-based optimization.

The Poincaré ball model with constant negative curvature :math:`-1 / k(k>0)` corresponds to the

Riemannian manifold

:math:`\left(\mathbb{P}^{n,k}, g_{\mathbf{x}}^{\mathbb{P}}\right)`.

:math:`\mathbb{P}^{n,k} = \left\{\mathbf{x} \in \mathbb{R}^{n}: \| \mathbf{x}\|<1 \right\}` is an open :math:`n`-demensionsional unit ball,

where :math:`\|. \|` denotes the Euclidean norm. Its metric tensor is :math:`g_{\mathbf{x}}^{\mathbb{P}} = \lambda_{\mathbf{x}}^{2} g^{E}`,

where :math:`\lambda_{\mathbf{x}} = \frac{2} {1- k\|\mathbf{x}\|^{2} }` is the conformal factor and :math:`g^{E}=\mathbf{I}_{n}` is the Euclidean metric tensor.

For two points :math:`\mathbf{x}, \mathbf{y} \in \mathbb{P}^{n,k}`, we ues the Möbius addition :math:`\oplus` operate adding

by connecting the gyrospace framework with Riemannian geometry:

The distance between two points :math:`\mathbf{x}, \mathbf{y} \in \mathbb{P}^{n,k}` is calculated by integration of the metric tensor, which is given as:

Denote point :math:`\mathbf{z} \in \mathcal{T}_{\mathrm{x}} \mathbb{P}^{n,k}` the tangent (Euclidean) space centered at any point :math:`\mathbf{x}` in the hyperbolic space.

For the tangent vector :math:`\mathbf{z} \neq \mathbf{0}` and the point :math:`\mathbf{y} \neq \mathbf{0}`,

the exponential map :math:`\exp _{\mathbf{x}}: \mathcal{T}_{\mathbf{x}} \mathbb{P}^{n,k} \rightarrow \mathbb{P}^{n,k}` and

the logarithmic map :math:`\log_{\mathbf{x}}: \mathbb{P}^{n,k} \rightarrow \mathcal{T}_{\mathbf{x}} \mathbb{P}^{n,k}` are given for

:math:`\mathbf{y} \neq \mathbf{x}` by:

and

It is noted that our initial data are on Euclidean space and need to be converted to embeddings on hyperbolic space, so first project the data on the previously obtained Euclidean space onto the hyperbolic manifold space

in order to use the Spectral-based hypergraph hyperbolic convolutional network to learn the information to update the node embeddings.

Set :math:`t:=\{\sqrt{K}, 0, 0, \dots, 0\}\in\mathbb{P}^{d, K}` as a reference point to perform tangent space operations,

where :math:`-1/K` is the negative curvature of hyperbolic model.

The above premise makes :math:`\langle(0, \mathbf{x}^{0, E}), t\rangle=0` hold,

so :math:`(0, \mathbf{x}^{0, E})` can be regarded as the initial embedding representation of the hypergraph structure on the tangent plane

of the hyperbolic manifold space :math:`\mathcal{T}_t\mathbb{P}^{d, K}`. The initial hypergraph structure embedding is

then mapped onto the hyperbolic manifold space :math:`\mathbb{P}` using the following equation:

The hyperbolic operation is accomplished by means of a feature mapping between Euclidean space and Hyperbolic space.