Fast and Flexible Large Graph Embedding Based on Anchors

Weizhong Yu, Feiping Nie, Fei Wang, Rong Wang, Xuelong Li

Abstract: Dimensionality reduction is one of the most fundamental topic in machine learning. A range of methods focus on dimensionality reduction have been proposed in various areas. Among the unsupervised dimensionality reduction methods, graph-based dimensionality reduction has begun to draw more and more attention due to its effectiveness. However, most existing graph-based methods have high computation complexity, which is not applicable to large-scale problems. To solve this problem, an unsupervised graph-based dimensionality reduction method called fast and flexible large graph embedding (FFLGE) based on anchors is proposed. FFLGE uses an anchor-based strategy to construct an anchor-based graph and design similarity matrix and then perform the dimensionality reduction efficiently. The computational complexity of the proposed FFLGE reduces to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(ndm)$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of samples, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> is the number of dimensions and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> is the number of anchors. Furthermore, it is interesting to note that locality preserving projection and principal component analysis are two special cases of FFLGE. In the end, the experiments based on several publicly large-scale datasets proves the effectiveness and efficiency of the method proposed.