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Graph matching involves combinatorial optimization based on edge-to-edge affinity matrix, which can be generally formulated as Lawler's Quadratic Assignment Problem (QAP). This paper presents a QAP network directly learning with the affinity matrix (equivalently the association graph) whereby the matching problem is translated into a constrained vertex classification task. The association graph is learned by an embedding network for vertex classification, followed by Sinkhorn normalization and a cross-entropy loss for end-to-end learning. We further improve the embedding model on association graph by introducing Sinkhorn based matching-aware constraint, as well as dummy nodes to deal with unequal sizes of graphs. To our best knowledge, this is one of the first network to directly learn with the general Lawler's QAP. In contrast, recent deep matching methods focus on the learning of node/edge features in two graphs respectively. We also show how to extend our network to hypergraph matching, and matching of multiple graphs. Experimental results on both synthetic graphs and real-world images show its effectiveness. For pure QAP tasks on synthetic data and QAPLIB benchmark, our method can perform competitively and even surpass state-of-the-art graph matching and QAP solvers with notable less time cost. We provide a project homepage at http://thinklab.sjtu.edu.cn/project/NGM/index.html.

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Graph matching involves combinatorial optimization based on edge-to-edge affinity matrix, which can be generally formulated as Lawler's Quadratic Assignment Problem (QAP). This paper presents a QAP network directly learning with the affinity matrix (equivalently the association graph) whereby the matching problem is translated into a constrained vertex classification task. The association graph is learned by an embedding network for vertex classification, followed by Sinkhorn normalization and a cross-entropy loss for end-to-end learning. We further improve the embedding model on association graph by introducing Sinkhorn based matching-aware constraint, as well as dummy nodes to deal with unequal sizes of graphs. To our best knowledge, this is one of the first network to directly learn with the general Lawler's QAP. In contrast, recent deep matching methods focus on the learning of node/edge features in two graphs respectively. We also show how to extend our network to hypergraph matching, and matching of multiple graphs. Experimental results on both synthetic graphs and real-world images show its effectiveness. For pure QAP tasks on synthetic data and QAPLIB benchmark, our method can perform competitively and even surpass state-of-the-art graph matching and QAP solvers with notable less time cost. We provide a project homepage at http://thinklab.sjtu.edu.cn/project/NGM/index.html.

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