Projection-free Graph-based Classifier Learning using Gershgorin Disc Perfect Alignment

• URL: http://arxiv.org/abs/2106.01642v1
• Date: Thu, 3 Jun 2021 07:22:48 GMT
• Title: Projection-free Graph-based Classifier Learning using Gershgorin Disc Perfect Alignment
• Authors: Cheng Yang, Gene Cheung, Wai-tian Tan, Guangtao Zhai
• Abstract summary: In graph-based binary learning, a subset of known labels $hatx_i$ are used to infer unknown labels. When restricting labels $x_i$ to binary values, the problem is NP-hard. We propose a fast projection-free method by solving a sequence of linear programs (LP) instead.
• Score: 59.87663954467815
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In semi-supervised graph-based binary classifier learning, a subset of known labels $\hat{x}_i$ are used to infer unknown labels, assuming that the label signal $x$ is smooth with respect to a similarity graph specified by a Laplacian matrix. When restricting labels $x_i$ to binary values, the problem is NP-hard. While a conventional semi-definite programming (SDP) relaxation can be solved in polynomial time using, for example, the alternating direction method of multipliers (ADMM), the complexity of iteratively projecting a candidate matrix $M$ onto the positive semi-definite (PSD) cone ($M \succeq 0$) remains high. In this paper, leveraging a recent linear algebraic theory called Gershgorin disc perfect alignment (GDPA), we propose a fast projection-free method by solving a sequence of linear programs (LP) instead. Specifically, we first recast the SDP relaxation to its SDP dual, where a feasible solution $H \succeq 0$ can be interpreted as a Laplacian matrix corresponding to a balanced signed graph sans the last node. To achieve graph balance, we split the last node into two that respectively contain the original positive and negative edges, resulting in a new Laplacian $\bar{H}$. We repose the SDP dual for solution $\bar{H}$, then replace the PSD cone constraint $\bar{H} \succeq 0$ with linear constraints derived from GDPA -- sufficient conditions to ensure $\bar{H}$ is PSD -- so that the optimization becomes an LP per iteration. Finally, we extract predicted labels from our converged LP solution $\bar{H}$. Experiments show that our algorithm enjoyed a $40\times$ speedup on average over the next fastest scheme while retaining comparable label prediction performance.

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