Matrix factorization is used  to learn a low-dimensional approximation of a node-node similarity matrix S, where S generalizes the adjacency matrix and captures some user defined notion of node-node similarity.

San Jose State University

Node embeddings is a way of graph representation. The goal of node embeddings is to map the nodes in the graph into a low-dimensional vector space, where the relations of the nodes in graph can be reflected by the geometric relations in the embedding space. Compared with hand-engineered features, node embeddings are more flexible and they are learned through learning processes.

Node Embeddings

Hamilton, W.L. Graph Representation Learning. *Synthesis Lectures on Artificial Intelligence and Machine Learning*. 2020, 14(3), 1–159. https://doi.org/10.2200/S01045ED1V01Y202009AIM046

Graph Representation Learning by William Hamilton

Summary of Shallow Embedding Algorithms.

Encoder and decoder graph framework is used for learning node embeddings for graph representation learning.

Encoder maps node in a graph to a low dimensional vector space.

Decoder uses the low dimensional node embeddings to reconstruct the node's position in the original graph.

Encoder and Decoder Graph Framework 

Encoder-Decoder idea based on Matrix
Factorization

Node embeddings are optimized so that two nodes have similar embeddings
if they tend to co-occur on short random walks over the graph.

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