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AMES: A Differentiable Embedding Space Selection Framework for Latent Graph Inference
Nov 20, 2023
In real-world scenarios, although data entities may possess inherent relationships, the specific graph illustrating their connections might not be directly accessible. Latent graph inference addresses this issue by enabling Graph Neural Networks (GNNs) to operate on point cloud data, dynamically learning the necessary graph structure. These graphs are often derived from a latent embedding space, which can be modeled using Euclidean, hyperbolic, spherical, or product spaces. However, currently, there is no principled differentiable method for determining the optimal embedding space. In this work, we introduce the Attentional Multi-Embedding Selection (AMES) framework, a differentiable method for selecting the best embedding space for latent graph inference through backpropagation, considering a downstream task. Our framework consistently achieves comparable or superior results compared to previous methods for latent graph inference across five benchmark datasets. Importantly, our approach eliminates the need for conducting multiple experiments to identify the optimal embedding space. Furthermore, we explore interpretability techniques that track the gradient contributions of different latent graphs, shedding light on how our attention-based, fully differentiable approach learns to choose the appropriate latent space. In line with previous works, our experiments emphasize the advantages of hyperbolic spaces in enhancing performance. More importantly, our interpretability framework provides a general approach for quantitatively comparing embedding spaces across different tasks based on their contributions, a dimension that has been overlooked in previous literature on latent graph inference.
Notes on Low-rank Matrix Factorization
May 06, 2016Low-rank matrix factorization (MF) is an important technique in data science. The key idea of MF is that there exists latent structures in the data, by uncovering which we could obtain a compressed representation of the data. By factorizing an original matrix to low-rank matrices, MF provides a unified method for dimension reduction, clustering, and matrix completion. In this article we review several important variants of MF, including: Basic MF, Non-negative MF, Orthogonal non-negative MF. As can be told from their names, non-negative MF and orthogonal non-negative MF are variants of basic MF with non-negativity and/or orthogonality constraints. Such constraints are useful in specific senarios. In the first part of this article, we introduce, for each of these models, the application scenarios, the distinctive properties, and the optimizing method. By properly adapting MF, we can go beyond the problem of clustering and matrix completion. In the second part of this article, we will extend MF to sparse matrix compeletion, enhance matrix compeletion using various regularization methods, and make use of MF for (semi-)supervised learning by introducing latent space reinforcement and transformation. We will see that MF is not only a useful model but also as a flexible framework that is applicable for various prediction problems.