# 计算机代写|机器学习代写machine learning代考|CS7641

## 计算机代写|机器学习代写machine learning代考|Geometric spatial methods

Monti et al. [Mon $+17$ ] propose MoNet, a general framework that works particularly well when the node features lie in a geometric space, such as 3D point clouds or meshes. MoNet learns attention patches using parametric functions in a pre-defined spatial domain (e.g. spatial coordinates), and then applies convolution filters in the resulting graph domain.

MoNet generalizes spatial approaches which introduce constructions for convolutions on manifolds, such as the Geodesic CNN (GCNN) [Mas+15] and the Anisotropic CNN (ACNN) [Bos+16]. Both GCNN and ACNN use fixed patches that are defined on a specific coordinate system and therefore eannot generalize to graph-struetured data. However, the MoNet framework is more general; any pseudo-coordinates (i.e. node features) can be used to induce the patches. More formally, if $\mathbf{U}^s$ are pseudo-coordinates and $\mathbf{H}^{\ell}$ are features from another domain, the MoNet layer can be expressed in our notation as:
$$\mathbf{H}^{\ell+1}=\sigma\left(\sum_{k=1}^K\left(\mathbf{W} \odot g_k\left(\mathbf{U}^s\right)\right) \mathbf{H}^{\ell} \Theta_k^{\ell}\right),$$
where $g_k\left(U^s\right)$ are the learned parametric patches, which are $N \times N$ matrices. In practice, MoNet uses Gaussian kernels to learn patches, such that:
$$g_k\left(\mathbf{U}^s\right)=\exp \left(-\frac{1}{2}\left(\mathbf{U}^s-\boldsymbol{\mu}_k\right)^{\top} \boldsymbol{\Sigma}_k^{-1}\left(\mathbf{U}^s-\boldsymbol{\mu}_k\right)\right),$$
where $\boldsymbol{\mu}_k$ and $\boldsymbol{\Sigma}_k$ are learned parameters, and $\boldsymbol{\Sigma}_k$ is restricted to be diagonal.

## 计算机代写|机器学习代写machine learning代考|Non-Euclidean Graph Convolutions

As we discussed in Section 23.3.3, hyperbolic geometry enables learning of shallow embeddings of hierarchical graphs which have smaller distortion than Euclidean embeddings. However, one major downside of shallow embeddings is that they do not generalize well (if at all) across graphs. On the other hand, Graph Neural Networks, which leverage node features, have achieved good results on many inductive graph embedding tasks.

It is natural then, that there has been recent interest in extending Graph Neural Networks to learn non-Euclidean embeddings. One major challenge in doing so again revolves around the nature of convolution itself. How should we perform convolutions in a non-Euclidean space, where standard operations such as inner products and matrix multiplications are not defined?

Hyperbolic Graph Convolution Networks (HGCN) [Cha+19a] and Hyperbolic Graph Neural Networks (HGNN) [LNK19] apply graph convolutions in hyperbolic space by leveraging the Euclidean tangent space, which provides a first-order approximation of the hyperbolic manifold at a point. For every graph convolution step, node embeddings are mapped to the Euclidean tangent space at the origin, where convolutions are applied, and then mapped back to the hyperbolic space. These approaches yield significant improvements on graphs that exhibit hierarchical structure (Figure 23.6).

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Geometric spatial methods

MoNet 概括了引入流形卷积构造的空间方法，例如贬地线 CNN (GCNN) [Mas+15] 和各向异性 CNN (ACNN) [Bos+16]。GCNN 和 ACNN 都使用在特定坐标系上定义的固定补丁，因此不能泛化到图结构数 据。然而，MoNet 框架更通用；可以使用任何伪坐标（即节点特征) 来诱导补丁。更正式地说，如果 $\mathbf{U}^s$ 是伪坐标和 $\mathbf{H}^{\ell}$ 是来自另一个领域的特征，MoNet 层可以用我们的符号表示为:
$$\mathbf{H}^{\ell+1}=\sigma\left(\sum_{k=1}^K\left(\mathbf{W} \odot g_k\left(\mathbf{U}^s\right)\right) \mathbf{H}^{\ell} \Theta_k^{\ell}\right),$$

$$g_k\left(\mathbf{U}^s\right)=\exp \left(-\frac{1}{2}\left(\mathbf{U}^s-\boldsymbol{\mu}_k\right)^{\top} \boldsymbol{\Sigma}_k^{-1}\left(\mathbf{U}^s-\boldsymbol{\mu}_k\right)\right)$$

## 计算机代写|机器学习代写machine learning代考|Non-Euclidean Graph Convolutions

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