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The Random Dot Product Graph (RDPG) is a popular generative graph model for relational data. RDPGs postulate there exist latent positions for each node, and specifies the edge formation probabilities via the inner product of the corresponding latent vectors. The embedding task of estimating these latent positions from observed graphs is usually posed as a non-convex matrix factorization problem. The workhorse Adjacency Spectral Embedding offers an approximate solution obtained via the eigendecomposition of the adjacency matrix, which enjoys solid statistical guarantees but can be computationally intensive and is formally solving a surrogate problem. In this paper, we bring to bear recent non-convex optimization advances and demonstrate their impact to RDPG inference. We develop first-order gradient descent methods to better solve the original optimization problem, and to accommodate broader network embedding applications in an organic way. The effectiveness of the resulting graph representation learning framework is demonstrated on both synthetic and real data. We show the algorithms are scalable, robust to missing network data, and can track the latent positions over time when the graphs are acquired in a streaming fashion.
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Tracking the Adjacency Spectral Embedding for Streaming Graphs
The popular Random Dot Product Graph (RDPG) generative model postulates that each node has an associated (latent) vector, and the probability of existence of an edge between two nodes is their inner-product (with variants to consider directed and weighted graphs). In any case, the latent vectors may be estimated through a spectral decomposition of the adjacency matrix, the so-called Adjacency Spectral Embedding (ASE). Assume we are monitoring a stream of graphs and the objective is to track the latent vectors. Examples include recommender systems or monitoring of a wireless network. It is clear that performing the ASE of each graph separately may result in a prohibitive computation load. Furthermore, the invariance to rotations of the inner product complicates comparing the latent vectors at different time-steps. By considering the minimization problem underlying ASE, we develop an iterative algorithm that updates the latent vectors' estimation as new graphs from the stream arrive. Differently to other proposals, our method does not accumulate errors and thus does not requires periodically re-computing the spectral decomposition. Furthermore, the pragmatic setting where nodes leave or join the graph (e.g. a new product in the recommender system) can be accommodated as well. Our code is available at https://github.com/marfiori/efficient-ASE
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- PAR ID:
- 10443086
- Date Published:
- Journal Name:
- 2022 56th Asilomar Conference on Signals, Systems, and Computers
- Page Range / eLocation ID:
- 847 to 851
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/IEEECONF56349.2022.10051861
